The concepts of plane of contact, pole, and polar plane are related to the study of conic sections in geometry.
Plane of Contact:
The plane of contact is a plane that passes through the points of intersection of a cone and a plane. In other words, when a plane intersects a cone, the points where the plane and cone touch determine a plane of contact. This plane is important in determining the properties of the conic section that is formed by the intersection.
Pole:
The pole is a point that is used to define a polar plane. It is a point that is not on the plane and is located on the axis of the cone. When a plane passes through the pole, it intersects the cone in a conic section. The pole is a useful concept in determining the properties of the conic section.
Polar Plane:
The polar plane is a plane that passes through the pole and is perpendicular to the plane of contact. It is the plane that contains the polar axis, which is the axis of the cone passing through the pole. The polar plane is important in determining the properties of the conic section that is formed by the intersection of the cone and the plane of contact.
In summary, the plane of contact, pole, and polar plane are all concepts that are used in the study of conic sections. The plane of contact is a plane that passes through the points of intersection of a cone and a plane. The pole is a point on the axis of the cone that is used to define a polar plane. The polar plane is a plane that passes through the pole and is perpendicular to the plane of contact. These concepts are important in determining the properties of conic sections and are used in various applications, including optics, astronomy, and engineering.
Answer:
Step-by-step explanation:
Sure! Let's start with the plane of contact.
Plane of Contact:
The plane of contact is a term commonly used in geometry and physics. When two objects come into contact with each other, their surfaces touch at a particular area. This area of contact is known as the plane of contact. It is a two-dimensional surface that represents the point of contact between the two objects.
Imagine two books placed on a table. The area where the bottom surface of each book touches the table is the plane of contact. It is like a flat sheet that separates the objects and defines the region where they interact with each other.
Pole and Polar Plane:
The concepts of pole and polar plane are closely related to the plane of contact. Let's explore them:
Pole:
In geometry, the pole is a point that represents the point of contact between a line and a plane. When a line is in contact with a plane, the point of contact is called the pole. It is the specific point where the line intersects the plane.
For example, if you have a line placed on a tabletop, the point where the line touches the surface is its pole.
Polar Plane:
The polar plane is the plane that contains all the lines passing through a specific point called the pole. In other words, the polar plane is a plane that is perpendicular to the line at its pole.
Continuing with the previous example, if you have a line touching the tabletop at a certain point, the polar plane would be a plane that is perpendicular to the line at that point. It includes all the lines passing through that point.
The pole and polar plane are connected in such a way that every line passing through the pole lies on the polar plane, and the polar plane contains the pole.
In summary, the plane of contact represents the area where two objects touch each other, while the pole represents the point of contact between a line and a plane. The polar plane is a plane that contains all the lines passing through the pole.
Hope it helps!!
What is the soultion to the system of equations? {-3x-y+z=8 -3x-y+3z=0x -3z= 3
The solution to the system of equations is x = -9; y = 15 and z = -4
How to solve an equation?An equation is an expression that can be used to show the relationship between two or more numbers and variables using mathematical operators.
Given the system of equations:
-3x - y + z = 8 (1)
-3x - y + 3z = 0 (2)
and:
x - 3z = 3 (3)
Solving the three equations simultaneously gives:
x = -9; y = 15 and z = -4
The solution is x = -9; y = 15 and z = -4
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please help me i suck at math hurry
Part 1:
To see why, notice that when x increases by 1, y increases by 2. This corresponds to the coefficient of x in the equation being 2. Also, when x is 0, y is 2, which corresponds to the y-intercept of 2 in the equation. Therefore, the equation that fits the given table is y = 2x + 2.
Part 2:
(a) 2.35 x 10⁻³ can be written in standard form as 0.00235
(b) 2.35 x 10³ can be written in standard form as 2350
Part 3:
A linear equation is an equation where the highest power of the variable is 1. In other words, the graph of a linear equation is always a straight line.
Part 4:
(b) (2,4):
For y=x+3, when x=2, y=5.
For -2x+y=1, when x=2, y=5.
Since (2,4) satisfies both equations, it is a solution.
(c) (2,5):
For y=x+3, when x=2, y=5.
For -2x+y=1, when x=2, y=5.
Since (2,5) satisfies both equations, it is a solution.
Part 5:
Using the Pythagorean Theorem:
a² + b² = c²
8² + b² = 10²
64 + b² = 100
b² = 100 - 64
b² = 36
b = 6
Use the formula for the area of a triangle:
A = (1/2)bh
A = (1/2)(8)(6)
A = 24 cm²
Therefore, the area of triangle ABC is 24 cm². Answer: (b) 24 cm²
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is x^3 an exponential function
HELP ASAP! ( I MEAN 10 MINUTES ) ( 15 POINTS )
Jeremiah needed dog food for his new puppy. He compared the prices and sizes of three types of dog food.
Canine Cakes Bark Bits Woofy Waffles
Size (pounds) 30 40 28
Cost $54 $70 $42
Part A: Calculate the corresponding unit rate for each package. (9 points)
Part B: Determine the best buy using the unit rates found in Part A. Explain your answer. (3 points)
Part A
Canine Cakes: $1.8 per pound
Bark Bits: $1.75 per pound
Woofy Waffles: $1.5 per pound
Part B:
Woofy Waffles has the best buy. The unit rate is the least of three.
What is known by the term rate?The term "rate" generally refers to the measure of change in one quantity with respect to another quantity. It describes how one quantity changes in relation to another quantity, often expressed as a ratio or a fraction.
What is the buying price?The buying price, also known as the purchase price or the cost price, is the amount of money that is required to purchase an item, product, or service. It is the price at which a buyer acquires a product or service from a seller or a vendor.
Canine cakes= 30pounds
cost =$54
Unit rate= $54/30 = $1.8 per pound
Bark Bits= 40 pounds
cost =$70
Unit rate= $70/40= $1.75 per pound
Woofy Waffles = 28pounds
cost =$42
Unit rate=$42/28= $1.5 per pound
The best buy is 28 pounds of Woofy Waffles for $42. The rate is least of the three.
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HELP ASAP (I chose 34 as random number)
The table shows the grading scale for Ms. Gray's social studies class.
A 90%–100%
B 80%–89%
C 70%–79%
Part A: Pick a number between 28 and 39. This number will represent how many points you earned. If you have a pop quiz worth a total of 40 points, using the number you selected, calculate the percentage you earned on the test. Show each step of your work. (8 points)
Part B: Based on the percentage found in Part A, would you earn a grade of A, B, or C using the grading scale provided? Explain your answer. (4 points)
Part A: if you earned 14 points out of 17 points on the quiz, your percentage score would be 82.35%.
Part B: the grade earned for the quiz is a B.
What is the percentage?
A percentage is a means to represent a piece of 100 as a ratio or percentage. The sign for it is % (percent), which stands for "per hundred." If your state, for instance, that 50 out of 100 people like chocolate, then means that 50 out of 100 people, or 0.5 (50/100), like chocolate overall. In several disciplines, including finance, business, mathematics, and statistics, percentages are frequently used.
Here, we have
Given: Ms. Gray's social studies class.
A 90%–100%
B 80%–89%
C 70%–79%
Part A: Let's say we pick the number 17 as the number of points earned on the quiz. If the quiz is worth a total of 17 points, and you earned 14 points, then your percentage score is calculated as follows:
Percentage score = (points earned / total points) x 100%
Percentage score = (14 / 17) x 100%
Percentage score = 82.35%
Hence, if you earned 14 points out of 17 points on the quiz, your percentage score would be 82.35%.
Part B: Based on the percentage score of 82.35%, you would earn a B grade using the grading scale provided.
According to the grading scale, a B grade is earned for a percentage score between 80%-89%, and the percentage score obtained in part A is within this range.
Hence, the grade earned for the quiz is a B.
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You drop a ball from a height of 1.5 meters. Each curved path has 71% of the height of the previous path.
a. Write a rule for the sequence using centimeters. The initial height is given by the term n = 1.
b. What height will the ball be at the top of the sixth path?
a) The rule that represents the height as a function of the number of bounces is described by H(n) = 1.5 · (71 / 100)ˣ⁻¹. (Correct choice: B)
b) The height at the top of the sixth path is equal to 0.27 meters. (Correct choice: B)
How to represent a bounces of a ball by geometric sequence formula
In this problem we need to derive the equation that represents maximum height as a function of the number of bounces:
H(n) = a · (r / 100)ˣ⁻¹
Where:
a - Initial height, in meters.r - Height ratio, in percentage. x - Number of bounces.If we know that a = 1.5 m and r = 71, then the rule for the sequence is:
H(n) = 1.5 · (71 / 100)ˣ⁻¹
And the height at the top of the sixth path:
H(6) = 1.5 · (71 / 100)⁶⁻¹
H(6) = 0.27 m
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y = |x - 5| + |x + 5| if x >5
Answer:
y = 2x
Step-by-step explanation:
You want the simplified form of y = |x -5| +|x +5| if x > 5.
Turning pointsThe graph of the whole function will have turning points where the absolute value expressions are 0:
x -5 = 0 ⇒ x = 5
x +5 = 0 ⇒ x = -5
For values of x > 5, we are concerned with that portion of the graph that is to the right of both of these turning points. Hence, both absolute value expressions are positive and unchanged by the absolute value bars.
y = (x -5) +(x +5) . . . . . . if x > 5
y = 2x . . . . . . . . . . . . . . collect terms
The simplified function is y = 2x.
__
Additional comment
The attached graph shows y=2x and the given function for x > 5. They are identical. (The y=2x graph is shown dotted, so you can see the red graph of the given function.)
If you transform y = 2x2 into y = 10x2, which option below describes the effect of this transformation on the graph of the quadratic function along the y-axis?
(b) The transformation stretches the graph by a factor of 5 describes the effect of this transformation on the graph of the quadratic function along the y-axis.
The transformation from y = 2x² to y = 10x² changes the coefficient of the x² term from 2 to 10. This change in the coefficient affects the vertical scaling of the graph of the quadratic function along the y-axis. When we multiply the entire function by a constant, it causes a vertical stretch or compression of the graph depending on the magnitude of the constant.
In this case, the transformation stretches the graph along the y-axis by a factor of 5 since 10 is 5 times greater than 2. This means that the vertical distance between the points on the graph of the function is now 5 times greater than before the transformation.
Therefore, the correct option is (b) - The transformation stretches the graph by a factor of 5.
Correct Question :
If you transform y = 2x2 into y = 10x2, which option below describes the effect of this transformation on the graph of the quadratic function along the y-axis?
a) The transformation shrinks the graph by a factor of 25.
b) The transformation stretches the graph by a factor of 5.
c) The transformation stretches the graph by a factor of 25.
d) The transformation shrinks the graph by a factor of 5.
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What is the range of the function in the graph?graph on the f-e axis, between the points (6, 40) and (12, 100)
A. 6≤e≤12
B. 40≤f≤100
C. 6≤f≤12
D. 40≤e≤100
For the function in the graph the range is option B: 40≤f≤100.
What is range of function?The collection of all potential output values (y-values) that a function could produce is known as the function's range. It is, in other words, the entire set of values that the function is capable of accepting as its input changes across its domain. The collection of all potential output values (y-values) that a function could produce is known as the function's range. It is, in other words, the entire set of values that the function is capable of accepting as its input changes across its domain.
The range of the function is the output values, or the y-coordinates of the function.
For the given graph, the output of the function is from 40 to 100 thus, the range is:
40≤f≤100
Hence, for the function in the graph the range is option B: 40≤f≤100.
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You invest $1850 in an account paying 5.2% interest compounded daily. What is the account's effective annual yield?
Answer: 5.36%
Step-by-step explanation:
The formula for the effective annual yield (EAR) when the interest is compounded daily is given by:
(1 + r/365)^365 - 1
where r is the annual interest rate.
In this case, the annual interest rate is 5.2% or 0.052. Substituting this value into the formula, we get:
(1 + 0.052/365)^365 - 1 = 0.0536
Multiplying this value by 100 gives the effective annual yield as a percentage:
0.0536 x 100 = 5.36%
Therefore, the effective annual yield of the account is 5.36%.
The formula for the volume of a right circular cylinder is V=πr2h. If r=2b and h=5b+3, what is the volume of the cylinder in terms of b?
The volume of the cylinder in terms of b is expressed as: (20b³ + 12b²)π cubic units.
What is the Volume of a Cylinder?Where r represents the radius of a cylinder and h represents its height, the volume of the cylinder can be calculated using the formula:
V = πr²h.
Given the following:
Radius (r) = 2b
Height (h) = 5b + 3
Volume (V) = π * (2b)² * (5b + 3)
Volume (V) = π * 4b² * (5b + 3)
Volume (V) = π * 20b³ + 12b²
Volume (V) = (20b³ + 12b²)π cubic units
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Video Wa community into four adve graph shows the total number of each zone for a three-week period. 1,000 900 800 700 600 500 400 300 200 100 0 Number of Customers Zone 1 Video Warehouse Zone 2 Zone 3 A. 3:5 OB. 3:2 OC. 2:1 OD. 1:2 Zone 4 1. During the three weeks, how many customers came from Zones 3 and 4? A. between 900 and 1,000 B. between 1,000 and 1,100 C. between 1,100 and 1,200 D. between 1,200 and 1,300 2. Approximately what is the ratio of custo from Zone 1 to customers from Zone 3
The total number of each zone for a three-week period is between 1200 and 1300, 1:2 is the ratio of customer from Zone 1 to customers from Zone 3.
1) During 3 weeks bound
Customers from zone 3 = 900
Customers from zone 4 = 320
So, total customers = (900 + 320)
= 1220
Hence, option D is correct i.e between 1200 and 1300.
2) Customer from zone 1 = 450
Customers from zone 3 = 900
So, Customer from zone 1 / Customer from zone 3 = 450/900
= 1/2
Hence, option 2 i.e., 1:2 is correct.
3) 3/5 pound of clay is used to make = 1 bowl
So, 1 bound of clay is used to make = 5/3 bowl
By unitary method,
10 pounds of clay is used to make = 10 *(5/3)
= 16.66
= 17 bowls (approx)
Hence, option d is correct answer.
Therefore, The total number of each zone for a three-week period is between 1200 and 1300, 1:2 is the ratio of customer from Zone 1 to customers from Zone 3.
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If f(x)=2x^2+3 and g(x)=x^2-7, find (f-g)(x).
Answer:
To find (f-g)(x), we need to subtract g(x) from f(x):
(f-g)(x) = f(x) - g(x)
Given f(x) = 2x^2 + 3 and g(x) = x^2 - 7, we can substitute these expressions into the formula above:
(f-g)(x) = f(x) - g(x)
= (2x^2 + 3) - (x^2 - 7)
Now we simplify by combining like terms:
(f-g)(x) = 2x^2 + 3 - x^2 + 7
= x^2 + 10
Therefore, (f-g)(x) = x^2 + 10.
Answer: To find (f-g) (x)
(f-g)(x)= f(x) -g(x)
Given f(x) = 2x^2 + 3 and g(x) = x^2 - 7, we can substitute these expressions into the formula above:
(f-g)(x) = f(x) -g(x)
=(2x^2+3) - (x^2-7)
Now we can simplify the equation:
(f-g)(x)= 2x^2+3 - x^2+7
= x^2 + 10
Hence, (f-g)(x) = x^2+10
Expert answer:
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which expression is equivalent to
Answer:
B. w - 1
Step-by-step explanation:
Using PEMDAS, we start see there are no expressions inside parenthesis or exponents we move to multiplication/division. The expression 1/4(4) can be rewritten as 1/4 * 4 or as 4/4, both of which equal 1.
So w - 1 is an equivalent expression.
A small college has 10 professors in the Mathematics Department. The department teaches pure math, applied math, and
statistics. Professors are apportioned using Hamilton's apportionment method, according to the number of majors in each
field. There are 4 pure math majors, 12 applied math majors, and 12 statistics majors.
If the department receives a grant to hire 1 more professor, will the Alabama paradox occur? Why or why not?
Answer:
Yes, because while the total number of professors increases, the number of pure math professors decreases.
Step-by-step explanation:
Using Hamilton's method:
pure math 4/28 = 0.142857
applied math 12/28 = 0.42857
statistics = 12/28 = 0.42857
10 × 0.142857 = 1.42857
10 × 0.42857 = 4.2857
10 × 0.42857 = 4.2857
numbers of professors:
pure math 1
applied math 4
statistics 4
total 9
Add 1 to pure math
Final original numbers of professors using Hamilton's method
pure math 2
applied math 4
statistics 4
total 10
Add 1 professor to department.
11 × 0.142857 = 1.5714
11 × 0.42857 = 4.71428
11 × 0.42857 = 4.71428
numbers of professors:
pure math 1
applied math 4
statistics 4
total 9
Add 1 to applied math and 1 to statistics
Final new numbers of professors using Hamilton's method
pure math 1
applied math 5
statistics 5
total 11
Despite the addition of 1 professor to the department, the field of pure mathematics went from 2 professors to 1 professor. This is an example of the Alabama paradox.
Answer: Yes, because while the total number of professors increases, the number of pure math professors decreases.
answer pls!! quick asp
Answer:
ZR = 145°
Step-by-step explanation:
the secant- secant angle ZAR is half the difference of the measures of the intercepted arcs , that is
∠ ZAR = [tex]\frac{1}{2}[/tex] ( ZR - KV )
30 = [tex]\frac{1}{2}[/tex] (5x + 10 - (3x + 4) ) ← multiply both sides by 2 to clear the fraction
60 = 5x + 10 - 3x - 4
60 = 2x + 6 ( subtract 6 from both sides )
54 = 2x ( divide both sides by 2 )
27 = x
Then
ZR = 5x + 10 = 5(27) + 10 = 135 + 10 = 145°
Goran used 2 1/2 gallons of gas on Sunday and 1/4 gallons of gas on Monday. How many gallons did he use on the two days combined? Write your answer as a mixed number in simplest form.
Okay, here are the steps to solve this problem:
* Goran used 2 1/2 gallons on Sunday
* Goran used 1/4 gallons on Monday
* So on Sunday he used 2 1/2 gallons and on Monday he used 1/4 gallons
* To find the total gallons used on both days:
** 2 1/2 gallons (used on Sunday)
+ 1/4 gallons (used on Monday)
= 2 3/4 gallons (total used on both days)
So in simplest form as a mixed number, the total gallons Goran used on both days combined is:
2 3/4
[tex]\sf 2\dfrac{3}{4}.[/tex]
Step-by-step explanation:To find this answer, all we need to do is add up both of the fractions, that will give us the total amount of gas used on both days. Let's calculate:
1. Convert the first fraction into an improper fraction.[tex]\sf 2\dfrac{1}{2} =\\ \\\dfrac{2}{2}+\dfrac{2}{2}+\dfrac{1}{2}=\dfrac{5}{2}[/tex]
2. Write the sum of the two fractions that express the daily gas consumption.[tex]\sf \dfrac{5}{2}+ \dfrac{1}{4}[/tex]
3. Using the formula from the attached image, rewrtite the fraction addition.[tex]\sf \dfrac{5}{2}+ \dfrac{1}{4}= \dfrac{(5*4)+(2*1)}{2*4}= \dfrac{(20)+(2)}{8}=\dfrac{22}{8}=\dfrac{11}{4}[/tex]
4. Convert the resulting improper fraction into a mixed fraction.[tex]\sf \dfrac{11}{4}=2.75[/tex]
Take the entire part of the decimal number (2) and write it as the whole number on the mixed number. Also, since the fraction has a denominator of 4, a unit of this fraction would be 4/4, then, the 2 units that we're going to express as a whole number would be 8/4. So, subtract 8/4 from 11/4 and express in the following fashion:
[tex]\sf 2(\dfrac{11}{4}-\dfrac{8}{4} )\\ \\\\ \sf 2(\dfrac{3}{4}) \\ \\ \\\ 2\dfrac{3}{4}[/tex]
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Solve 9h2+9h=−2 by using the quadratic formula. Give an exact answer and simplify any fractions. If there are multiple answers, list them separated by a comma, e.g. 1,2. If there is no solution, enter ∅.
The value of h is ( -1/3, -2/3)
What is quadratic equation?Quadratic equation is a type of equation in which the highest power of variable is 2. A quadratic equation is represented as ax²+bx + c = 0
Solving, 9h²+9h = -2
9h²+9h+2 = 0
9h² +6h +3h +2 = 0
(9h²+6h) ( 3h +2) = 0
3h( 3h + 2)+1 ( 3h+2) = 0
(3h+2)(3h+1) = 0
Therefore ;
3h+2 = 0
3h = -2
h = -2/3 or
3h+1 = 0
3h = -1
h = -1/3
therefore the value of h is ( -1/3, -2/3)
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(A.BA) The area of a rectangular trampoline is 112 f2. The length of the trampoline is 6 ft greater than the width of the trampoline. This situation can be regresented by the equation W^2+ 6w = 112. What is the value b when the equation is written in standard form?
The value of b when the equation is written in standard form is determined as: 6.
What is an Equation in Standard Form?An equation can be expressed in standard form as:
Ax² + Bx + C = 0, where A, B, and C are constants.
Given that the situation is expressed by the equation, w² + 6w = 112, we can expressed this in standard form as follows:
w² + 6w = 112
Subtract 112 from both sides:
w² + 6w - 112 = 112 - 112
w² + 6w - 112 = 0
The standard form is therefore, w² + 6w - 112 = 0, and the value of b is equal to 6.
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The town's emergency response planning committee wants to place four emergency response centers at the four corners of town Each would serve the people who live within 3 mi of the response center Sketch the loci of points for the areas served What are the problems with this idea ? What is one potential solution ?
The town's emergency response planning committee has proposed to place four emergency response centers at the four corners of the town, with each center serving the people who live within 3 miles of the respective response center. The idea is to provide coverage to the entire town and ensure prompt emergency response for the residents. However, there are some problems with this idea.
Unequal coverage: Placing the emergency response centers at the four corners of the town may result in unequal coverage for the residents. Depending on the size, shape, and population distribution of the town, some areas may be farther away from the response centers, resulting in longer response times and reduced effectiveness in emergency situations.
Overlapping coverage: Placing four response centers in a small town may result in overlapping coverage areas, where the coverage areas of multiple response centers overlap with each other. This may lead to duplication of resources and inefficiencies in emergency response efforts.
Limited reach: Placing response centers only at the four corners of the town may result in limited reach for certain areas, especially those located in the middle or farther away from the corners. This may leave some residents outside the 3-mile coverage radius without access to timely emergency response services.
One potential solution to address these problems could be to use a more strategic approach to determine the locations of the emergency response centers. This could involve conducting a thorough analysis of the town's population density, geographical features, road network, and existing emergency resources. Based on this analysis, the response centers could be strategically placed at locations that provide the best coverage to the entire town, considering factors such as response time, resource allocation, and accessibility.
For example, instead of placing all the response centers at the corners of the town, they could be distributed more evenly across the town to ensure more equitable coverage. Additionally, the use of advanced GIS (Geographical Information System) technology and modeling techniques could help in identifying optimal locations for the response centers, taking into account various factors such as population density, road network, and travel time.
Furthermore, collaboration and coordination among the emergency response centers, along with proper communication and information sharing, can help in avoiding duplication of resources and improving the efficiency of emergency response efforts.
In conclusion, while the idea of placing four emergency response centers at the four corners of town may seem simple, there are potential problems such as unequal coverage, overlapping coverage, and limited reach. A more strategic and data-driven approach, considering factors such as population density, geographical features, and existing resources, can help in identifying optimal locations for the response centers and ensuring effective emergency response services for all residents of the town.
If you invest $3900 at a 7.84 simple annual interest rate, approximately how long will it take for you to have a total of $10,000
It would take approximately 19.94 years to accumulate a total of $10,000 from an initial investment of $3,900 at a simple annual interest rate of 7.84%.
To determine how long it will take to accumulate $10,000 from an initial investment of $3,900 at a simple annual interest rate of 7.84%, we can use the formula for simple interest:
I = Prt
where I is the interest earned, P is the principal or initial investment, r is the annual interest rate as a decimal, and t is the time in years.
To find the time required to reach a final amount of $10,000, we can rearrange the formula:
t = (I/P) / r
First, we need to calculate the interest earned on the initial investment:
I = Prt
I = (3900)(0.0784)t
I = 305.76t
To reach $10,000, we need to earn an additional $10,000 - $3,900 = $6,100 in interest. So we can set up the equation:
6100 = 305.76t
Solving for t, we get:
t = 6100 / 305.76
t ≈ 19.94 years
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A business has 40,000 to spend on advertising in an upcoming sale. The money is to be divided between television (x), radio (y) and newspapers (z). The business manager has decided to spend three times as much money on television as on radio. The manager has also decided to spend 8000 less on radio advertising than on newspapers. Find out, using matrix inversion method, the total amount divided between television, radio and newspapers.
Okay, let's break this down step-by-step:
* The business has $40,000 total to spend on advertising
* Some will go to TV (x), some to radio (y), and some to newspapers (z)
* 3 times as much will go to TV as radio, so x = 3y
* They will spend $8,000 less on radio than newspapers, so y = z - 8,000
* We have:
x = 3y (1)
y = z - 8,000 (2)
x + y + z = 40,000 (3)
To solve this using matrix inversion:
1) Turn the equations into a matrix:
3 1 0 y
1 -1 1 z
1 1 1 x
2 0 40,000
2) Invert the matrix:
0.3333 0.3333 0.3333
-0.25 0.75 0
0.125 0.125 0.750
3) Plug in the values from (2) and (3):
y = 0.3333(z - 8,000)
x = 0.125z + 0.125(40,000 - z)
4) Solve for z, the amount for newspapers. We get:
z = 40,000 * (0.75) = 30,000
5) Plug z = 30,000 back into the other equations:
x = 0.125 * 30,000 + 0.125 * 10,000 = 12,000
y = 0.3333 * (30,000 - 8,000) = 8,000
z = 30,000
So in total:
TV (x) = $12,000
Radio (y) = $8,000
Newspapers (z) = $30,000
Does this make sense? Let me know if you have any other questions!
A ladder leans against the side of a house. The angle of elevation of the ladder is 61 degrees , and the top of the ladder is 5 m above the ground. Find the distance from the bottom of the ladder to the side of the house. Round your answer to the nearest tenth.
Answer:
50.8
Step-by-step explanation:
O is the center of the regular decagon below. Find its area. Round to the nearest tenth if necessary. 2
The area of Regular Decagon is 325 square units.
How to find the area of a regular decagon?The formula for the area of a regular polygon is :
Area (A) = [number of sides (n) × length of side (l) × apothem (a)] /2
Apothem Formula is given by:
For the decagon, n = 10 (10 sides) and a = 10
Now, we calculate the length of the decagon:
a = l / 2tan(180/n)°
10 = l / 2tan(180/10)°
10 = l / 2 tan18°
l = 10 × 2 × tan18°
l = 6.50 units
We have to find the area of decagon:
A = [number of sides (n) × length of side (l) × apothem (a)] /2
A = (10 × 6.50 × 10)/2
A = 325 square units
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Foe complete question, to see the attachment.
VERY EASY 20 POINTS
The graph of y = f(x) is shown below, in red. Find the equation that corresponds to the blue graph.
Answer:
y = f(x) + 3
Hope this helps!
Step-by-step explanation:
Because the graph went up 3...
Answer:
y = f(x) + 3
Hope this helps!
Step-by-step explanation:
Because the graph went up 3...
Use separation of variables to find the general solution to the following differential equation.
Therefore, the general solution to the differential equation is
Y = e⁽ˣ⁾2+X+C) - 1 or Y = -e⁽ˣ⁾2+X+C) - 1
What exactly is a different equation?A differential equation is an equation that connects the derivatives of one or more unknown functions. It is an equation that uses the derivatives of a function or functions, in other words. Many physical processes, including the motion of objects under the influence of forces, the movement of fluids, and the spread of disease, are modelled using differential equations in science and engineering. Ordinary differential equations (ODEs) and partial differential equations (PDEs) are the two primary categories of differential equations.
To solve this differential equation using separation of variables, we first need to separate the variables Y and X on opposite sides of the equation:
dY / (Y + 1) = (2X + 1) dX
Following that, we incorporate both sides of the problem:
∫ dY / (Y + 1) = ∫ (2X + 1) dX
The integral on the left side can be evaluated using the substitution
u = Y + 1 and du = dY:
ln|Y + 1| = ∫ dY / (Y + 1) = ln |u| + C1
where C1 is the constant of integration.
The integral on the right side can be evaluated using the power rule of integration:
∫ (2X + 1) dX = X² + X + C2
where C2 is another constant of integration.
Putting these results together gives the general solution to the differential equation:
ln|Y + 1| = X² + X + C
where C = C1 + C2 is the combined constant of integration.
To solve for Y, we exponentiate both sides of the equation:
|Y + 1| = e⁽ˣ⁾2+X+C)
Taking into account the absolute value, we have two cases:
Case 1: Y + 1 = e⁽ˣ⁾2+X+C)
Y = e⁽ˣ⁾2+X+C) - 1
Case 2: Y + 1 = -e⁽ˣ⁾2+X+C)
Y = -e⁽ˣ⁾2+X+C) - 1
Therefore, the general solution to the differential equation DY/DX=(Y+1)(2X+1) is:
Y = e⁽ˣ⁾2+X+C) - 1 or Y = -e⁽ˣ⁾2+X+C) - 1
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The monthly cost (in dollars) of a long-distance phone plan is a linear function of the total calling time (in minutes). The monthly cost for 39 minutes of calls is 18.45 and the monthly cost for 56 minutes is $20.66. What is the monthly cost for 50 minutes of calls?
Step-by-step explanation:
We can use the two data points given to find the equation of the line, which gives the monthly cost (y) in terms of the calling time in minutes (x).
First, we can find the slope of the line:
slope = (change in y) / (change in x)
slope = (20.66 - 18.45) / (56 - 39)
slope = 0.219
Next, we can use one of the data points and the slope to find the y-intercept (b). Let's use the data point (39, 18.45):
y - y1 = m(x - x1)
y - 18.45 = 0.219(x - 39)
y - 18.45 = 0.219x - 8.541
y = 0.219x + 9.909
So the equation for the monthly cost is y = 0.219x + 9.909.
To find the monthly cost for 50 minutes of calls, we plug in x = 50:
y = 0.219(50) + 9.909
y ≈ $21.44
Therefore, the monthly cost for 50 minutes of calls is approximately $21.44.
Simplify.
8x^3/2 × -7x^2/3
And pls add an explanation. I know the answer but not how to get there.
The expression 8x^3/2 × -7x^2/3 can be simplified to give
-56 x^(13/6).How to simplify the expressionTo reduce the given expression, we can employ the principles of exponents and carry out the multiplication.
The given expression is:- 8x^(3/2) * (-7x^(2/3))
Utilizing this property, combining two terms with exponents entails adding the exponents when they feature the same bases. Thus, we establish:
8 * (-7) = -56 (coefficient multiplication)
x^(3/2 + 2/3) results in x^(13/6) (exponent addition)
Bringing it all together, we have:
8x^(3/2) * (-7x^(2/3)) = -56x^(13/6)
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5. - 1 5/7 divided by 1/2
Answer: -3 3/7
Step-by-step explanation:
First, you can convert -1 5/7 to an improper fraction by multiplying the denominator with the whole number, then add the answer to the numerator. Like this:
7 × 1 = 7
7 + 5 = 12
12 will be your numerator and 7 will be your denominator. When you're done converting, bring down the negative sign. The fraction should be:
-12/7
Now, since -1 5/7 is -12/7, we can divide.
To divide, we can use, KCF (Keep, Change, Flip)
First, keep -12/7.
Second, change division to multiplication.
Third, flip 1/2.
Your problem will be set up like:
-12 /7 × 2/1
Now, we can solve. Multiply both the numerator AND the denominator. Since you're multiplying, you don't have to change the denominator to match each other. The answer will be:
-24 / 7
However, this fraction is improper. We can use long division to make this fraction proper.
Divide 24 and 7. You'll get 3, with a remainder of 3.
The remainder will be the new numerator and the answer will be the whole number.
3 - Remainder (Whole Number)
3 - Answer (New Numerator)
7 - Dividend (Original Denominator) Will not be changed
Hence, your final answer is:
-3 3/7
Reply below if you have any questions or concerns.
You're Welcome!
- Nerdworm
[tex]-\dfrac{24}{7} }.[/tex]
Step-by-step explanation:1. Turn the mixed fraction into an improper fraction.[tex]\sf -1\dfrac{5}{7}=\\\\ \\ -1+\dfrac{5}{7}=\\\\ \\ -(\dfrac{7}{7}+\dfrac{5}{7})=\\ \\ \\-(\dfrac{12}{7})[/tex]
2. Write the division.[tex]\dfrac{-\dfrac{12}{7} }{\dfrac{1}{2} }[/tex]
3. Use the properties of fraction to rewrite the division (check attached image).[tex]-\dfrac{12}{7} }*\dfrac{2}{1} =\\ \\ \\-\dfrac{12}{7} }*2=\\ \\ \\-\dfrac{12*2}{7} }\\ \\ \\-\dfrac{24}{7} }[/tex]
Showing results for a rectangular glass dish has a measurements of 2.5 inches high, 6.75 inches wide and 8.5 inches long. the density of the glass in the dish is 2.23 grams per cubic centimeter and the mass of the dish is about 0.9 kilograms, what is the thickness of the glass?