Using the fundamental identities, the equivalent expressions are hereby matched as follows:
a. 2 csc (θ) cot (θ) = 2 sin²(θ) / (sin(θ) + cos(θ))
b. 2 sec (θ) tan (θ) = (2 sec²(θ)) - 2 sec(θ)
c. 2 cot (θ) = 2 / sin(θ) - csc(θ)
d. 2 csc² (θ) = 2 / (csc(θ) - 1)(csc(θ) + 1)
How did we arrive at these expressions?Using the fundamental identities, rewrite each expression as follows:
a. 2 csc (θ) cot (θ) = 2 / sin(θ) * cos(θ) = 2 cos(θ) / sin(θ) = 2 / sin(θ) - 2 / sin(θ) + 2 cos(θ) / sin(θ) = 2 / sin(θ) - 2 / (1 + cot(θ)) = (2 - 2 sin(θ)) / (sin(θ) + cos(θ)) = (2 sin²(θ)) / (sin(θ) + cos(θ))
b. 2 sec (θ) tan (θ) = 2 / cos(θ) * sin(θ) / cos(θ) = 2 sin(θ) / cos²(θ) = 2 / cos²(θ) - 2 / cos(θ) = (2 sec²(θ)) - 2 sec(θ)
c. 2 cot (θ) = 2 cos(θ) / sin(θ) = 2 / sin(θ) - 2 / sin²(θ) = 2 / sin(θ) - csc(θ)
d. 2 csc² (θ) = 2 / sin²(θ) = 2 / (1 - cos²(θ)) = 2 / (1 - cos(θ))(1 + cos(θ)) = 2 / (csc(θ) - cot(θ))(csc(θ) + cot(θ)) = 2 / (csc(θ) - 1)(csc(θ) + 1)
Therefore, the matches are:
a. 2 csc (θ) cot (θ) = 2 sin²(θ) / (sin(θ) + cos(θ))
b. 2 sec (θ) tan (θ) = (2 sec²(θ)) - 2 sec(θ)
c. 2 cot (θ) = 2 / sin(θ) - csc(θ)
d. 2 csc² (θ) = 2 / (csc(θ) - 1)(csc(θ) + 1)
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Which equations represent circles that have a diameter of 12 units and a center that lies on the y-axis? Select two options. x2 + (y – 3)2 = 36 x2 + (y – 5)2 = 6 (x – 4)² + y² = 36 (x + 6)² + y² = 144 x2 + (y + 8)2 = 36
The equations which represents circles that have a diameter of 12 units and a center that lies on the y-axis are; x² + (y – 3)² = 36 and x² + (y + 8)² = 36.
In which equations is the diameter and center of the circle as described?Recall, the equation of a circle takes the form;
(x - h)² + (y – k)² = r² where r = radius = diameter/2 and (h, k) is the center.
Therefore, if diameter is; 12, the radius of the circle is; 6 so that; r² = 36 and if the center lies on the y-axis, the value of h must be 0.
Ultimately, the equations that satisfy the given conditions are; x² + (y – 3)² = 36 and x² + (y + 8)² = 36.
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9. Maria is twice the age of Jonas, who is
twice the age of Franklin. Franklin is 2. yr
less than half Maria's age. How old are
Maria, Jonas, and Franklin?
Answer: Maria is 12 years old, Jonas is 6 years old, and Franklin is 3 years old.
Step-by-step explanation: Let's start by assigning variables to represent the ages of Maria, Jonas, and Franklin.
Let's use the variable "M" to represent Maria's age.
Let's use the variable "J" to represent Jonas's age.
Let's use the variable "F" to represent Franklin's age.
From the problem statement, we know that:
Maria is twice the age of Jonas: M = 2J
Jonas is twice the age of Franklin: J = 2F
Franklin is 2 years less than half of Maria's age: F = 0.5M - 2
We can use these equations to solve for the ages of Maria, Jonas, and Franklin.
First, we can substitute J = 2F into the equation M = 2J to get M = 2(2F) = 4F.
Next, we can substitute this expression for M into the equation F = 0.5M - 2 to get F = 0.5(4F) - 2, which simplifies to 2F = 6 and therefore F = 3.
Now that we know Franklin's age, we can use the equation J = 2F to find Jonas's age: J = 2(3) = 6.
Finally, we can use the equation M = 2J to find Maria's age: M = 2(6) = 12.
Therefore, Maria is 12 years old, Jonas is 6 years old, and Franklin is 3 years old.
The function h(t)=-4t^2+14t+8 gives the height h, in feet, of a football as a function of time t, in seconds, after it is kicked. How long does it take for the football to hit the ground?
Answer:
t = 4
Step-by-step explanation:
Use the quadratic formula to find the zeros of the function, when h(t) equals 0 that means the ball is touching the ground, giving the maximum time it took for the ball to hit the ground.
[tex]\frac{-b\frac{+}{-} \sqrt{b^2-4*a*c} }{2*a} \\\frac{-14\frac{+}{-} \sqrt{14^2-4*-4*8} }{2*-4} \\t = -0.5,4[/tex]
It takes 4 seconds, since time can not be negative
=
Suppose that you decide to borrow $14,000 for a new car. You can select one of the following loans, each requiring
regular monthly payments.
Installment Loan A: three-year loan at 5.1%
Installment Loan B: five-year loan at 4.8%
PA
[1-(1-+-:)]
Use PMT=
to complete parts (a) through (c) below.
a. Find the monthly payments and the total interest for Loan A.
The monthly payment for Loan A is S
(Do not round until the final answer. Then round to the nearest cent as needed.)
mlm
*****
H
√
V
1,
(0,0)
More
Answer:
$2,602.44
Step-by-step explanation:
To find the monthly payments and the total interest for Loan A, we can use the formula for the present value of an installment loan:
PV = PMT x [1 - (1 + r/n)^(-nt)] x (n/r)
where PV is the present value of the loan, PMT is the monthly payment, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the number of years of the loan.
For Loan A, we have:
PV = $14,000
r = 0.051 (5.1% as a decimal)
n = 12 (monthly payments)
t = 3
Substituting these values into the formula and solving for PMT, we get:
PMT = PV / ([1 - (1 + r/n)^(-nt)] x (n/r))
= 14000 / ([1 - (1 + 0.051/12)^(-12*3)] x (12/0.051))
= $417.79
So the monthly payment for Loan A is $417.79.
To find the total interest paid over the life of the loan, we can simply multiply the monthly payment by the total number of payments and subtract the original loan amount:
Total interest = PMT x (nt) - PV
= 417.79 x (3 x 12) - 14000
= $2,602.44
Therefore, the total interest paid for Loan A is $2,602.44.
A sprinkler set in the middle of a lawn sprays in a circular pattern. The area of the lawn that gets sprayed by the sprinkler can be described by the equation (x+7)2+(y−2)2=225
.
What is the greatest distance, in feet, that a person could be from the sprinkler and get sprayed by it?
Answer:
Step-by-step explanation:
Answer:
Step-by-step explanation:
Find all possible zeros
The possible zeros from the function would be zero.
How to find the possible zeros ?To find all possible zeros of the polynomial f(x) = 2x³ + 3x² - 10x + 7, we can use the Rational Root Theorem.
We can test these possible zeros by plugging them into the polynomial:
f(1) = 2(1)³ + 3(1)² - 10(1) + 7 = 2
f(-1) = 2(-1)³ + 3(-1)² - 10(-1) + 7 = 12
f(7) = 2(7)³ + 3(7)² - 10(7) + 7 = 776
f(-7) = 2(-7)³ + 3(-7)² - 10(-7) + 7 = -732
f(1/2) = 2(1/2)³ + 3(1/2)² - 10(1/2) + 7 = -0.5
f(-1/2) = 2(-1/2)³ + 3(-1/2)² - 10(-1/2) + 7 = 8.5
f(7/2) = 2(7/2)³ + 3(7/2)² - 10(7/2) + 7 = 89.75
f(-7/2) = 2(-7/2)³ + 3(-7/2)² - 10(-7/2) + 7 = -83.75
None of the possible rational zeros are actual zeros of the polynomial f(x) = 2x³ + 3x² - 10x + 7.
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The full question is:
Find all possible zeros. (Enter your answers as a comma-separated list.) f(x) = 2x3 + 3x2 − 10x + 7
Adeline has a box full of notebooks. If she gives 13 notebooks to each of her children, she will have 8. notebooks left. if she gives 15 notebooks to each childish will have zero notebooks left.how many notebooks and how many children does Adeline have
Answer:
Adeline has 60 notebooks and 4 children.
Step-by-step explanation:
Let's assume that Adeline has 'x' notebooks and 'y' children.
According to the given information, if Adeline gives 13 notebooks to each child, she will have 8 notebooks left. This can be represented by the equation:
x - 13y = 8 ...(1)
Similarly, if she gives 15 notebooks to each child, they will have zero notebooks left, which can be represented by the equation:
x - 15y = 0 ...(2)
We now have two equations with two variables. To solve for 'x' and 'y', we can use the method of elimination. Multiplying equation (1) by 15 and equation (2) by 13, we get:
15x - 195y = 120 ...(3)
13x - 195y = 0 ...(4)
Subtracting equation (4) from equation (3), we get:
2x = 120
x = 60
Substituting the value of 'x' in equation (2), we get:
60 - 15y = 0
y = 4
Therefore, Adeline has 60 notebooks and 4 children.
A box with a square base 2.5 feet wide on a side is 5 feet high. What is its volume?
Answer: the volume of the box is 31.25 cubic feet.
Step-by-step explanation: The volume of the box can be found by multiplying the area of the square base by its height:
Volume = Base Area x Height
Since the base of the box is a square with 2.5 feet wide on a side, its area can be calculated as:
Base Area = Side Length x Side Length = 2.5 ft x 2.5 ft = 6.25 sq. ft
And since the height of the box is 5 feet, the volume can be calculated as:
Volume = Base Area x Height = 6.25 sq. ft x 5 ft = 31.25 cubic feet
Therefore, the volume of the box is 31.25 cubic feet.
A cylinder is sliced at an angle, leaving the shape shown at right. The shorter height
is 12 cm while the longer height is 18 cm. The radius of the base is 4 cm.
What is the volume of this sliced cylinder?
Answer:
12 × 4 × 18 = 864
( Use the volume formal L × W × H to help in all volume equations )
please help me with these 5 math problems please hurry
1) A and D are linear equations while B and C are non linear
2) The rule of the equation is given by option C
3) The solution is option C
4) The area is option B
5) The standard form expressions are shown by option C
What is a linear equation?We know that;
y = x + 3
-2x + y = 1
Then
x - y = -3 ---- (1)
-2x + y = 1 ---- (2)
x = -3 + y --- (3)
Substitute (3) into (2)
-2(-3 + y) + y = 1
6 - 2y + y = 1
6 -y = 1
-y = 1 -6
y = 5
Substitute y = 5 into (1)
x - 5 = -3
x = -3 + 5
x = 2
Thus the solution is (2, 5)
a^2 = 10^2 - 8^2
a = √100 - 64
a = 6
A = 1/2bh
A = 0.5 * 8 * 6
A = 24cm^2
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write 2/100 as a percentage
Answer:
2%
Step-by-step explanation:
2/100 = 0.02
Then we can convert the decimal to a percentage by multiplying by 100:
0.02 x 100 = 2%
So 2/100 as a percentage is 2%. I hope this helps!
Danielle likes to purchase items at estate sales, clean them up, then resale them online for a profit. She recently purchased an antique dresser with a mirror at an estate sale, cleaned it up, and advertised it for 250% of her purchase price. Danielle sold the dresser for $255, which was 15% less than the advertised price.
To the nearest whole dollar, how much did Danielle purchase the antique dresser for and what was her initial advertised price?
A. purchase price: $120, advertised price: $300
B. purchase price: $108, advertised price: $270
C. purchase price: $130, advertised price: $325
D. purchase price: $117, advertised price: $293
To the nearest whole dollar, the antique dresser was purchased by Danielle as follows: A. purchase price: $120, advertised price: $300.
How the purchase price is determined?The purchase price of the antique dresser can be determined by reversing the solution from the discounted price.
The discounted price is divided by the discount factor to calculate the advertised or marked price.
Let the purchase price of the antique dresser = x
Markup = 250%
Discount = 15%
Discounted price = $255
Discount factor = 0.85 (1 - 0.15)
Marked up or advertised price = $300 ($255 ÷ 0.85)
Purchase price = $120 ($300 ÷ 2.5)
Thus, the correct option is Option A.
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Solve for x.
37°
10 cm
x = [?] cm
Round to the nearest hundredth.
X
I really need help and can you tell me on how to solve it
Answer: 6.02
Step-by-step explanation:
use SOH CAH TOA, or in this case SOH
the Sine equals Opposite over Hypotenuse, which in this case is sin37=x/10.
Plug into a calculator and you get 6.02 after rounding.
Hope this helps :)
(and thanks for answering one of my questions)
The value of x is 6.01815 cm.
What is Trigonometry?Trigonometry is a discipline of mathematics that studies the relationship between the sides of a triangle (right triangle) and their angles. There are six trigonometric functions that define the relationship between sides and angles.
We have,
Angle = 37
Hypotenuse = 10 cm
Using Trigonometry
sin 37 = P/ H
0.601815 = x/ 10
x= 6.01815 cm
Thus, the value of x is 6.01815 cm.
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Log5x-log4=2 solve the equation
Answer:
x = 80
Step-by-step explanation:
using the rules of logarithms
log x - log y = log ([tex]\frac{x}{y}[/tex] )
[tex]log_{b}[/tex] x = n ⇒ x = [tex]b^{n}[/tex]
given
log5x - log4 = 2
log ([tex]\frac{5x}{4}[/tex] ) = 2
note that log is actually [tex]log_{10}[/tex]
then
[tex]log_{10}[/tex] ( [tex]\frac{5x}{4}[/tex] ) = 2
[tex]\frac{5x}{4}[/tex] = 10² = 100 ( multiply both sides by 4 to clear the fraction )
5x = 400 ( divide both sides by 5 )
x = 80
What is the area of a sector when 0=pi/2 radians and r=8/3
[tex]\textit{area of a sector of a circle}\\\\ A=\cfrac{\theta r^2}{2} ~~ \begin{cases} r=radius\\ \theta =\stackrel{radians}{angle}\\[-0.5em] \hrulefill\\ \theta =\frac{\pi }{2}\\[1em] r=\frac{8}{3} \end{cases}\implies A=\cfrac{1}{2}\cdot \cfrac{\pi }{2}\cdot\left( \cfrac{8}{3} \right)^2 \\\\\\ A=\cfrac{1}{2}\cdot \cfrac{\pi }{2}\cdot \cfrac{64}{9}\implies A=\cfrac{16\pi }{9}\implies A\approx 5.59[/tex]
Angles in a Triangle
Answer:
a = 40°
Step-by-step explanation:
An isosceles triangle has two sides that are the same size. The angles opposite from those sides are equal (they are called base angles.) And, the three angles in a triangle add up to 180°
So,
100 + a + a = 180
combine like terms
100 + 2a = 180
subtract 100
2a = 80
divide by 2
a = 40
What is the ratio of squares to circles in this picture?
Answer:
Step-by-step explanation:
since there is one square and 4 circles, the answer is 1/4
4. The Federal Reserve System
a. lends money to banks
Ob. functions as the bank for the government
c. aids the check-clearing process
Od. all of these
5. What portion of its revenues does the United States spend on defense
a.
about 10 percent
b.
about 20 percent
c. about 40 percent
d. about 50 percent
Economics
November 2000
I
Page 2 of 3
4. The Federal Reserve System does d. all of these central bank functions.
Lends money to banksFunctions as the bank for the governmentAids the check-clearing process5. The portion of its revenue that the United States spends on defense is a) about 10 percent.
What is the federal reserve?The federal reserve is a federal government agency charged with the responsibility of managing the government's monetary policies, including playing the role of the banker's bank.
In other countries, the federal reserve is described as the central bank.
What is defense spending?Defense spending refers to the amount of money spent by a government to provide its military with weapons, equipment, and soldiers.
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you roll a 6 sided die; what is P(4 or less than 3)
Answer:
1/2
Step-by-step explanation:
The probablility of any number on a die is 1/6.
So the numbers 4, 2, and 1 each have a probability of 1/6 meaning that each have a total probability of: 1/2. 3 is not accounted for due to our restriction.
A lunch box is 24 cm long, 15 cm wide and 5 cm thick. What is
the volume of the lunch box?
Answer:
( Use this formula below to help you find the volume of all equations using volume)
Volume formal = L × W × H
Volume formal = 24 × 15 × 5
Answer = 24 × 15 × 5 = 1,800
Please help don't know how to do
Answer:
Step-by-step explanation:
Step 1: FOIL
Step 2: Simplification
Step 3: Getting ready to combine like terms
Find the measure of 3x-46+14=90
Answer:
Step-by-step explanation:
3x = 90 + 46 - 14
3x = 122
x=[tex]\frac{122}{3}[/tex]≈40,7
Answer: 40.6 repeated
Step-by-step explanation: combine -46 and 14 which is -32. then add 32 to 90. then divide 3 on both sides. 122/3 is 40.6 repeated
based on an exponential model of the data, what is the predicted value of variable 2 when variable 1=2? round your answer to the nearest whole number.
a. variable 2 ≈ 128
b. variable 2≈ 166
c. variable 2≈ 181
d. variable 2≈ 263
Based on an exponential model of the data, the predicted value of variable 2 when variable 1 = 2 is: A. variable 2 ≈ 128.
How to write an equation of the line of best fit for the data set?In order to determine an exponential equation for the line of best fit that models the data points contained in the graph (scatter plot), we would have to use a graphing calculator (Microsoft Excel).
Based on the scatter plot (see attachment) which models the relationship between the x-values (variable 1) and y-values (variable 2), an exponential equation for the line of best fit is given by
[tex]y = 137.93e^{-0.03x}[/tex]
Now, we can determine the predicted value of variable 2 when variable 1 = 2;
[tex]y(2) = 137.93e^{-0.03\times 2}\\\\y(2) = 137.93e^{-0.06}[/tex]
y(2) ≈ 128.
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to graph -5,5 start at 0,0 and then go?
Answer:
No
Step-by-step explanation:
to grap (-5,5) do not start at (0,0)
-5 is at the x axis so start it at (-0) then for 5 its juts (0) so (-0,0)
The employees of a jewelry store/
receive a monthly salary plus a
commission on their total monthly sales.
The equation representing an
employee's monthly earnings is y =
0.16x + 1250 where x is the monthly
sales and y is the monthly earnings.
Which of the following is true?
If the equation representing an employee's monthly earnings is y = 0.16x + 1250 where x is the monthly sales and y is the monthly earnings, the true statement is c) Employees receive 16% commission on their total monthly sales..
What is an equation?An equation is an algebraic statement of the equality or equivalence of mathematical expressions.
While equations use the equal symbol (=), mathematical expressions combine variables with numbers, constants, values, and mathematical operands.
Let the monthly earnings of an employee = y
Let the monthly sales = x
Equation:y = 0.16x + 1,250
Thus, from this equation, 0.16 represents 16% commission on sales, therefore, the correct option is Option C.
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Question Completion with Answer Options:a) The monthly salary of an employee is $1,250.
b) The slope is 1,250 and the y-intercept is 0.16.
c) Employees receive 16% commission on their total monthly sales.
d) The monthly salary of an employee is 12.50%.
In 1852, a person sold a house to a lady for $30. If the lady had put the $30 into a bank account paying 6% interest, how much would the investment have been worth in the year 2012 if interest were compounded in the following ways?
Suppose that 7 green balls and 9 purple balls are placed in an urn. Two balls are then drawn in succession. What is the probability that the second ball drawn is a purple ball if the first ball is replaced before the second is drawn?
Answer: 0.2461
Step-by-step explanation:
There are 16 balls in total, and 9 of them are purple. If the first ball drawn is replaced before the second is drawn, then the number of purple balls and the number of total balls in the urn remain the same for the second draw.
The probability of drawing a purple ball on the first draw is 9/16. Since the ball is replaced, the probability of drawing a purple ball on the second draw is also 9/16.
The probability that the second ball drawn is a purple ball, given that the first ball drawn was replaced and was a green ball:
(7/16) * (9/16) = 63/256 ≈ 0.2461
What is the domain of the function in the graph?
graph on the k-j axis, between the points (6, 120) and (11, 80)
A. 80≤j≤120
B. 6≤k≤11
C. 6≤j≤11
D. 80≤k≤120
Answer:
The domain of the function in the graph is option B, 6≤k≤11.
In the given graph, the horizontal axis represents k and the vertical axis represents j. The graph shows a line segment connecting the points (6, 120) and (11, 80). The domain of a function is the set of all possible values of the independent variable (input) for which the function is defined. In this case, k is the independent variable and j is the dependent variable (output).
The line segment in the graph represents a linear function, where the value of j depends on the value of k. The function is defined only between the values of k that correspond to the endpoints of the line segment, which are k = 6 and k = 11. Therefore, the domain of the function is 6≤k≤11.
5.8.PS-20
Question Help
Todd plans to run at least 3 miles each week for his health. Todd has a circular route in the neighborhood to run. Once
around that route is 420 yards. If Todd runs that route 10 times during the week, will he cover at least 3 miles? Explain.
Click the icon to view the customary units.
Select the correct answer, and fill in the answer boxes to complete the explanation.
(Type integers, fractions, or mixed numbers.)
OA. No, because 3 miles is
•B. Yes, because 3 miles is
yards, and 10 times around the route is
yards, and 10 times around the route is
yards, which is less than 3 miles.
yards, which is the greater than 3 miles.
Answer:
No
Step-by-step explanation:
No, because 10 times around the route is 4200 yards, which is less than 3 miles
I need help with questions 2 and 3 please.
Answer:
1.x=7 or x=-7
2.x=root of 65 or x=-root of 65
Step-by-step explanation:
I have put what I mean by root 65