At the given rate, we can say that they will walk 6 laps and run 10 laps for a total of 16 miles.
What is a direct proportional relationship?In a direct proportional relationship, we can say that the output variable is found by the multiplication of the input variable and the constant of proportionality k, as follows:
y = kx.
For the first question, we have that out of a total distance of 8 miles, and we can see that;
Walk is 3/8 of the distance.
Run is 5/8 of the distance.
Thus, the proportional relationships for the distances are given as follows:
Walked = 3/8 x Total Distance.
Ran = 5/8 x Total Distance.
We are told that the total distance is 16 miles and so;
Walked: 3/8 * 16 = 6 laps
Ran: 5/8 * 16 = 10 laps
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What is the solution to the system of equations below x y 7 and x 2y 11?
The solution to the system of equations is x = 3 and y = 4.
this question is related to system of equations ,
the equations are,
x + y = 7 .......[1]
x + 2y = 11
x= 11 - 2y
now, place the value of x in equation 1,
(11 - 2y) + y = 7
11 - 2y +y = 7
11 - y = 7
- y = 7 - 11
- y = - 4
y = 4
now to find the value of x, place the value of y in equation 1
x + 4 = 7
x = 7 - 4
x = 3.
so, the value of x and y is 3 and 4.
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A triangle has an angle that measures 126°. The other two angles are in a ratio of 11:16. What are the measures of those two angles?
Therefore, the measures of those two angles are 33° and 24°.
Define angles.An angle is a figure in plane geometry that is created by two rays or lines that have a shared endpoint. The Latin word "angulus," which means "corner," is the source of the English term "angle."
Define geometry.Geometry is a branch of mathematics that studies the sizes, shapes, positions, angles, and dimensions of things.
We know, the sum of the angles in a triangle is always 180 degrees.
126 + x + y = 180
x + y = 54
Given that the two angles are in a ratio of 11:16, the proportion:
x/y = 11/16
Solving for x and y, we get:
x = 11y/16
We substitute this equation into the equation x + y = 54 and we get:
11y/16 + y = 54
Solving for y we get y = 24. Then we can use the proportion to find x:
x = 11 * 24 / 16 = 33.
So the two angles are 33° and 24°.
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y
Which of the following best represents R = A - B?
A)
B)
R
The closing side of the triangle serves as the best representation of the resultant vector for the resultant R=A+B, so selecting option C would be the appropriate response.
what is vector ?A vector in mathematics is a quantity that has both magnitude and direction but no specific location. Illustrations of such numbers include velocity and acceleration. The term "vector" refers to something that has both magnitude and direction. When viewed geometrically, a vector can be thought of as a directed reference line, the direction of which is indicated by an arrow and whose length corresponds to the vector's magnitude. A quantity or phenomenon known as a vector has two distinct qualities that are its size and direction. Also included in the term is a description of how these values are represented geometrically or mathematically. Vectors are present in electromagnetic fields, as well as in weight, velocity, force, and velocity.
given
As stated in the issue, we must determine the resultant vector by adding vectors A and B.
The triangle law of vector addition, which states that the total is supplied by the triangle's closing side, can be used to calculate the vector's outcome.
Option C is the proper response, thus.
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Your neighbor pays you $17 for every two hours of work how many hours do you work if you earn $42.50
Answer:
To find the number of hours you work, you need to set up an equation using the information given. Let's say that the number of hours you work is x. The equation would be:
17(x/2) = 42.50
The parentheses around x/2 indicate that the division should be done before the multiplication. So, x/2 = 2.5. Multiplying both sides of the equation by 2 gives us:
x = 5
So, you work 5 hours.
Step-by-step explanation:
So, we know that you earn $17 per 2 hours. We also know that we earned $42.50. Now, we just need to find the hours! We can do this by using a cross multiply.
$17 is directly correlated to 2 hours of work. Therefore, to find the hourly wage we should divide it by 2:
$8.5 = 1 hour
$42.50 = x hours?
Now cross multiply:
$8.5x = $42.50
x = 5 hours
Therefore, your answer is 5 hours!
Hope this helps!
How do you solve a 3 sided triangle?
It is impossible to solve a 3-sided triangle because triangles require 3 sides and 3 angles in order to be solved.
A triangle is a three-sided polygon, and the sum of its angles is always 180 degrees. In order to solve a triangle, you must know the length of all three sides and the measure of all three angles. If you know two sides and the angle between them (the angle opposite to one of the sides), you can use the Law of Sines to determine the other angle. If you know two angles and a side, you can use the Law of Cosines to determine the other two sides. With all three sides and angles known, you can use the Pythagorean Theorem to check your work. However, if you only have three sides, it is impossible to solve the triangle because you do not have enough information.
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Warren is tiling his bathroom. He wants to cut three different sizes of square tiles. Tile A must be square inches, tile B must be square inches, and tile C must be square inches. He needs to know the side length of each tile. Part A In the table, note your guesses and checks to find the length of tile A, which has an area of square inches. Estimate the value to five decimal places
The length of tile A is 2.36068 inches.
The quantity of square units required to completely fill a square is known as the area of a square.
The region that lies inside the confines of a flat item or a two-dimensional figure is generally referred to as the area.
The measurement is made using square units, with square meters serving as the standard unit (m^2).
The square unit serves as the unit of area.
Let us recall that the area of a square is given by;
A = side^2
This is because, all the sides of a square are equal hence
When A = 5 square inches
5 = side^2
side = √5
side = 2.36068 inches
Therefore, The length of tile A is 2.36068 inches.
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A rigid body exists in an n-dimensional space. How many coordinates are needed to specify the position and orientation of this body in this space
The number of coordinates needed to specify the position and orientation of a rigid body in an n-dimensional space is n.
A rigid body is an object that maintains its shape and size while in motion. In order to specify the position and orientation of a rigid body in an n-dimensional space, we need to know its coordinates in each of the n dimensions. The coordinates describe the location of the body in the space and the orientation of the body relative to the space's coordinate system. Thus, n coordinates are required to fully specify the position and orientation of a rigid body in an n-dimensional space.
However, it is important to note that if the rigid body is placed in a space that's non-Euclidean, other parameters could be necessary to fully describe the rigid body position and orientation.
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For the three-part question that follows, provide your answer to each question in the given workspace. Identify each part with a coordinating response. Be sure to clearly label each part of your response as Part A, Part B, and Part C.
Use the image below for Part A, Part B, and Part C.
Two triangles: GHI and JKL
Both have one similar angle but different side lengths
Part A: Sharon says the triangles above are similar using the AA Postulate. Is she correct?
Part B: Explain the answer from Part A.
Part C: If HI=7, find KL. Show your work.
Please Help!
Answer:
see explanation
Step-by-step explanation:
A
for the triangles to be similar by the AA postulate then 2 angles in both triangles must be congruent.
In this case we are only given 1 pair of congruent angles.
Thus not able to say similar by AA postulate, therefore Sharon is incorrect in her assumption.
B
the sides are of different lengths but the ratios of corresponding sides are equal, that is
[tex]\frac{JK}{GH}[/tex] = [tex]\frac{8}{4}[/tex] = 2 and [tex]\frac{JL}{GI}[/tex] = [tex]\frac{10}{5}[/tex] = 2
If the ratios of 2 pairs of corresponding sides of 2 triangles are equal and the included angles are congruent , then the triangles are similar, so
Δ GHI and Δ JKL are similar by the SAS postulate
C
the ratios of corresponding sides are equal , then
[tex]\frac{KL}{HI}[/tex] = [tex]\frac{JL}{GI}[/tex]
[tex]\frac{KL}{7}[/tex] = [tex]\frac{10}{5}[/tex] = 2 ( multiply both sides by 7 to clear the fraction )
KL = 14
9 people audition for a choir. The choir director must choose one soprano, one alto,
and one tenor.
In how many ways can the director fill these positions?
The numbers of ways that the director fill these positions is 504 ways.
What is the position about?There are a few different ways to approach this problem, but one possible method is to use the combination formula.
To fill the position of soprano, the director has 9 choices. After selecting one soprano, the director is left with 8 singers to choose from for the alto position. Then, the director has 7 singers to choose from for the tenor position.
So, the total number of ways the director can fill these positions is:
9 × 8 × 7
= 504 ways.
Alternatively, One can also use the permutation formula to solve the question. The formula of permutation is (n!/(r!(n-r)!)), where n is the total number of items to choose from, and r is the number of items to be chosen.
For this case n is 9, and r is 3, so the permutation would be:
(9!/(3!(9-3)!)
= (987)/(321)
= 504
Therefore, In both cases, the answer is 504 ways.
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Can the inverse of a function be the same function?
Answer:
Yes
Step-by-step explanation:
y = x has the inverse x = y
How do you solve a linear equation step by step?
Linear equations are mathematical equations that involve one or more variables. These equations can be solved by following a few simple steps:
How to solve linear equationsIdentify the constant and the coefficient of the variable.Use the coefficient to divide both sides of the equation by the same number.Subtract the constant from both sides of the equation.Divide both sides of the equation by the coefficient of the variable.Check your solution by substituting it back into the original equation.Linear equations can be used to solve a wide range of mathematical problems and are an essential part of any mathematics curriculum.
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A building with a height of 48 meter casts a shadow that is 30 meter long. A person standing next to the building casts shadow that is 0.8 meter long. How tall is the person
The height of the person is approximately 0.426 meters or 42.6 cm.
We can use the concept of similar triangles to solve this problem. The height of the building and the length of its shadow form a right triangle, with the height as the opposite side and the shadow as the hypotenuse. The person and their shadow also form a similar right triangle, with the height of the person as the opposite side and the shadow as the hypotenuse.
We can set up the following proportion:
(Height of building)/(length of shadow) = (height of person)/(length of shadow)
48/30 = h/0.8
Solving for h, we get:
h = (48/30) * 0.8 = 0.533 * 0.8 = 0.426
Therefore, the height of the person is approximately 0.426 meters or 42.6 cm.
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Which expression is not equivalent to logp 36? 1. 6 log 2 2. log 9+ Iog, 4 O 3 2 logb 6 4. log 72 logp 2
log 30+ log 6 is not equivalent to log 36
What is equivalent expressions?Equivalent expressions are expressions that perform the same function not with standing their appearance. If two algebraic expressions are equivalent, they have the same value when the same value(s) for the variable are used (s).
The original phrase is as follows
Log 36
Following that, we write the alternatives (a) log 2+ log 18
Apply the logarithmic rule log 2+ log 18 log (218)
Log 2 plus log 18 log (36)
Remove the bracket
Log 2 plus log 18 log36
(a) Log 2+ Log 18 is the same as log 36.
(b) Log 30 plus log 6
Use the logarithms’ law. 30 logs or more 6 logs (30 x 6)
Log 30 plus log 6 Equals log (180)
Remove the bracket
Log 30-plus log 6-log180
Hence,
(c) Log 30+log 6 does not equal log 36.
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Aiden is a spice trader.
The price he charges for an order of cumin (in dollars) as a function of the size of the order (in kilograms) is graphed.
What is the approximate average rate at which the total cost increases, as the order size increases from 100 to 600 kilograms?
Approximately $2.40 per kilogramme is the average rate that the total cost rises as the order size does, from 100 to 600 kg.
How much is the average rate?The average rate of change is the average change rate of one thing in respect to another. Simply explained, a computation that divides the amount of change in one thing by the comparable amount of change in another is known as an average rate of change function.
How are average rates calculated?Divide the y-value change by the x-value change to determine the average rate of change. When analyzing changes in observable parameters like average speed or average temperature, finding that average rate of change is especially helpful.
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Sabreena is playing a card game, and the odds of her winning after drawing the next card are 5:7. What is the probability of her not winning? Enter your answer as a decimal rounded to the nearest thousandth, like this: 0.423
The probability that Sabreena is going to lose is given as 0.5833
How to solve for the probabilityProbability is a measure of the likelihood of a specific event or outcome occurring. It is typically expressed as a decimal number between 0 and 1, or as a percentage between 0% and 100%.
A probability of 0 indicates that an event is impossible and will never occur, while a probability of 1 indicates that an event is certain and will always occur.
If the odds that Sabreena has of winning in this game is 5: 7, this is written as 5 / 7
In this case, the odds of Sabreena winning are 5:7,
so the probability of winning is 5/(5+7)
the probability of not winning is 1 - 5/12
The probability of not winning is approximately 0.5833
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what is 0.133 with a bar over 33
Answer:
2/15
Step-by-step explanation:
make x= 0.13
multiply it by 10^1 .
10x = − 1.33
subtract x from 10x
10x−x =
1.33 - 0.13
9x = 1.2
divide both sides by 9 to get x as a fraction
x = 1.2/9 =
12/90 =
2/15
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A bar over a digit in a decimal number, such as 0.133 with a bar over the 3, indicates that the number is a repeating decimal. In this case, 0.133 is a repeating decimal and the digit 3 is repeating infinitely.
What is repeating decimal?A repeating decimal is a decimal number whose digits are periodic and repeat infinitely. They are also called recurring decimals. For example, 0.333... is a repeating decimal, with the digit 3 repeating indefinitely. The same goes for 0.666... and so on. They can also be represented in the form of a fraction, where the numerator is the repeating decimal and the denominator is a power of 10 with a 1 followed by as many zeroes as the number of digits in the repeating block. Repeating decimals can also be represented with a bar over the repeating block of digits, for example 0.133 with a bar over the 3, indicating that the digit 3 is repeating infinitely. It's also written as 0.133(3) where (3) means the repeating decimal. Another way is to use a dot above the repeating digit.To learn more about decimal refer:
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HELPPPPP
Jada rode her bike 5 days in a row. She rode the same distance each day. This model represents the situation. Each column represents one mile and the shaded parts of each column represent the fraction of a mile that Jada rode her bike each day.
Which expression can be used to determine the total distance in miles Jada rode her bike over the 5 days?
(A: 5 × 4/5
(B: 5 + 4/5
(C: 5 × 1/4
(D: 5 + 1/4
PLSSSS HELP
Answer: 5 x 4/5
Step-by-step explanation:
In each column there is 4/5 shaded . There are 5 columns .
So , 5 x 4/5
Answer:
A) 5 × 4/5
Step-by-step explanation:
The shaded part each column in the model represents what fraction of a mile that Jada rode her bike each day.
There are 5 columns.
Therefore, to determine the total distance in miles that Jada rode over the days, we can multiply the distance that she rode each day by the number of days that she rode that distance (5 days):
A) [tex]5 \textrm{ days} \times \dfrac{4}{5} \textrm{ mile each day}[/tex]
Hassan plans to repaint some classroom bookcases. He has 1010 gallons of paint. All of the bookcases are the same size and each requires \tfrac{7}{8}
8
7
gallon of paint. What is the maximum number of complete bookcases he can paint?
Hassan can paint maximum 11 bookcases with 10 gallons of paint.
What are Fractions?A fraction represents a numerical value, which defines the parts of a whole.
Generally, the fraction can be a portion of any quantity out of the whole thing and the whole can be any specific things or value.
The basics of fractions explain the top and bottom numbers of a fraction.
Suppose a number has to be divided into four parts, then it is represented as x/4. So the fraction here, x/4, defines 1/4th of the number x. Hence, 1/4 is the fraction here.
Given,
Hassan has 10 gallons of paints.
1 bookcase requires 7/8 gallons of paint
Let Hassan paints x bookcases.
According to the question
7/8x = 10
7x = 80
x = 80/7
x = 11.43
But no. of bookcases should be in whole.
Number of bookcases painted with 10 gallons of paint = 11.
Hence, Maximum 11 bookcases can be painted by Hassan with 10 gallons of paint
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How do you find the value of 3 7?
6x + y = 20
x - 4y = -5
The value of [tex]x[/tex] is [tex]3[/tex] and [tex]y[/tex] is [tex]2[/tex]
The given equations are
[tex]6x+y=20\\x-4y=-5[/tex]
We need to find the value of [tex]x[/tex] and [tex]y[/tex]
First let's take equation [tex]2[/tex] and we can simplify it
[tex]x-4y=-5\\x=-5+4y[/tex]
Now, let's substitute the value of [tex]x[/tex] in equation [tex]1[/tex], that is
[tex]6x+y=20\\6(-5+4y)+y=20[/tex]
We can open the bracket and simplify
[tex]=-30+24y+y=20\\=-30+25y=20\\=25y=20+30\\=25y=50\\=y=\frac{50}{25} \\y=2[/tex]
Now we can substitute the value of [tex]y[/tex] in equation [tex]x-4y=-5[/tex]
[tex]x-4(2)=-5\\x-8=-5\\x=-5+8\\x=3[/tex]
The complete question is find the value of [tex]x[/tex] and [tex]y[/tex] from the given equations.
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How do you solve the equation 5t 3 3t 5?
By solving the equation 5t - 3 = 3t - 5 we get the equation is true when t = -1. So the solution is t = -1.
To solve the equation 5t - 3 = 3t - 5, you can use algebraic manipulation to isolate the variable on one side of the equation.
The first step is to get all the terms with the variable on one side of the equation and all the constant terms on the other side.
5t - 3 = 3t - 5
adding 3 to both sides:
5t = 3t - 2
subtracting 3t from both sides:
2t = -2
then divide both sides by 2:
t = -1
So the solution is t = -1, meaning that when t = -1, the equation is true.
--The question is not clear, answering to the question below--
"How do you solve the equation 5t - 3 = 3t - 5?"
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Ron killed 8 times as many spiders as Harry,H (Write an expression to represent the following situations)
Answer:
Ron killed 8 times as many spiders as Harry: R = 8 * H
Step-by-step explanation:
This expression represents the situation in which Ron killed 8 times as many spiders as Harry.
How do you solve this?.....Katherine has a loyalty card good for a 3% discount at her local grocery store. What number should she multiply the prices on the tags by to find the price she would have to pay, before tax, in one step?
Answer:
Step-by-step explanation:
To find the number Katherine should multiply the prices on the tags by to find the price she would have to pay before tax, we need to calculate the inverse of the discount percentage, which is 1 - (discount percentage / 100).
Given that the discount is 3% and we want to find the price after the discount, that means:
1 - (3/100) = 1 - 0.03 = 0.97
So, the number Katherine should multiply the prices on the tags by to find the price she would have to pay, before tax, in one step, is 0.97. Multiplying the price of an item by 0.97 is equivalent to subtracting 3% from the original price, which is the same as finding the final price after the discount.
For example, if an item is priced at $10, Katherine can multiply this by 0.97 to find that the final price she would have to pay before tax is $9.7
10*0.97 = 9.7
The temperature was recorded at midnight for 5 consecutive days. The
recorded temperatures are: -2°, 4°, 5°, -6°, and 9°. What is the average
temperature?
I need help this makes no sense
Answer:
∠ SQT = 63°
Step-by-step explanation:
since QS bisects ∠ RQT , then
∠ RQS = ∠ SQT = 7x + 7
then
∠ RQS + ∠ SQT = ∠ RQT , that is
7x + 7 + 7x + 7 = 9x + 54
14x + 14 = 9x + 54 ( subtract 9x from both sides )
5x + 14 = 54 ( subtract 14 from both sides )
5x = 40 ( divide both sides by 5 )
x = 8
Then
∠ SQT = 7x + 7 = 7(8) + 7 = 56 + 7 = 63°
What is undecidable problem give two examples?
An undecidable problem is one for which it has been demonstrated that it is impossible to develop an algorithm that always leads to the correct yes-or-no answer. Two examples are:
The Whitehead problem The halting problemWhat the halting problem?The halting problem is the problem of determining whether a computer programme will finish running or continue to run indefinitely based on a description of the programme and an input. Alan Turing demonstrated in 1936 that a general algorithm for solving the halting problem for all possible program-input pairs does not exist.
A "pathological" programme g, when called with some input, can pass its own source and input to f and then specifically do the opposite of what f predicts g will do. This case cannot be handled by any f. A key part of the proof is a mathematical definition of a computer and programme, known as a Turing machine; the halting problem is intractable over Turing machines.
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Consider the differential equation dy/dx=xy^4.
(b) Find ⅆ2yⅆx2 in terms of x and y. Determine the concavity of all solution curves for the given differential equation in Quadrant II. Give a reason for your answer.
(c) Find the particular solution y=f(x) to the given differential equation with initial condition f(4)=−1.
The differential equation [tex]\frac{dy}{dx} = xy^{4}[/tex] is considered and the solutions for the following questions are found.
What is a differential equation?
A differential equation is an equation that links the derivatives of one or more unknown functions. Differential equations contain derivatives, which might be either partial derivatives or ordinary derivatives.
The differential equation describes a relationship between the variable that is constantly varying with respect to the change in another quantity, and the derivative represents a rate of change.
Given,
[tex]\frac{dy}{dx} = xy^{4}[/tex]
a) Now finding [tex]\frac{d^{2} y}{dx^{2} }[/tex] = [tex]y^{4} + 4xy^{3} \frac{dy}{dx}[/tex] = [tex]y^{4} + 4xy^3 (xy^4) = y^4 + 4x^2y^7[/tex]
In quadrant II, the values of x < 0 and the values of y >0.
So [tex]\frac{d^{2} y}{dx^{2} }[/tex] = [tex]y^4 + 4x^2y^7[/tex] > 0 for all values of x and y in quadrant II.
Therefore the concavity is up for of all solution curves of the given differential equation.
b)
[tex]\frac{dy}{dx} = xy^{4}\\\\\frac{dy}{y^4} =x dx\\\\\int {\frac{dy}{y^4} } \, = \int {x} \, dx \\\\\frac{y^{-4+1} }{-4+1} = \frac{x^{2} }{2} + c\\\\\frac{y^{-3} }{-3} = \frac{x^{2} }{2} +c[/tex]
Now it is given that initial condition is f(4) = -1
So substituting x =4 and y = -1 is the above equation to find c, we get
[tex]\frac{-1^{-3} }{-3} = \frac{4^2}{2} +c \\\\\frac{1}{3} = 8 + c\\\\c = \frac{1}{3} - 8 = \frac{-23}{8}[/tex]
Therefore the solution y = f(x) is to the given differential equation with initial condition f(4)=−1.
[tex]\frac{y^{-3} }{-3} = \frac{x^{2} }{2} -\frac{23}{8}\\ \\ \frac{1}{3y^3} = \frac{46 - 8x^2 }{16} = \frac{23 - 4x^2 }{8}\\\\y = \sqrt[3]{\frac{8}{3(23-4x^2)} }[/tex]
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The vertices of a convex pentagon are $(-1, -1), (-3, 4), (1, 7), (6, 5)$ and $(3, -1)$. What is the area of the pentagon
On solving the provided question, we can say that vertices of pentagon are (-1, -1), (-3, 4), (1, 7), (6, 5) and (3, -1); Area = 30 sq unit
what is a pentagon?A pentagon is a polygon with five sides that is used in geometry. The inner angles of a straightforward pentagon add up to 540°. Simple or self-intersecting pentagons are both possible. A pentagram is a self-intersecting regular pentagon. Five angles and all sides of a regular pentagon are equal. It is referred to as an irregular pentagon if the side lengths and angle measurements do not match. All outside angles, as far as we are aware, balance out inside angles. The polygon's exterior angle total is therefore equal to n(360°/n). There are five sides to a pentagon, therefore n must equal five. Consequently, 5(360°/5) = 360° is the sum of the pentagon's outer angles.
vertices of a convex pentagon are (-1, -1), (-3, 4), (1, 7), (6, 5) and (3, -1).
Area = 1/2 [ 24+3+15+0-3] - [3-30-3+9-0]
Area = 1/2| 60 |
Area of pentagon = 30 sq unit
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Determine whether the lines tangent to the curve at the x-intercepts of the curve are parallel. Show the analysis that leads to your conclusion.
For the curve [tex]x^{2} +xy+y^{2} = 27[/tex] , the line tangents to the curve at the x intercepts of the curve are parallel .
The Slope of tangent line to a function at any point is denoted by [tex]\frac{dy}{dx}[/tex] .
The Curve is given as ⇒[tex]x^{2} +xy+y^{2} = 27[/tex] ;
On differentiating both sides of the curve with respect to "x" ,
we get ;
the slope of the line as ⇒ [tex]Slope = \frac{dy}{dx} = -\frac{2x+y}{x+2y}[/tex] ;
Now for the x- intercepts , the value [tex]y = 0 ,[/tex]
On substituting [tex]y=0[/tex] in the curve [tex]x^{2} +xy+y^{2} = 27[/tex] ,
we get ;
[tex]x^{2} +x(0)+0^{2} =27[/tex]
On simplifying further ,
we get ;
[tex]x=\pm 3\sqrt{3}[/tex] ;
So , the two x intercepts of the curve are [tex](3\sqrt{3},0)[/tex] and [tex](-3\sqrt{3},0)[/tex] ;
to find the slopes , we substitute the values in the slope equation .
On substituting the values in the slope equation ,
we get ;
[tex]Slope_{(3\sqrt{3} ,0)} = -\frac{2(3\sqrt{3}) +9}{3\sqrt{3} +2(0)}[/tex] = [tex]-2[/tex] and
[tex]Slope_{(-3\sqrt{3} ,0)} = -\frac{2(-3\sqrt{3}) +9}{-3\sqrt{3} +2(0)}[/tex] = [tex]-2[/tex].
we see that the slope at both the intercepts is same ,
Therefore , the lines tangents of the curve at the x intercepts are parallel .
The given question is incomplete , the complete question is
Consider the curve defined by [tex]x^{2} +xy+y^{2} = 27[/tex] . Determine whether the lines tangent to the curve at the x - intercept of the curve are parallel .
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2. Write a factored form function, f(x) for a cubic function that has zeros (- 7, 0) and (3, 0)
The cubic equation in factor form f(x) = (x + 7) · (x - 3) · (x - r) contains zeros (- 7 , 0) and (3, 0), for r = 0: f(x) = (x + 7) · (x - 3) · (x - r).
How to derive a cubic equation in factor form
Cubic functions are polynomials with grade 3 and whose standard form is described by the following form:
y = a · x³ + b · x² + c · x + d
Where a, b, c, d are real coefficients. The factor form of a cubic function is described by the following definition:
y = a · (x - r₁) · (x - r₂) · (x - r₃)
Where:
a - Lead coefficient.r₁, r₂, r₃ - Real zeros.Replace all known variables in factor form of cubic equation: (a = 1, r₁ = - 7, r₂ = 3, r₃ = r)
f(x) = (x + 7) · (x - 3) · (x - r)
Variable r means that there is more than one answer concerning the expression, if we assume that r = 0, then the particular solution is f(x) = (x + 7) · (x - 3) · x.
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