The tangent line has the largest slope at x = 3√2 and x = -3√2 on the curve y = 1 + 60x³ − 2x⁵.
First, let's find the derivative of the given function y = 1 + 60x³ − 2x⁵ using the power rule for differentiation:
dy/dx = 0 + 3(60)x² - 5(2)x⁴
= 180x² - 10x⁴
To find the critical points, we set the derivative equal to zero and solve for x:
180x² - 10x⁴ = 0
Factoring out common terms, we get:
10x²(18 - x²) = 0
Setting each factor equal to zero, we have:
10x² = 0 or 18 - x² = 0
From the first equation, we find x = 0.
From the second equation, we have:
18 - x² = 0
x² = 18
Taking the square root, we get:
x = ±√18
= ±3√2
So the critical points are x = 0, x = 3√2, and x = -3√2.
Now we need to evaluate the slope at these critical points. We can do this by plugging each x-value into the derivative:
When x = 0:
dy/dx = 180(0)² - 10(0)⁴ = 0
When x = 3√2:
dy/dx = 180(3√2)² - 10(3√2)⁴ = 180(18) - 10(216) = 3240 - 2160 = 1080
When x = -3√2:
dy/dx = 180(-3√2)² - 10(-3√2)⁴ = 180(18) - 10(216) = 3240 - 2160 = 1080
The slope is 0 when x = 0 and 1080 when x = 3√2 or x = -3√2.
Therefore, the tangent line has the largest slope at x = 3√2 and x = -3√2 on the curve y = 1 + 60x³ − 2x⁵.
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The complete question is as follows:
At which points on the curve y = 1 + 60x³ − 2x5⁵ does the tangent line have the largest slope?
Quick algebra 1 question for 10 points!
Only answer if you know the answer, quick shout-out to tariqareesha2 and MrBrainly, tysm for the help!
Answer:
Situation 2 (B)
Step-by-step explanation:
I think it would be B since the power of the gear increases proportionally with the radius.
The system of equations shown is solved using the linear combination method
StartLayout 1st row 1st column 6 x minus 5 y = negative 8 right-arrow 2nd column 6 x minus 5 y = negative 8 right-arrow 6x minus 5 y = negative 8 2nd row 1st column negative 24 x + 20 y = 32 right-arrow one-fourth (negative 24 x + 20 y = 32) right-arrow negative 6 x + 5 y = 8 with Bar Underscript 3rd row 3rd column 0 = 0 EndLayout
What does 0 = 0 mean regarding the solution to the system?
The 0 = 0 in the solution to the system implies that the system of equations involving the equations 6x - 5y = -8 and 6x - 5y = -8 have infinitely many solutions
What are linear equations?
Linear equations are equations that have constant average rates of change, slope or gradient
How to determine the linear combination to the system?A system of linear equations is a collection of at least two linear equations.
In this case, the system of equations is given as
6x - 5y = -8
6x - 5y = -8
Subtract the equation 6x - 5y = -8 from the equation 6x - 5y = -8
6x - 5y = -8
- 6x - 5y = -8
----------------------
0 = 0
This in other words means that the difference between both equations is 0
When the solution to a system of equation is 0 = 0, it means that the system of equations have infinitely many solutions
Hence, the 0 = 0 in the solution to the system implies that the system of equations involving the equations 6x - 5y = -8 and 6x - 5y = -8 have infinitely many solutions
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Answer: The answer is D for those who don't like to read extra long expert answers (no offense to experts, just shorten answers to the direct answer pls)
Step-by-step explanation: Edge 2022
Study island question below its 8th grade algebra
Answer:
376.8 [tex]units^{2}[/tex]
Step-by-step explanation:
There are a lot of moving parts. We need to find the area of the bottom of the circle the center and the top circle.
The top and the bottom will be the same.
The area of a circle:
a = [tex]\pi[/tex][tex]r^{2}[/tex]
a = 3.14([tex]5^{2}[/tex])
a = 3.14(25)
a =78.5
The center: Think of this as a label on soup can. If you tear the label off it would be a rectangle. To find the area we would multiply the height of the can by the circumference of the circle. They tell us the height is 7 and we can calculate the circumference.
C = [tex]\pi[/tex]d The d is the diameter it is 2 of the radiuses.
C = [tex]\pi[/tex]10
C = 3.14(10)
C = 31.4
Now we multiply 31.4 and 7 together to find the area of the center of the cylinder.
(31.4)(7) = 219.8
The total area:
top 78.5
center 219.8
bottom 78.5
Total 376.8
A moving company’s moving rates can be represented by the function f(x) = 3x 400, where x is the number of miles for the move. another moving company’s rates can be represented by the function g(x) = 5x 200. which function represents the difference between the moving companies' rates, h(x) = f(x) – g(x)? h(x) = 2x 200 h(x) = –2x 200 h(x) = 5x – 200 h(x) = 5x 600
The function that represents the difference between the moving companies' rates is:
h(x) = -2x + 200.
How to find the difference of two functions?The difference of two functions is found subtracting the like terms from the functions.
In this problem, the functions for the rates are given as follows:
f(x) = 3x + 400.g(x) = 5x + 200Hence the difference is given by:
h(x) = 3x + 400 - (5x + 200) = 3x - 5x + 400 - 200 = -2x + 200.
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I need help finding the quotient of this equation in simplified form.
by breaking it down u can write it as
15p^-4/_20p^-12 * q^-6/q^-3
15/-20=-3/4
using law of indices we have
-3/4p^-4+12 * q^-6+3
-3/4p^8q^-3
the answer is the first own
Mark bought gift tokens for her daughter, Sally, to use at the state fair. Sally used 13/20 of the tokens on rides. If she used 65 tokens, how many tokens had Mark bought?
Answer:
100
Step-by-step explanation:
Formulate a plan based on the conditions stated: 65/(13/20)
A fraction is divided by multiplying its reciprocal: 65x(20/13)
Remove the connecting element: 5x20
Determine the quotient or product: 100
100 tokens are bought by mark for his daughter.
What is ratio?The ratio can be defined as the number that can be used to represent one quantity as a percentage of another. Only when the two numbers in a ratio have the same unit can they be compared. Ratios are used to compare two objects.
Given, Mark bought gift tokens for her daughter, Sally, to use at the state fair. Sally used 13/20 of the tokens on rides.
let "x" be the total token she have
Since, total token she used is 65
Thus,
13/20 * x = 65
x = 100
Therefore, Mark bought for his, daughter 100 Tokens.
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Geometry: complete this proof, ASAP!!!
Answer:
By definition, angles A and 1 are corresponding angles and angles B and 1 are consecutive angles. By the corresponding angles postulate, angles A and 1 are congruent, and by the consecutive angles theorem, angles B and 1 are supplementary. By the definition of supplementary angles, measures of angle B and 1 add up to 180 degrees (m<B + m<1 = 180). By definition of congruent angles, angles A and 1 have same measurement (m<A = m<1). By substitution property of equality, measures of angles A and B add up to 180 degrees (m<A + m<B = 180). By definition of supplementary angles, angles A and B are supplementary.
ÐA in cos A = 0.7715 in degrees? (not sure if the way I worded that makes sense but yea)
The value of A such that cos A = 0.7715 is 39.5 degrees
How to solve for A in the equation cos A = 0.7715?The equation is given as:
cos A = 0.7715
Where the angle is in degrees
There are several ways to solve this equation, some of which include:
Using a scientific calculatorUsing a graph or a graphing toolUsing the cosine tableIn this case, we make use of the first option; using a scientific calculator
Recall that the equation is represented as:
cos A = 0.7715
Next, we take the arccos (or anti cosine) of the above equation.
This is represented as
arccos(cos A) = arccos(0.7715)
Evaluate the expression on the left-hand side
A = arccos(0.7715)
Evaluate the expression on the left-hand side using the scientific calculator
A = 39.5112207183
Approximate the above expression
A = 39.5
Hence, the value of A such that cos A = 0.7715 is 39.5 degrees
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‼️‼️‼️‼️HELP‼️‼️‼️‼️‼️
A certain item is available at 7 stores. Three stores sell it for $20, two stores sell it for $15, one store sells it for $13, and one sells it for $16. What is the average (arithmetic mean) of the median price and the mode price?
The average median and mode price is $18. Option C is correct.
Given that certain item is available at 7 stores, three stores sell them for $20, two stores sell them for $15, one store sells them for $13, and one sells them for $16.
The mean is the average of the given numbers and is calculated by dividing the sum of the given numbers by the total number of numbers.
Firstly, we will find the median of the given items by arranging the given numbers in ascending order, we get
13,15,15,16,20,20,20
To find the median use the formula (n+1)/2, where n is the number of values in your dataset.
(7+1)/2=8/2=4
In the ascending order numbers 4th term is 16.
So, median is 16
Mode is the highest repeating term in the set or numbers.
So, here mode is 20
Now, we will calculate the average of median and mode, we get
Average=(median +mode)/2
Average=(16+20)/2
Average=18
Hence, the average (arithmetic mean) of the median price and the mode price where certain item is available at 7 stores, three stores sell it for $20, two stores sell it for $15, one store sells it for $13, and one sells it for $16 is $18.
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Use the laplace transform to solve the given system of differential equations. dx dt + 3x + dy dt = 1 dx dt − x + dy dt − y = et x(0) = 0, y(0) = 0
Let [tex]X(s)[/tex] and [tex]Y(s)[/tex] denote the Laplace transforms of [tex]x(t)[/tex] and [tex]y(t)[/tex].
Taking the Laplace transform of both sides of both equations, we have
[tex]\dfrac{dx}{dt} + 3x + \dfrac{dy}{dt} = 1 \implies \left(sX(s) - x(0)\right) + 3X(s) + \left(sY(s) - y(0)\right) = \dfrac1s \\\\ \implies (s+3) X(s) + s Y(s) = \dfrac1s[/tex]
[tex]\dfrac{dx}{dt} - x + \dfrac{dy}{dt} = e^t \implies \left(sX(s) - x(0)\right) - X(s) + \left(sY(s) - y(0)\right) = \dfrac1{s-1} \\\\ \implies (s-1) X(s) + s Y(s) = \dfrac1{s-1}[/tex]
Eliminating [tex]Y(s)[/tex], we get
[tex]\left((s+3) X(s) + s Y(s)\right) - \left((s-1) X(s) + s Y(s)\right) = \dfrac1s - \dfrac1{s-1} \\\\ \implies X(s) = \dfrac14 \left(\dfrac1s - \dfrac1{s-1}\right)[/tex]
Take the inverse transform of both sides to solve for [tex]x(t)[/tex].
[tex]\boxed{x(t) = \dfrac14 (1 - e^t)}[/tex]
Solve for [tex]Y(s)[/tex].
[tex](s - 1) X(s) + s Y(s) = \dfrac1{s-1} \implies -\dfrac1{4s} + s Y(s) = \dfrac1{s-1} \\\\ \implies s Y(s) = \dfrac1{s-1} + \dfrac1{4s} \\\\ \implies Y(s) = \dfrac1{s(s-1)} + \dfrac1{4s^2} \\\\ \implies Y(s) = \dfrac1{s-1} - \dfrac1s + \dfrac1{4s^2}[/tex]
Taking the inverse transform of both sides, we get
[tex]\boxed{y(t) = e^t - 1 + \dfrac14 t}[/tex]
Which of the following functions is graphed below?
The definition of the given piecewise function is:
[tex]y = x^3 - 3, x \leq 2[/tex][tex]y = x^2 + 6, x > 2[/tex]What is a piecewise function?A piecewise function is a function that has different definitions, depending on the input.
In this problem, for x until 2, the function is the cube function shifted down 3 units, hence the definition is:
[tex]y = x^3 - 3, x \leq 2[/tex]
For x greater than 2, the function is the square function shifted up 6 units, hence the definition is:
[tex]y = x^2 + 6, x > 2[/tex]
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Select the correct answer. what is the completely factored form of this expression? 3x2 − 17x − 28 a. (3x 4)(x 7) b. (3x 4)(x − 7) c. 3x2 − 17x − 28 d. (3x2 4)(x − 7)
Answer:
A. (3x + 4)(x-7)
Step-by-step explanation:
3x times x = 3x^2
3x times -7 = -21x
4 times x = 4x
4 times -7 = -28
-21x + 4x = -17x
3x^2 - 17x - 28
For s=5 and t=8, 3st²-s²:
3st²-s² =
Step-by-step explanation:
Given,
s=5
t=8
solution,
= 3st² - s²
= 3*5*8² - 5²
= 3*5*64 - 25
= 960 - 25
= 935
prove:- sin^2A-cos^2B=sin^2B-cos^2A
Step-by-step explanation:
1: change 'sin²A' and 'cos²B' into (1-cos²A)and (1-sin²A) respectively.
2:open the brackets and put signs accordingly.
3: cancel (-1) and (+1).
4: remaining is your answer.
How many solutions can be found for the equation 4z 2(z − 4) = 3z 11? none one two infinitely many
There is only one solution for the equation 4z + 2(z -4) = 3z + 11 because the exponent for the power of z is 1.
What is an equation?In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign =.
What is the Solution?A solution is any value of a variable that makes the specified equation true.
According to the given information:
4z + 2(z-4)= 3z+11
Solve the equation,
4z+2z-8=3z+11
6z-3z=11+8
3z =19
z=
Hence,
Number of solution that can be found for the equation 4z + 2(z-4)= 3z+11 is option(2) one
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Complete the equation of the line through
(-10, 3) and (-8, -8).
The answer is y = -5.5x - 52.
First, let's find the slope of the line.
m = Δy/Δx
m = -8 - 3 / -8 - (-10)m = -11 / -8 + 10m = -11/2m = -5.5Now, substitute the slope and one of the given points in the point slope equation.
y - y₁ = m (x - x₁)
y - 3 = -5.5 (x - (-10))y - 3 = -5.5 (x + 10)y - 3 = -5.5x - 55y = -5.5x - 52Answer: y=-5,5x-52.
Step-by-step explanation:
Equation of a straight line
[tex]\displaystyle\\\boxed {\frac{x-x_1}{x_2-x_1}=\frac{y-y_1}{y_2-y_1} }[/tex]
[tex](-10;3)\ \ \ \ \ (-8;-8).\\\displaystyle\frac{x-(-10)}{-8-(-10)} =\frac{y-3}{-8-3}\\ \frac{x+10}{-8+10}=\frac{y-3}{-11} \\ \frac{x+10}{2}=\frac{y-3}{-11} \\-11*(x+10)=2*(y-3)\\-11x-110=2y-6\\2y=-11x-104\ |:2\\y=-5,5x-52.[/tex]
Using synthetic division what is (3x² + 7x - 18) = (x - 3)
Answer:
3x + 16 + 30/x - 13
Step-by-step explanation:
x^-19x-39=0 complete the square
The resulting equation on completing the square is (x+19/2)^2 = 39 + (19/2)^2
Completing the square method for a quadQuadratic equation are equations that has a leading degree of 2. Most quadratic equations have 3 terms.
Given the quadratic equation x^2-19x-39 = 0
Add 39 to both sides to have:
x^2 - 19x - 39 + 39 = 0 + 39
x^2 -19x = 39
Complete the square of the resulting
x^2-19x+(19/2)^2 = 39 + (19/2)^2
(x+19/2)^2 = 39 + (19/2)^2
x + 19/2 = ±√39 + (19/2)^2
x = ±√39 + (19/2)^2 - 19/2
Hence the resulting equation on completing the square is (x+19/2)^2 = 39 + (19/2)^2
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Describe the design and original dimensions of the photo you chose for Francesca to give as a gift.
Step-by-step explanation:
Transcribed image text: Choosing a Photo Francesca wants to give one of her photographs to a friend as a gift. Help her choose the photograph and determine the style and size of the mat she can afford 1. Describe the design and original dimensions of the photo you chose for Francesca to give as a gift. 2 points: 1 point for describing the photo and 1 point for including the dimensions) Photo description: The photo Shous Francasca and has Friend huagina. Dimensions (including units): The still Photo is a Xg meaning its width is indas and Langth is gind Finding the Total Area of the Photo and Mat 2. Francesca wants to glue her photo on top of a mat that will increase the length of each side. The mat border has a width of x, so the length of each side would increase by 2x inches. Represent the length and width of the mat with an expression. (2 points) Length: Width: 3. The area of the mat is the product of the length and width. Write an expression for area as the product of two binomials. Do not multiply.(1 point)
Bank a charges a $35 monthly fee for a checking account debit card with the limited train fashion bank b charges a $15 monthly fee for a checking account and debit card plus $0.70 for each transaction 25 transaction is giving a month
Total charges from bank A exists $15.
Total charges on Bank B spending exists $45.
How to estimate the Total charges on bank B?Given: Bank A charges a $15 monthly fee for a checking account and debit card, with unlimited transactions.
Shade 25 transactions.
Total charges from bank A = $15 monthly
Bank B charged a $35 monthly fee for a checking account and debit card, plus $ 0.70 for each transaction.
She made 25 transactions.
Total charges on bank B = $35 + (0.40)25
Total charges on bank B = $35+10
Total charges on bank B = $45
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Please answer this question ASAP!
The traffic lights at three different road crossings change after every 48, 72 and 108 seconds respectively. If they change simultaneously at 8 am, at what time will they change to again?
Answer:
8:07:12 am
Step-by-step explanation:
The lights will change simultaneously after any multiple of the least common multiple of their cycle times.
LCMThe least common multiple of a set of numbers is the product of the highest powers of their prime factors.
48 = 2⁴×3
72 = 2³×3²
108 = 2²×3³
The highest power of the factor 2 is 4. The highest power of the factor 3 is 3, so the LCM of the numbers 48, 72, and 108 is ...
LCM = 2⁴×3³ = 16×27 = 432
The lights will change simultaneously after each period of 7 minutes 12 seconds. They will change simultaneously again at 8:07:12 am.
Helppppppp show the steps to your answer
The numeric value of the expression -a² - 2bc - |c| for a = -3, b = -5 and c = 2 is of 9.
How to find the numeric value of an expression?The numeric value of an expression is found replacing each letter by it's attributed value.
In this problem, the expression is:
-a² - 2bc - |c|
The attributed values are:
a = -3, b = -5 and c = 2
Hence the numeric value will be given by:
-a² - 2bc - |c| = -(-3)² - 2(-5)(2) - |2| = -(9) + 20 - 2 = -9 + 18 = 9.
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Five pounds is equal to 2.268 kilograms. How many kilograms is 2 pounds?
Mass is the amount of matter that a body possesses and its unit in the international system of units is the kilogram.
How many kilograms is 2 pounds?Knowing that 1 pound is 2.205 kilograms.Therefore ⇒ 2 lb * 1 kg / 2.205 lb = 0.907 kgAnswer: In 2 pounds there are 0.907 kilograms.
Suppose you invested $1000 every 3 months over a 15 year period. If money earns an annual rate of 6.5% compounded quarterly, how much would be available at the end of the time period. How much is the interest earned? Show all your calculations
Using the future value formula, it is found that $100,336.67 will be available at the end of the time period, of which $40,336.67 was of interest earned.
What is the future value formula?The future value formula is given by:
[tex]F = P\frac{(1 + i)^n - 1}{i}[/tex]
In which:
P is the periodic payment.i is the equivalent yearly interest rate.n is the number of periods.For this problem, considering the situation described, especially the quarterly compoundings, the parameters are given as follows:
P = 1000, i = 0.065/4 = 0.01625, n = 15 x 4 = 60.
Hence the amount available is given by:
[tex]F = P\frac{(1 + i)^n - 1}{i}[/tex]
[tex]F = 1000\frac{(1 + 0.01625)^{60} - 1}{0.01625}[/tex]
F = $100,336.67
The amount deposited was:
15 x 4 x $1000 = $60,000.
Hence the amount earned in interest was:
$100,336.67 - $60,000 = $40,336.67
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The following figure shows the entire
graph of a relationship.
Does the graph represent a function?
[tex] {\qquad\qquad\huge\underline{{\sf Answer}}} [/tex]
The given graph doesn't represent a function, as it has two values of y for a single value of x ( that is : for x = 3 it has two solutions, -3 and -6 )
therefore it can't be considered as a function ~
Cube a and cube b are similar solids. the volume of cube a is 27 cubic inches, and the volume of cube b is 125 cubic inches. how many times larger is the base area of cube b than the base area of cube a?
The base area of cube B is 25/9 times larger than the base area of cube A.
What is area cubic?
Surface area of cube is the sum of areas of all the faces of cube, that covers it. The formula for surface area is equal to six times of square of length of the sides of cube. It is represented by 6a2, where a is the side length of cube. It is basically the total surface area.We know that volume of cube with each side of units is equal to .
First of all, we will find the each side of cube A and B as:
[tex]A^{3} = 27[/tex]
[tex]\sqrt[3]{A^{3} } = \sqrt[3]{27}[/tex]
A = 3
[tex]B^{3} = 125[/tex]
[tex]\sqrt[3]{B^{3} } = \sqrt[3]{125}[/tex]
B = 5
Now, we will find base area of both cubes as:
[tex]\frac{Base area of B}{Base area of A} = \frac{B^{2} }{A^{2} }[/tex]
[tex]\frac{Base area of B}{Base area of A} = \frac{5^{2} }{3^{2} }[/tex]
[tex]\frac{Base area of B}{Base area of A} = \frac{25}{9}[/tex]
Therefore, the base area of cube B is 25/9 times larger than the base area of cube A.
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which parent function is represented by the table?
Answer:
x^2
Step-by-step explanation:
So there's very two obvious ones we can elimination immediately, that's f(x) = x, and f(x) = |x|. The reason for this is because we can just look at the first ordered pair of (-2, 4). If it was f(x) = x, then the ordered pair would be (-2, -2), but it isn't. Very similar reasoning for f(x) = |x|, except for this function each ordered pair should have the same magnitude or absolute value. So the f(-2) should output 2, since it's the absolute value, but of course this isn't the case since it's 4.
So let's look at 2^x
[tex]2^{-2} = \frac{1}{2^2} = \frac{1}{4}[/tex]
Since the y value is 4 and not 1/4, this is not the parent function
It should be the second option (x^2), and we can double check
(-2)^2 = 4
(-1)^2 = 1
(0)^2 = 0
(1)^2 = 1
(2)^2 = 4
It outputs the same y-values as the table so this is the parent function
Mike repairs televisions. his revenue, in dollars, can be modeled by the equation y = 25 30x, where x is the number of hours spent repairing televisions. his overhead cost, in dollars, can be modeled by the equation y=5x2−10, where x is the number of hours spent repairing televisions. after how many hours does he break even?
The number of hours spent repairing televisions. after how many hours does he break even is 7 hours
System of equationsThese are equations that contains several equations with unknowns. Given the function that modelled the revenue generated from television repair as;
y = 25 + 30x
where:
x is the number of hours spent repairing televisions.
If his overhead cost, in dollars, is modelled by the equation y=5x^2−10, the company breaks even when;
25 + 30x = 5x^2 -10
Divide through by 5
5 + 6x = x^2 - 2
Equate to zero to have:
x^2 - 6x -2 - 5 = 0
x^2 - 6x - 7 = 0
x^2 - 7x + x - 7 = 0
x(x-7)+1(x-7) =0
x = -1 and 7
Hence the number of hours spent repairing televisions. after how many hours does he break even is 7 hours
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What is the area of the triangle shown below?
Answer: the area of the triangle is 5.
Step-by-step explanation:
[tex]A(0;0) \ \ \ \ B(1;3) \ \ \ \ C(4;2) \ \ \ \ S_{ABC}=?\\Use \ the\ formula:\\\displaystyle\\\boxed{S=\frac{1}{2}*|[(x_A-x_C)*(y_B-y_C)-(x_B-x_C)*(y_A-y_c)] |}\\x_A=0\ \ \ \ x_B=1\ \ \ \ x_C=4\ \ \ \ y_A=0\ \ \ \ \ y_B=3\ \ \ \ \ y_C=2.\\S=\frac{1}{2}*|[(0-4)*(3-2)-(1-4)*(0-2)]|\\S=\frac{1}{2}*| [(-4)*1-(-3)*(-2)]|=\\ S=\frac{1}{2}*| (-4-6)|\\S=\frac{1}{2}*|(-10)|\\S= \frac{1}{2} *10\\S=5.[/tex]
True or False: When x-coordinates are selected to create a table of values for a function, negative values of x may not be used.
[tex] \sf \blue{True✅}[/tex]
Step-by-step explanation:Yes, it's true sometimes we don't use negative value of 'x'.
So, we conclude it's True.
Hope it help you