Final answer:
To find the critical numbers, differentiate the function using the product rule, set the derivative equal to zero, and solve for x. Critical numbers are where the derivative is zero or undefined, provided they are within the domain of the function.
Explanation:
To find the critical numbers of the function f(x) = x4/5(x − 6)2, you need to locate the values of x where the first derivative of the function is either zero or undefined. The first derivative can be calculated using the product rule and the power rule.
First, let's find the derivative:
f'(x) = d/dx [x4/5] * (x - 6)2 + x4/5 * d/dx [(x - 6)2]
After simplifying, you will get a derivative function where you can then set it equal to zero to find the critical points. The points where the derivative is zero are potential local maxima, minima, or points of inflection. Additionally, points where the derivative is undefined can also be critical points, if they are within the domain of the function.
Once you calculate and simplify the derivative, set it equal to zero and solve for x. You might find that you get explicit values of x, which are the critical numbers of the function. If the function's derivative does not exist at some point, that will also be a critical number.
Remember, critical numbers are only relevant if they are within the domain of the original function.
Nancy randomly scattered one handful of seeds in her garden. The garden is circular and has a radius of 5 feet. Somewhere in the garden is a 2 feet by 2 feet square patch that has especially good soil for the seeds. If it is equally likely for a seed to land anywhere in the garden, what is the probability that a seed will land in the 2 feet by 2 feet square?
TWO QUESTION MUST ANSWER BOTH TO RECEIVE POINTS (PLEASE TAKE IT SERIOUSLY)
1) Line IJ has an equation of a line y = −3x − 8. Which of the following could be an equation for a line that is parallel to line IJ?
A) y = 3x + 7
B) y = 1 over 3x + 7
C) y = −3x + 7
D) y = −1 over 3x + 7
2) Find the perimeter of the shape below:
A) 12.4 units
B) 13.4 units
C)15.1 units
D) 16.8 units
A farmer has 300 ft of fencing with which to enclose a rectangular pen next to a barn. The barn itself will be used as one of the sides of the enclosed area.
What is the maximum area that can be enclosed by the fencing?
Enter your answer in the box.
ft²
Answer
Maximum area A_max = 11250 ft^2
Step-by-Step Explanation
Declaring Variables:-
The length of the rectangle = y
The width of the rectangle = x
Solution:-
- The perimeter of a rectangle can be expressed using the above two variables as follows:
Perimeter (P) = 2*Length + 2*Width
= 2* ( x + y )
- Since the barn is used as one of the sides (let's say y) we can subtract y
we don't need fencing for this side. The length of the fence required L is:
Length (L) = P - y
= 2*x + y
- We are given 300 feet of fencing. So we equate the length equal to 300 and develop a linear relationship between width and length of the barn.
300 = 2x + y
y = 300 - 2x
- The area (A) of the rectangle is given by the following expression:
A = Length*width
A = x*y
- Substitute the relationship developed between x and y in the Area (A) expression above. Then we have:
A = x*(300 - 2x)
A = 300x - 2x^2
- We will take first derivative of the expression of area (A) developed with respect to x and find the critical point of the area function by setting the first derivative A'(x) = 0.
A(x) = 300x - 2x^2
A'(x) = 300 - 4x
0 = 300-4x
x = 300 / 4 = 75 ft
- The critical point of the given function lies for the width (x) of 75 ft. We will plug in the critical value x = 75 ft back into the original function of Area and find the maximum area.
A(x) = 300x - 2x^2
A(75) = 300 (75) - 2(75)^2
A_max = A(75) = 11250 ft^2
- The maximum area that can be enclosed by the fencing is 11250 ft²
Answer:11250
Step-by-step explanation:
The vertex of a quadratic function is located at (1, 4), and the y-intercept of the function is (0, 1). What is the value of a if the function is written in the form y = a(x – h)2 + k?
a=____
Answer:
-3
Step-by-step explanation:
right on edge
Which sets of the real number system does − 2 belong to?
Write an expression to show the sum of fourteen and triple eight.
Larry wants to inscribe an equilateral triangle in a circle using only a compass and straightedge. He will first construct a circle and locate its center. He will then perform the following steps.
Step 1: Set the width of the compass to the diameter of the circle.
Step 2: Plot a point on the circle. Place the compass on this point and draw two arcs to cut the circle. These two points of intersection between the arcs and the circle represent two of the vertices.
Step 3: Draw a segment between the two vertices. Adjust the width of the compass to the length of the segment. Place the compass on one vertex and draw an arc to construct the third vertex.
Step 4: Draw a line from one vertex to the next to form an equilateral triangle.
What is the error in Larry's construction?
A. He did not draw a pair of perpendicular bisectors to construct two diameters of the circle.
B. He drew only two arcs.
C. He set the width of the compass to the diameter of the circle.
D. He did not ensure that the length of the sides of the triangle was the same as the diameter of the triangle.
The error in Larry's construction is:
C. He set the width of the compass to the diameter of the circle.
Step-by-step explanation:The steps for the construction of an inscribed equilateral triangle is as follows:
Step 1: Draw a circle on a piece of paper.Take a point anywhere on the circumference of the circle and take that point as a starting point.Step 2: Now without changing the span of the circle we have draw two arcs to cut the circle.These two points of intersection between the arcs and the circle represent two of the vertices. Step 3: Draw a segment between the two vertices. Adjust the width of the compass to the length of the segment. Place the compass on one vertex and draw an arc to construct the third vertex. Step 4: Draw a line from one vertex to the next to form an equilateral triangle.Hence, the error he that he set the width of compass equal to the diameter of circle.
Instead he should have set the width of the compass equal to the radius of the circle.
For model 5, the tip of the blade travels 0.18 mile per revolution. What is the approximate speed, in revolutions per minute,for model 5?
need the answer tooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
The approximate speed of model 5's 148-foot blade travels at 0.18 miles per revolution converted to feet is approximately 385.8 revolutions per hour.
To find the number of revolutions per minute for Model 5 with a blade length of 148 feet and a tip speed of 0.18 miles per revolution,
use the following calculation:
First, convert the tip speed from miles to feet:
0.18 miles × 5280 feet per mile = 950.4 feet per revolution
Now, divide this by the length of the blade in feet:
950.4 feet per revolution / 148 feet = 6.43 revolutions per minute
Finally, multiply this by 60 minutes per hour to get the number of revolutions per hour:
6.43 revolutions per minute × 60 minutes per hour = 385.8 revolutions per hour
Therefore, the required speed of model 5 has a blade that makes approximately 385.8 revolutions per hour.
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35 POINTS!!
The scatter plot shows the relationship betweeen the number of hours students spend watching television and the numer of hours they spend playing outdoor sports each week.
A graph is titled Television and Sport. On the x axis, the label is Weekly Hours of
What is the y-intercept of the line of best fit and what does it represent?
A 10.4 hours; the number of hours students play outdoor sports in a week when they do not watch television
B 7.2 hours; the number of hours students play outdoor sports in a week when they do not watch television
C 10.4 hours; the number of hours students watch television in a week when they do not participate in any outdoor sports
D 7.2 hours; the number of hours students watch television in a week when they do not participate in any outdoor sports
Answer:
D 7.2 hours; the number of hours students watch television in a week when they do not participate in any outdoor sports
Step-by-step explanation:
Given is a scatter plot which shows the relationship betweeen the number of hours students spend watching television and the numer of hours they spend playing outdoor sports each week.
From the graph showing the regression line, the regression line has y intercept as 7.2
(When x=0, y =7.2)
Hence correct answer is
D 7.2 hours; the number of hours students watch television in a week when they do not participate in any outdoor sports
Suppose that a car rental agency charges a fixed amount per day plus an amount per mile for renting a car. heidi rented a car one day and paid $58 for 190 miles. on another day she rented a car from the same agency and paid $86 for 330 miles. find the linear function f(x) (where x is in miles and f(x) is in dollars) that the agency could use to determine its daily rental charges.
The linear function f (x) hat the agency could use to determine its daily rental charges will be;
⇒ f (x) = 0.2x + 20
Where, f (x) is in dollars and x is in miles.
What is linear expression?
A linear expression is an algebraic statement where each term is either a constant or a variable raised to the first power.
Given that;
Heidi rented a car for one day and paid $58 for 190 miles.
And, On another day she paid $86 for 330 miles.
Now,
We find the linear function for f (x), where f (x) is in dollars and x is in miles as;
Here the two situations are given for miles and dollars as;
(190, 58) and (330, 86)
Thus, The linear function with two points (190, 58) and (330, 86) is defined as;
⇒ y - y₁ = (y₂ - y₁)/(x₂ - x₁) (x - x₁)
⇒ y - 58 = (86 - 58) / (330 - 190) (x - 190)
⇒ y - 58 = 28 / 140 (x - 190)
⇒ y - 58 = 1/5 (x - 190)
⇒ y - 58 = 0.2 (x - 190)
⇒ y - 58 = 0.2x - 38
⇒ y = 0.2x - 38 + 58
⇒ y = 0.2x + 20
⇒ f (x) = 0.2x + 20
Thus, The linear function f (x) hat the agency could use to determine its daily rental charges will be;
⇒ f (x) = 0.2x + 20
Where, f (x) is in dollars and x is in miles.
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To determine the daily rental charges at a car rental agency based on fixed and per mile fees, the linear function f(x) is found to be $40 + $0.15 per mile.
Linear function f(x):
Define variables: Let d = daily rental charge, m = per mile charge.Form equations based on given data: $58 = d + 190m and $86 = d + 330m.Solve the system of equations to find f(x): d = $40 and m = $0.15. Therefore, f(x) = 40 + 0.15x.Your school raise 125% of its fundraising goal. The goal raised $6,750. What was the goal?
Look at the rational expression below. x^2-4/2-x Which value of x makes the expression undefined?
Answer:
given rational function is undefined for x = 2
Step-by-step explanation:
The given rational function is
[tex]\frac{x^2-4}{2-x}[/tex]
The rational function is undefined if the denominator is zero.
Hence, the given rational function is undefined if
[tex]2-x=0[/tex]
Solve the equation for x
[tex]x=2[/tex]
Therefore, the given rational function is undefined for x = 2
Classify the model as exponential growth or exponential decay. Identify the growth or decay factor AND the percent of increase or decrease per time period.
y=2(1/2)^t
Plz help!
What is the value of e^In7x a.1 b.7e c.7x d.7
HELP FAST!!!
What is the value of x? Enter your answer in the box. x = Triangle with angles labeled x minus 4 degrees, 3 x degrees, and 100 degrees.
To find the value of x in the triangle, set up an equation using the triangle angle sum theorem and solve for x. The value of x is 21 degrees.
The problem involves finding the value of x in a triangle with given angle measures. According to the triangle angle sum theorem, the sum of the angles in any triangle is always 180 degrees. In this case, we are given two expressions for the angles in terms of x: x - 4 degrees and 3x degrees, plus a given angle of 100 degrees.
To solve for x, set up an equation using the given angles:
x - 4 + 3x + 100 = 180
Combine like terms to solve for x:
4x - 4 + 100 = 180
4x + 96 = 180
4x = 180 - 96
4x = 84
x = 84 / 4
x = 21
An object with reflectional symmetry can be created by reflecting it about an axis called the _____.
A. Point of symmetry
B. Line of symmetry
C. Symmetrical half
D. Equator of symmetry
Reflection symmetry is a symmetry with respect to the reflection and is also know as mirror symmetry, line symmetry or mirror-image symmetry.
When a figure undergoes a reflection and does not change , then the figure is said to has the reflectional symmetry.
An object with reflectional symmetry can be created by reflecting it about an axis called the Line of Symmetry.
Hope this helps ..!!
Thank you :)
Which expression has an equivalent value to x^2 + 9x + 8 for all values of x?
(x + 1)(x + 8)
(x + 2)(x + 6)
(x + 4)(x + 4)
(x + 5)(x +4)
can you please help me with this I will crown you brainly
There are 454 grams in a pound. There are 16 ounces in a pound. How many grams are in an ounce?
Answer: There are 28.375 grams in an ounce.
Step-by-step explanation:
Given: Total grams in a pound =454 grams
Total ounces in a pound = 16 ounces
Therefore, 16 ounces = 454 grams
To find grams in an ounce, we need to divide 454 grams by 16, we get
The total grams in an ounce=[tex]\frac{454}{16}=28.375\ grams[/tex]
∴The total grams in an ounce= 28.375 grams
Hence, there are 28.375 grams in an ounce.
28.375 grams in an ounce.
To calculate how many grams are in an ounce, divide 454 (grams in a pound) by 16 (ounces in a pound) to get 28.375 grams in an ounce.
To find out how many grams are in an ounce:
Given: 1 pound = 454 grams and 1 pound = 16 ounces.Since 1 pound = 454 grams and 1 pound = 16 ounces, divide 454 by 16 to find out how many grams are in an ounce.454 grams ÷ 16 = 28.375 grams in an ounce.What type of number results when a negative number is multiplied by a positive number and vice versa?
A. A negative number
B. Zero
C. A positive number
A sphere has a radius of 26 inches. a horizontal plane passes through the center of the sphere describe the cross section formed by the plane and the sphere.
When a horizontal plane passes through the center of a sphere with a radius of 26 inches, the cross-section formed is a circle with a radius of 26 inches.
let's break down the problem into two parts:
Describe the cross-section: We'll describe the shape formed when a horizontal plane passes through the center of the sphere.
Calculate the dimensions of the cross-section: We'll calculate the dimensions of the cross-section, such as its radius or diameter.
1. Describe the cross-section:
When a horizontal plane passes through the center of a sphere, it divides the sphere into two equal halves. The cross-section formed by this plane and the sphere will be a circle. Since the plane passes through the center of the sphere, the circle will be perfectly centered.
2. Calculate the dimensions of the cross-section:
Since the radius of the sphere is given as 26 inches, the radius of the cross-section (which is also the radius of the circle formed) will also be 26 inches.
So, the cross-section formed by the plane and the sphere is a circle with a radius of 26 inches.
A soccer ball has a radius of 6 inches, what is the approximate volume of the ball? Pi =3.14 *
What is the ratio of x to y?
3x = 8y
Final answer:
The ratio of x to y, given the equation 3x = 8y, is 8:3. This is found by dividing both sides of the equation by 3y and simplifying.
Explanation:
To find the ratio of x to y, given the equation 3x = 8y, you can divide both sides of the equation by 3 to solve for x in terms of y, or vice versa.
Dividing both sides by 3y yields:
x/y = (8y)/(3y)
Assuming y is not zero, the y terms cancel out on the right side of the equation, giving us:
x/y = 8/3
Method 2: Using fraction notation:
Rewrite the equation with x isolated:
x = (8/3) × y
This also shows that the ratio of x to y is 8:3.
Both methods lead to the same answer: x : y = 8 : 3.
Thus, the ratio of x to y is 8:3.
A wedding planner purchased small and large lanterns for a wedding reception. The small lanterns cost $25 each, and the large lanterns cost $40 each. The planner purchased a total of 40 lanterns for a total of $1180.
Answer:
The wedding planner must have purchased 28 small lanterns and 12 large lanterns.
Step-by-step explanation:
Let the number of small and large lanterns purchased be S and L respectively.
S + L = 40 ........ Equation 1
L = 40 - S
25S + 40L = 1180 ....... Equation 2
25S + 40(40 - S) = 1180
25S + 1600 - 40S = 1180
-15S = 1180 - 1600
-15S = -420
S = 28
L = 40 - S
L = 40 - 28
L = 12
The wedding planner must have purchased 28 small lanterns and 12 large lanterns.
Charlie used these steps to solve the equation 6x+5=1+2(3x+2)6x+5=1+23x+2. Which choice describes the meaning of his result, 5=5
Answer:
All values of x make the equation true. It is an infinite solution.
Samuel has a collection of toy cars. his favorites are the 272727 red ones which make up 60\%60%60, percent of his collection.how many toy cars does samuel have?
Answer:
Khan Academy said 45 hope it helps! :)
Find the supplement to the angle whose measure is 70 degrees. Show all your work
HELP!!!!!!!!!Which geometric series converges?
Answer:
its b
Step-by-step explanation:
PLEASE HELP ASAP!!! 42 POINTS
Really need help! Been trying to solve for a while.
1. Solve the system by using a table. 3y=4x+7 , -4x-4y=28. These are the possible solutions:(–4, –3),(–3, –4), (4, 3),(3, 4).
2.Solve by graphing. -3x-y=-10, 4x-4y=8. possible solutions:(-1,3), (3-1), (1,3), (3,1)