Use the rules to find derivatives of the following functions at the specified values. a. f(x) = 2x³ at x = 2 f' (2)= g(z) =13x ½ at x = 3 g' (3)= h(z)= hx at x = 4 j(z) 130x¯¹ at x = 5 j' (5)=

The critical numbers of f are x = -5 and x = 5.

(A) To find the derivative f'(x), we differentiate f(x) = x³ - 75x - 2 with respect to x:
f'(x) = 3x² - 75

(B) There are no partition numbers for f' as partition numbers are related to integer partitions, which are not applicable in this context.

(C) To find the critical numbers of f, we set f'(x) equal to 0 and solve for x:
3x² - 75 = 0
x² = 25
x = ±5

So the critical numbers of f are x = -5 and x = 5.

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Answer:We can start by resolving the forces acting on the ornamental star in the vertical direction and in the horizontal direction. Since the star is stationary, these forces must balance each other out.

Let's call the tension in the higher chain T1 and the tension in the lower chain T2. We can use trigonometry to find the vertical and horizontal components of each tension force.

For T1:

The vertical component is T1cos(40°) and the horizontal component is T1sin(40°).

For T2:

The vertical component is T2cos(30°) and the horizontal component is T2sin(30°).

Now, resolving the forces in the vertical direction, we have:

T1cos(40°) + T2cos(30°) - mg = 0

where m is the mass of the star and g is the acceleration due to gravity.

Substituting the values:

T1cos(40°) + T2cos(30°) - (40 kg)(9.81 m/s^2) = 0

T1cos(40°) + T2cos(30°) = 392.4 N

Next, resolving the forces in the horizontal direction, we have:

T1sin(40°) = T2sin(30°)

Now we have two equations with two unknowns:

T1cos(40°) + T2cos(30°) = 392.4 N

T1sin(40°) = T2sin(30°)

Solving these equations simultaneously, we get:

T1 = 266.4 N

T2 = 205.1 N

Therefore, the tension in the higher chain (T1) is 266.4 N and the tension in the lower chain (T2) is 205.1 N.

Step-by-step explanation:

Answer:

If Mandy completed 70% of her walk in 3 hours, then we can find her walking rate as follows:

Let's assume that the entire walk takes t hours. Then, 70% of the walk would take 0.7t hours. We know that Mandy completes 70% of her walk in 3 hours, so we can set up the following equation:

0.7t = 3

Solving for t, we get:

t = 3 ÷ 0.7 ≈ 4.29

So, the entire walk will take approximately 4.29 hours. Since Mandy has already walked for 3 hours, the remaining time she needs to complete her walk is:

4.29 - 3 = 1.29 hours

Therefore, the answer is closest to option A, 3 4/5 hours.

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The decay factor corresponding to each percent decrease is 0.87. The correct option is c. The customer would save a total of $17.35, or 17.35% off the original price.

To determine the decay factor corresponding to each percent decrease, we need to use the formula:

decay factor = (100% - percent decrease)/100%

For the first discount of 13%, the decay factor would be:

decay factor = (100% - 13%)/100% = 0.87

For the additional neighborhood discount of 5%, the decay factor would be:

decay factor = (100% - 5%)/100% = 0.95

So the correct answer is option C, with a decay factor of 0.87 for the 13% discount and 0.95 for the neighborhood discount.

These decay factors represent how much of the original price is left after the discount has been applied. For example, if the original price was $100, after the 13% discount, the price would be:

price after first discount = $100 x 0.87 = $87

Then, after the additional 5% discount, the final price would be:

final price = $87 x 0.95 = $82.65

So the customer would save a total of $17.35, or 17.35% off the original price.

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Answer:

A, C, F

Step-by-Step:

The amplitude is whatever the coefficient is behind the trig function

Answer:

72.5 square inches

Step-by-step explanation:

See attachment.

The areas of the 2 shapes are in blue, but when added together:

60+12.5=72.5

Hope this helps!

The rewritten expression of 12 using the distributive property is 3(2 + 2)

From the question, we have the following parameters that can be used in our computation:

12 distributive property

This means that

12

Express as 6 + 6

So, we have

12 = 6 + 6

Factor out 3 from the equation

So, we have

12 = 3(2 + 2)

The above equation has been rewritten using the distributive property.

Hence, the rewritten expression using the distributive property is 3(2 + 2)

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Answer:

Step-by-step explanation:

The possible outcomes of rolling a fair six-sided die are the numbers 1, 2, 3, 4, 5, and 6, each of which has an equal probability of $\frac{1}{6}$ of appearing.

The probability of rolling a number greater than 1 is $\frac{5}{6}$, since there are five out of six possible outcomes that satisfy this condition (namely, 2, 3, 4, 5, and 6).

The probability of rolling a number less than 3 is $\frac{2}{6}=\frac{1}{3}$, since there are two out of six possible outcomes that satisfy this condition (namely, 1 and 2).

To find the probability of both events happening (rolling a number greater than 1 and then rolling a number less than 3), we can multiply their respective probabilities:

$\frac{5}{6}\cdot\frac{1}{3}=\frac{5}{18}$

Therefore, the probability of rolling a number greater than 1 and then rolling a number less than 3 is $\boxed{\frac{5}{18}}$.

Answer:

-4 = 5

0 = -1

2 = -4

4 = -7

(-4, 5)

(0, -1)

(2, -4)

(4, -7)

Step-by-step explanation:

First, let's identify what each term represents.

y-intercept: -1

slope: 3/2

Then, fill out the table.

x = -4

-1 - (3/2 · -4)

-1 - (-12/2)

-1 - (-6)

-1 + 6

y = 5

x = 0

-1 - (3/2 · 0)

-1 - 0

y = -1

x = 2

-1 - (3/2 · 2)

-1 - (6/2)

-1 - (3)

y = -4

x = 4

-1 - (3/2 · 4)

-1 - (12/2)

-1 - (6)

y = -7

Then, plot the points in the function on the graph.

(-4, 5)

(0, -1)

(2, -4)

(4, -7)

The equation to find the number of hours, t, until Patty and Carol are at the same location is: 70t + 50t = 300.


1. Patty and Carol start driving at the same time, towards each other on the same highway.
2. The distance between their cities is 300 miles.
3. Patty drives at an average speed of 70 mph, so in t hours she covers 70t miles.
4. Carol drives at an average speed of 50 mph, so in t hours she covers 50t miles.
5. As they drive towards each other, the sum of the distances they cover should equal the total distance between their cities.
6. Therefore, combining the distances covered by Patty and Carol, we get: 70t (Patty's distance) + 50t (Carol's distance) = 300 (total distance).

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Answer:p(x)=3x+5

Step-by-step explanation:

f(x)=2x+3

g(x)=x+2

p(x)=(2x+2)+(x+2)= 3x+5

Answer:

p(x)=ax + b

Step-by-step explanation:

The answer is B) - (11x² - 6x)ln(3) + 2ln(3) - 6ln(x) + C, where C is the constant of integration.

To solve this problem, we are given a function h(x) and asked to find its antiderivative or indefinite integral, which is denoted by H(x) and is defined as the function whose derivative is h(x). We are also given a specific value of H(x) at x = 6 and asked to use it to find the constant of integration, denoted by C.

The given function is h(x) = 22x - 3x² - 6/x, and we want to find H(x), which is the antiderivative of h(x). Using the power rule of integration, we can integrate each term of h(x) separately:

where C is the constant of integration. Note that we need to include an absolute value sign around x in the natural logarithm term because the function is not defined for x = 0.

Next, we are given that H(6) = 31, which means that when x = 6, the value of H(x) is 31. Substituting x = 6 and H(x) = 31 into the above equation, we get:

Simplifying, we get:

Therefore, the antiderivative of h(x) is:

We can simplify the expression by using the identity ln|a/b| = ln|a| - ln|b|:

where C is the constant of integration. Thus, the final answer is:

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The quadratic equation given the points (6,0), (-1,0), and (7,4).

The correct answer is h(x) = 1/2(x + 1)(x - 6).

To find the quadratic equation given the points (6,0), (-1,0), and (7,4), we can use the general form of a quadratic equation, which is [tex]h(x) = ax^2 + bx + c.[/tex]

First, let's substitute the coordinates of the given points into the equation to create a system of equations:

For the point (6,0):

[tex]0 = a(6)^2 + b(6) + c ---- (1)[/tex]

For the point (-1,0):

[tex]0 = a(-1)^2 + b(-1) + c ---- (2)[/tex]

For the point (7,4):

[tex]4 = a(7)^2 + b(7) + c ---- (3)[/tex]

We now have a system of three equations with three unknowns (a, b, c). We can solve this system to find the values of a, b, and c.

Solving the system of equations (1), (2), and (3), we find:

a = 1/2

b = -3/2

c = 0

Thus, the quadratic equation that satisfies the given points is:

h(x) = 1/2(x + 1)(x - 6).

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Yes, there is a relationship like a straight line between the raises administrators at State University receive and their performance on the job

To obtain this interpretation, we consider the equation of the regression line, which relates the predicted raise amount (y hat) to the job performance rating (x). The y-intercept term in this equation is the value of y hat when x equals zero. Therefore, the estimated y-intercept of $14,000 represents the predicted raise amount for an administrator whose job performance rating is zero, which corresponds to the base raise amount at State University.

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Answer:

D

Step-by-step explanation:

Answer: Box and whisker plots are plots on number lines with a box and two lines off the edges, called whiskers. The box has a line at the upper quartile(1), one at the lower quartile(2), and one in the center of the box at the median(3). The two lines go to the ends of the data, one at the minimum(4) and one at the maximum(5).

  4     2     3       1             5

   |-----[___|____]-----------|

₀__₁__₂__₃__₄__₅__₆__₇__₈

I hope this helps.

The profit is maximized at $16,400 when the ticket price is $12.

To find the ticket price that maximizes the profit, we used the fact that the maximum or minimum value of a quadratic function occurs at its vertex. For a quadratic function in the form of P(x) = ax^2 + bx + c, the x-coordinate of the vertex can be found using the formula x = -b / 2a.

In this case, we were given the function [tex]P(x) = -100x^2 + 2400x - 8000,[/tex]where x represents the price per ticket. The coefficient of [tex]x^2[/tex] is negative, which tells us that the graph of this function is a downward-facing parabola. The vertex of this parabola represents the maximum value of the function.

Using the formula x = -b / 2a, we found the x-coordinate of the vertex to be x = -2400 / 2(-100) = 12. This means that a ticket price of $12 will maximize the profit.

To verify that this is indeed the maximum profit, we substituted x = 12 into the profit function P(x):

[tex]P(12) = -100(12)^2 + 2400(12) - 8000 = 16,400[/tex]

We can see that the profit is maximized at $16,400 when the ticket price is $12.

In summary, to find the ticket price that maximizes the profit, we used the formula x = -b / 2a to find the x-coordinate of the vertex of the quadratic function representing the profit from selling tickets to a musical. The maximum profit occurs at the ticket price that corresponds to the x-coordinate of the vertex.

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Answer:

Step-by-step explanation:

It's given that, The height of three basketball players is 210 cm, 220 cm and 191 cm.

h₁ = 210 cm

h₂ = 220 cm

h₃ = 191 cm

Total number of observations = 3.

Average = Sum of all observations/Total number of observations.

↠ Average = h₁ + h₂ + h₃/3

↠ Average = 210 +220 + 191 /3

↠ Average = 430 + 191/3

↠ Average = 621/3

↠ Average = 207

Therefore, The required average height is 207 cm.

Using regression analysis, we get the following quadratic function:

y = 557[tex]x^{2}[/tex] + 1,690x + 60,000

We can use the data given and fit a quadratic equation in the form of y = a[tex]x^{2}[/tex] + bx + c, where y represents the number of flu cases and x is the number of years since 2012.

x (years since 2012) y (number of flu cases)

0 60,000

1 62,000

2 63,000

3 64,000

4 65,000

5 66,000

6 67,000

7 68,000

8 69,000

9 70,000

10 71,000

11 72,000

12 73,000

13 74,000

14 75,000

15 76,000

Next, we can use this table to find the coefficients a, b, and c that give us the best-fit quadratic function.

Using a regression analysis, we get the following quadratic function:

y = 557[tex]x^{2}[/tex] + 1,690x + 60,000

Here, the coefficient of x is 1,690, which represents the linear term in the quadratic equation. It tells us how much the number of flu cases changes with each year since 2012, assuming a quadratic relationship.

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Year Number of Flu Cases

2000 20,000

2001 22,000

2002 25,000

2003 28,000

2004 32,000

2005 35,000

2006 38,000

2007 42,000

2008 45,000

2009 50,000

2010 55,000

2011 58,000

2012 60,000

2013 62,000

2014 63,000

2015 64,000

Find a quadratic function that models the number of cases of flu each year, where y is years since 2012. What is the coefficient of x?

After solving the word problem, the large bag contains enough sugar to make 40 cakes

Since the large bag is 2.5 times larger than the small bag, and the small bag contains enough sugar to make 9 cakes, the large bag contains enough sugar to make:

2.5 * 9 = 22.5 cakes

However, since there are only 30 cups of sugar in the large bag, and we don't know how much sugar is needed to make a single cake, we cannot determine the exact number of cakes that can be made with a large bag of sugar.

As for the small bag, if it had enough sugar to make 9 cakes and there are 0.75 cups of sugar remaining, then each cake requires:

(sugar in bag - remaining sugar) / number of cakes

= (9 - 0.75) / 9

= 0.75 cups of sugar

Therefore, the large bag contains enough sugar to make:

30 cups / 0.75 cups per cake

= 40 cakes

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Answer:

1

1

1

2

2

2

2

Step-by-step explanation:

7) Roy buys 5 whole pizzas.

8) There are 36 gifts altogether.

9) There is 5 cows and 8 chickens.

10) He travel 210 km.

We have the four parts of question:

Now, We have the information:

7) A whole pizza costs P 190.00 and P 40. 00 for every additional topping.

If he spent P 1070 for pizza with 3 sets of additional toppings.

So, The formation of equation is:

=> (1070 - 3 × 40) ÷ 190

=> 5

8) Total gifts are : 4

and Inside each large gift are 2 medium-sized gifts and inside each

medium-sized gift are 3 small gifts.

So, the formation of equation;

=> 4 × 2 = 8

=> 8 × 3 = 24

The total is:

8 + 24 + 4 = 36

9) There are 13 heads and 36 feet .

We know:

Cows have (C) 4 legs

Chickens have (c)  2 legs

So, The equation will be:

C + c=13   ....eq.(1)

Multiply by 4, we get:

4C+2c=36

Then,

4C+4c=52

4C+2c=36

--------------- (-)

2c=16

c = 8 chickens

We put the value of c in eq.(1), we get  

C=13 - 8

C = 5 cows

Hence, There is 5 cows and 8 chickens.

10) Alexander travelled at 6:00 a.m.

and, average speed is 70 km per hour.

So, He travelled 140 km in 2 hours.

After two hours of driving, he stopped for 30 minutes for a rest.

He continued driving and reach his hometown at 9:30 a.m.

So, therefore he continues driving at 8:30

then he arrived at his destination at 9:30

So after his break for driving he travels 1 more hour

So in total he travelled 3 hours to arrive at his destination

3 × 70 = 210.

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As you approach zero from the left on a number line, the integers become increasingly negative, but the absolute values of those integers remain positive.

A number line is a visual representation of numbers placed in order on a straight line. It is a graphical tool used to represent the real numbers, starting from negative infinity on the left side and extending to positive infinity on the right side. The number line is divided into equal intervals, and each point on the line corresponds to a specific value or number. The distance between any two points on the number line represents the numerical difference between the corresponding numbers. The number line is a fundamental tool in mathematics for understanding the order and magnitude of numbers, as well as for performing operations such as addition, subtraction, and comparison.

As you approach zero from the left on a number line, the integers become increasingly negative, but the absolute values of those integers remain positive.

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The correct options are:

Option B:  The football is thrown from 6 feet off the ground.

when t = 0

h(t) = 6 * 75 * 0 + 16 * 0

= 6 feet

Option C : - 16t² represents the effect of gravity

we have

[tex]h = ut - \frac{1}{2}gt^2[/tex]

[tex]\frac{1}{2}gt^2=-16t^2[/tex]

F. The vertex of the graph of h gives the maximum height of the foot

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The triangle that will always have the same area, no matter how it is drawn, is a right triangle that has two 45° angles. The correct answer is option A

This is because any right triangle with two 45° angles is a special type of triangle called an isosceles right triangle. In an isosceles right triangle, the two legs are congruent, which means they have the same length. Therefore, no matter how the triangle is drawn, its area will always be the same.

To see why this is true, let's use the formula for the area of a right triangle:

Area = (base x height) / 2

In an isosceles right triangle, the base and height are congruent, so we can write:

Area = (leg x leg) / 2

Simplifying this expression, we get:

Area = (leg^2) / 2

Since the legs of an isosceles right triangle are congruent, we can substitute leg for both legs in the formula:

Area = (leg^2) / 2 = (leg x leg) / 2

No matter how the triangle is drawn, the legs will have the same length, so the area of the triangle will always be the same.

Therefore, the correct answer is option A, a right triangle that has two 45° angles.

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Answer: it will need to be rewritten so that the variable is only on one side of the equation

Step-by-step explanation:f the equation contains fractions, you may elect to multiply both sides of the equation by the least common denominator

The number of solutions that are there for the pair of equations for lines Q and S is zero solution (no solution) because the lines are parallel with no point of intersection.

In Mathematics and Geometry, no solution is sometimes referred to as zero solution, and an equation is said to have no solution when the left hand side and right hand side of the equation are not the same or equal.

This ultimately implies that, a system of equations would have no solution when the line representing each of the equations are parallel lines and have the same slope i.e both sides of the equal sign are the same and the variables cancel out.

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The volume of the resulting solid is estimated 4,128.38 cubic units.

The volume of the ball isgitten as:

V_ball = (4/3)πr^3

r is the radius of the ball. In this scenario

r = 11, so:

V_ball = (4/3)π(11)^3

V_ball = (4/3)π(1331)

V_ball = 4,396.46 cubic units

The volume of the hole is gotten as :

V_hole = (4/3)πr^3

r is the radius of the hole.

In thisscenario, r = 4, so:

V_hole = (4/3)π(4)^3

V_hole = (4/3)π(64)

V_hole = 268.08 cubic units

In conclusion, the volume of the resulting solid is:

V_resulting_solid = V_ball - V_hole

V_resulting_solid = 4,396.46 - 268.08

V_resulting_solid = 4,128.38 cubic units

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