Question 1

Answer true or false. An object that is thrown upward and reaches a height of s(t) = 60 + 160t -16t² for 0 $\le$ t $\le$ 3. The object is highest at t = 4.

Accepted Answer

We cannot determine if the object is highest at t=4 without knowing the full equation of its motion, including the downward acceleration due to gravity.

Question 2

An electrical generator produces a current in amperes starting at t = 0 s and running until t = 6 $\pi$ s that is given by 4cos(4t). Find the maximum current produced.

Accepted Answer

The answer of An electrical generator produces a current in...

Question 3

Two numbers sum to 42. Find the two numbers whose product is maximum.

Accepted Answer

Let the two numbers be x and 42-x. 
Their product is P(x) = x(42-x) = -x^2 + 42x
This is a quadratic function with a=-1, b=42, and c=0. 
The maximum value of the product occurs at x = -b/2a = -42/(-2) = 21.
Therefore, the two numbers are 21 and 21. 
Their product is 21*21 = 441, which is the maximum possible product. 
Thus, the answer is choice D.

Question 4

Find the point on the curve x² + y² = 25 closest to (0, 9).

Accepted Answer

The curve described is a circle with a radius of 5 centered at the origin (0,0), given by the equation $x^2 + y^2 = 25$. The point (0, 9) is outside this circle, and the closest point on the circle to (0, 9) would be vertically above or below the center along the y-axis, since the x-coordinate of the point (0, 9) is 0. The closest point on the circle to (0, 9) along the y-axis, without going over the radius of 5, is (0, 5).

Question 5

Express the number 42 as a sum of two nonnegative terms whose product is as large as possible.

Accepted Answer

The answer of Express the number 42 as a sum...

Question 6

Suppose that the number of bacteria present in a culture bacteria at time t is given by N = -5000(t² + 4t - 32). Find the smallest number of bacteria in the culture during the time interval 0 $\le$ t $\le$ 3.

Accepted Answer

The answer of Suppose that the number of bacteria present...

Question 7

Express the number 54 as the sum of two nonnegative numbers whose product is as large as possible.

Accepted Answer

To maximize the product of two numbers which sum up to 54, they must be equal. Thus, $\frac{54}{2} = 27$. So the two non-negative numbers are $(27,27)$.

Question 8

Find the point on the curve x² + y² = 121 closest to (0, 12).

Accepted Answer

The answer of Find the point on the curve x²...

Question 9

An object moves a distance s away from the origin according to the equation s(t) = 4t³ - 6t + 6, where 0 $\le$ t $\le$ 11. At what time is the object farthest from the origin?

Accepted Answer

The answer of An object moves a distance s away...

Question 10

A rectangular lot is to be bounded by a fence on three sides and by a wall on the fourth side. Two kinds of fencing will be used with heavy duty fencing selling for $5 a foot on the side opposite the wall. The two remaining sides will use standard fencing selling for $4 a foot. What are the dimensions of the rectangular plot of greatest area that can be fenced in at a cost of $55,200?

Accepted Answer

The answer of A rectangular lot is to be bounded...

Question 11

Answer true or false. The rectangle with the largest area that can be drawn inside a semi-circle is a square.

Accepted Answer

The rectangle with the largest area that can be drawn inside a semi-circle is not a square; it is a rectangle with a width equal to the diameter of the semi-circle and a height equal to the radius.

Question 12

A right triangle has a perimeter of 56. What are the lengths of each side if the area contained within the triangle is to be maximized?

Accepted Answer

The answer of A right triangle has a perimeter of...

Question 13

A sheet of cardboard 30 in square is used to make an open box by cutting squares of equal size from the corners and folding up the sides. What size squares should be cut to obtain a box with largest possible volume?

Accepted Answer

The answer of A sheet of cardboard 30 in square...

Question 14

A rectangular sheet of cardboard 64 cm by 93 cm is used to make an open box by cutting squares of equal size from the four corners and folding up the sides. What size squares should be cut to obtain the largest possible volume?

Accepted Answer

Let x be the size of the squares cut from each corner. Then the length of the base of the box will be 93 - 2x, and the width of the base will be 64 - 2x. The height of the box will be x. Therefore, the volume of the box will be:
V = x(93 - 2x)(64 - 2x)
Expanding this expression and simplifying, we get:
V = -4x^3 + 314x^2 - 5952x
To find the value of x that maximizes the volume, we need to find the critical points of V. Taking the derivative of V with respect to x, we get:
V' = -12x^2 + 628x - 5952
Setting this expression equal to zero and solving for x, we get:
x = 13/3 or x = 48
Since x must be less than half the width and length of the cardboard sheet, we have x ≤ min{32, 46.5/3}. Therefore, the only critical point in the feasible region is x = 13/3.

To confirm that this critical point is a maximum, we need to check that V''(13/3) < 0. Differentiating V' with respect to x, we get:
V'' = -24x + 628
Substituting x = 13/3, we get:
V''(13/3) = 4/3 > 0
Therefore, x = 13/3 is a minimum point, and x = 48 is a maximum point but not feasible. Hence, the optimal size for the squares that should be cut is approximately 4.33333. Therefore, the closest choice is (B), 11ead0bc_8cf0_ed5b_99a0_df04747bad31_TB6988_11.

Question 15

Answer true or false. The point on the parabola y = 5x² closest to (0, 0.8) is (0, 0).

Accepted Answer

The point on the parabola closest to (0, 0.8) will lie on the axis of symmetry of the parabola, which is the vertical line x = 0. Therefore, the x-coordinate of the closest point must be 0, but the y-coordinate may not necessarily be 0. We would need to find the y-coordinate of the point by plugging in x = 0 into the equation y = 5x^2 and we get y = 0, so the x-coordinate is correct but the y-coordinate is not.

Question 16

A company has a cost of operation function given by C(t) = 0.02t² - 2t + 5,000 for 0 $\le$ t $\le$ 500. Find the minimum cost of operation.

Accepted Answer

The answer of A company has a cost of operation...

Question 17

Answer true or false. An object thrown upward reaches a height of s(t) = -9.8t² + 39.2t. The object is highest at t = 2, where t is in seconds?

Accepted Answer

At the maximum point of the object's height, the velocity is zero. We can find this point by taking the derivative of s(t) and setting it equal to zero: s'(t) = -9.8 + 39.2 = 0. Solving for t, we get t = 4 seconds. Therefore, the statement is false and the object is highest at t = 4.

Question 18

A storm is passing with the wind speed in mph changing over time according to v(t) = -t² + 22t -40, for 0 $\le$ t $\le$ 11. Find the highest wind speed that occurs.

Accepted Answer

The equation provided seems to be corrupted or incorrectly formatted, making it difficult to directly interpret or solve as given. However, the question asks for the highest wind speed, which typically involves finding the maximum value of the given function over the specified interval. Without the ability to accurately decipher the function, we cannot apply calculus tools (like differentiation to find critical points) directly. The answer provided here, therefore, cannot be mathematically verified based on the given information. The choice "E) 81 mph" is selected as a placeholder, indicating the need for clarification or correction of the function to accurately solve the problem.

Question 19

For a triangle with sides 6 m, 8 m, and 10 m, the smallest circle that contains the triangle has a diameter of

Accepted Answer

The triangle with sides 6 m, 8 m, and 10 m is a right-angled triangle (Pythagorean triple). The smallest circle that contains this triangle is the circumcircle, whose diameter is the hypotenuse of the triangle, which is 10 m.

Question 20

Answer true or false. The triangle with the largest area that can be drawn inside a circle is an equilateral triangle.

Accepted Answer

This is true due to the fact that an equilateral triangle inscribed in a circle has all its vertices on the circle and also has its centroid, which is also the center of the circle. This property results in the maximum possible area of the triangle.

Quiz 4: The Derivative in Graphing and Applications

Answer true or false. An object that is thrown upward and reaches a height of s(t) = 60 + 160t -16t² for 0 $\le$ t $\le$ 3. The object is highest at t = 4.

An electrical generator produces a current in amperes starting at t = 0 s and running until t = 6 $\pi$ s that is given by 4cos(4t) . Find the maximum current produced.

Two numbers sum to 42. Find the two numbers whose product is maximum.

Find the point on the curve x² + y² = 25 closest to (0, 9) .

Express the number 42 as a sum of two nonnegative terms whose product is as large as possible.

Suppose that the number of bacteria present in a culture bacteria at time t is given by N = -5000(t² + 4t - 32) . Find the smallest number of bacteria in the culture during the time interval 0 $\le$ t $\le$ 3.

Express the number 54 as the sum of two nonnegative numbers whose product is as large as possible.

Find the point on the curve x² + y² = 121 closest to (0, 12).

An object moves a distance s away from the origin according to the equation s(t) = 4t³ - 6t + 6, where 0 $\le$ t $\le$ 11. At what time is the object farthest from the origin?

Answer true or false. The rectangle with the largest area that can be drawn inside a semi-circle is a square.

A right triangle has a perimeter of 56. What are the lengths of each side if the area contained within the triangle is to be maximized?

A sheet of cardboard 30 in square is used to make an open box by cutting squares of equal size from the corners and folding up the sides. What size squares should be cut to obtain a box with largest possible volume?

A rectangular sheet of cardboard 64 cm by 93 cm is used to make an open box by cutting squares of equal size from the four corners and folding up the sides. What size squares should be cut to obtain the largest possible volume?

Answer true or false. The point on the parabola y = 5x² closest to (0, 0.8) is (0, 0).

A company has a cost of operation function given by C(t) = 0.02t² - 2t + 5,000 for 0 $\le$ t $\le$ 500. Find the minimum cost of operation.

Answer true or false. An object thrown upward reaches a height of s(t) = -9.8t² + 39.2t. The object is highest at t = 2, where t is in seconds?

A storm is passing with the wind speed in mph changing over time according to v(t) = -t² + 22t -40, for 0 $\le$ t $\le$ 11. Find the highest wind speed that occurs.

For a triangle with sides 6 m, 8 m, and 10 m, the smallest circle that contains the triangle has a diameter of

Answer true or false. The triangle with the largest area that can be drawn inside a circle is an equilateral triangle.

Quiz 4: The Derivative in Graphing and Applications

Answer true or false. An object that is thrown upward and reaches a height of s(t) = 60 + 160t -16t2 for 0 ≤\le≤ t ≤\le≤ 3. The object is highest at t = 4.

An electrical generator produces a current in amperes starting at t = 0 s and running until t = 6 π\piπ s that is given by 4cos(4t) . Find the maximum current produced.

Two numbers sum to 42. Find the two numbers whose product is maximum.

Find the point on the curve x2 + y2 = 25 closest to (0, 9) .

Express the number 42 as a sum of two nonnegative terms whose product is as large as possible.

Suppose that the number of bacteria present in a culture bacteria at time t is given by N = -5000(t2 + 4t - 32) . Find the smallest number of bacteria in the culture during the time interval 0 ≤\le≤ t ≤\le≤ 3.

Express the number 54 as the sum of two nonnegative numbers whose product is as large as possible.

Find the point on the curve x2 + y2 = 121 closest to (0, 12).

An object moves a distance s away from the origin according to the equation s(t) = 4t3 - 6t + 6, where 0 ≤\le≤ t ≤\le≤ 11. At what time is the object farthest from the origin?

Answer true or false. The rectangle with the largest area that can be drawn inside a semi-circle is a square.

A right triangle has a perimeter of 56. What are the lengths of each side if the area contained within the triangle is to be maximized?

A sheet of cardboard 30 in square is used to make an open box by cutting squares of equal size from the corners and folding up the sides. What size squares should be cut to obtain a box with largest possible volume?

A rectangular sheet of cardboard 64 cm by 93 cm is used to make an open box by cutting squares of equal size from the four corners and folding up the sides. What size squares should be cut to obtain the largest possible volume?

Answer true or false. The point on the parabola y = 5x2 closest to (0, 0.8) is (0, 0).

A company has a cost of operation function given by C(t) = 0.02t2 - 2t + 5,000 for 0 ≤\le≤ t ≤\le≤ 500. Find the minimum cost of operation.

Answer true or false. An object thrown upward reaches a height of s(t) = -9.8t2 + 39.2t. The object is highest at t = 2, where t is in seconds?

A storm is passing with the wind speed in mph changing over time according to v(t) = -t2 + 22t -40, for 0 ≤\le≤ t ≤\le≤ 11. Find the highest wind speed that occurs.

For a triangle with sides 6 m, 8 m, and 10 m, the smallest circle that contains the triangle has a diameter of

Answer true or false. The triangle with the largest area that can be drawn inside a circle is an equilateral triangle.

Answer true or false. An object that is thrown upward and reaches a height of s(t) = 60 + 160t -16t² for 0 $\le$ t $\le$ 3. The object is highest at t = 4.

An electrical generator produces a current in amperes starting at t = 0 s and running until t = 6 $\pi$ s that is given by 4cos(4t) . Find the maximum current produced.

Find the point on the curve x² + y² = 25 closest to (0, 9) .

Suppose that the number of bacteria present in a culture bacteria at time t is given by N = -5000(t² + 4t - 32) . Find the smallest number of bacteria in the culture during the time interval 0 $\le$ t $\le$ 3.

Find the point on the curve x² + y² = 121 closest to (0, 12).

An object moves a distance s away from the origin according to the equation s(t) = 4t³ - 6t + 6, where 0 $\le$ t $\le$ 11. At what time is the object farthest from the origin?

Answer true or false. The point on the parabola y = 5x² closest to (0, 0.8) is (0, 0).

A company has a cost of operation function given by C(t) = 0.02t² - 2t + 5,000 for 0 $\le$ t $\le$ 500. Find the minimum cost of operation.

Answer true or false. An object thrown upward reaches a height of s(t) = -9.8t² + 39.2t. The object is highest at t = 2, where t is in seconds?

A storm is passing with the wind speed in mph changing over time according to v(t) = -t² + 22t -40, for 0 $\le$ t $\le$ 11. Find the highest wind speed that occurs.