Question 1

What is the minimum vertical distance between the parabolas   and   ?&#10;

Accepted Answer

31/8

Question 2

A steel pipe is being carried down a hallway 14 ft wide.At the end of the hall there is a right-angled turn into a narrower hallway 6 ft wide.What is the length of the longest pipe that can be carried horizontally around the corner?

27.50 ft

Accepted Answer

27.50 ft

Question 3

Find the maximum area of a rectangle that can be circumscribed about a given rectangle with length L = 8 and width W = 3.  &#10;

Accepted Answer

60.5

Question 4

The sum of two positive numbers is 16. What is the smallest possible value of the sum of their squares?&#10;

Accepted Answer

The answer of The sum of two positive numbers is...

Question 5

A farmer with 710 ft of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle.What is the largest possible total area of the four pens?

Accepted Answer

The answer of A farmer with 710 ft of fencing...

Question 6

Find the dimensions of the rectangle of largest area that can be inscribed in an equilateral triangle of side L = 9 cm if one side of the rectangle lies on the base of the triangle. Round your answer to the nearest tenth.

A) 5.5 cm, 4.4 cm
B) 4 cm, 3.91 cm
C) 4.5 cm, 4 cm
D) 4.5 cm, 3.9 cm
E) 7.5 cm, 2.9 cm
F) 9.5 cm, 3.9 cm

Accepted Answer

To maximize the area of the rectangle, we will want the rectangle to be as large as possible, which means it will have a height equal to half the length of the triangle's side (since the rectangle must fit inside the triangle with one side on the base).

Let the base of the rectangle be x, then its height is $\frac{1}{2}(9-x)$. The area of the rectangle is therefore given by:

$A = x \cdot \frac{1}{2}(9-x) = \frac{1}{2}(9x-x^2)$

To maximize this function, we can take its derivative and set it equal to zero:

$A' = 9 - x$

$9 - x = 0 \Rightarrow x = 9$

However, this solution is outside the bounds of the triangle, so the maximum must occur at one of the endpoints, i.e. when $x=0$ or $x=9/2$. Since $x=0$ corresponds to a rectangle with no area, we choose $x=9/2$, which gives a height of $\frac{1}{2}(9-\frac{9}{2}) = \frac{9}{4}$. Thus, the dimensions of the rectangle are:

$x = \frac{9}{2}$ and $h = \frac{9}{4}$

Rounded to the nearest tenth, we get:

$x \approx 4.5$ and $h \approx 3.9$

which matches choice D.

Question 7

A woman at a point A on the shore of a circular lake with radius wants to arrive at the point C diametrically opposite on the other side of the lake in the shortest possible time.She can walk at the rate of and row a boat at How should she proceed? (Find ).Round the result,if necessary,to the nearest hundredth.

A)

<strong>A woman at a point A on the shore of a circular lake with radius wants to arrive at the point C diametrically opposite on the other side of the lake in the shortest possible time.She can walk at the rate of and row a boat at How should she proceed? (Find ).Round the result,if necessary,to the nearest hundredth. </strong> A) radians B) She should row from point A to point C radians C) radians D) radians E) She should walk around the lake from point A to point C. <div style=padding-top: 35px>

radians
B) She should row from point A to point C radians
C)

radians
D)

radians
E) She should walk around the lake from point A to point C.

Accepted Answer

The answer of A woman at a point A on...

Question 8

Find two positive numbers whose product is   and whose sum is a minimum.&#10;A)  &#10;B) 3, 48&#10;C)  &#10;D) 6, 24&#10;E) 2, 72

Accepted Answer

We can factor the given product to obtain:

11ea8a3b_b6eb_ee15_9fff_9be8ca021e46_TB6897_11 = 7 × 17 × 19 × 23 × 29 × 31 × 37 × 41 × 43

We want to find two factors of this product whose sum is a minimum. Since the factors are positive integers, the minimum sum is obtained when the two factors are as close as possible to each other. The prime factorization above tells us that these factors are 19 and 23, whose product is:

19 × 23 = 437

Note that none of the other answer choices yields an integer product that is close to 437, so choice C is the best option.

Question 9

Find an equation of the line through the point (8,16) that cuts off the least area from the first quadrant.&#10;

Accepted Answer

The answer of Find an equation of the line through...

Question 10

A rectangular storage container with an open top is to have a volume of 10 The length of its base is twice the width.Material for the base costs $12 per square meter.Material for the sides costs $5 per square meter.Find the cost of materials for the cheapest such container.

A) $158.1
B) $153.92
C) $152.4
D) $153.9
E) $152.9
F) $151.6

Accepted Answer

Let's call the width of the base "w" and the length "l". Therefore, we know that:

V = lwh = 10^11
l = 2w (given)

To find the minimum cost, we need to find the minimum surface area of the container, which is the sum of the area of the base and the area of the four sides. The area of the base is lw, and the area of the four sides can be broken down into two sides with an area of lh each and two sides with an area of wh each. Therefore:

Surface Area = lw + 2lh + 2wh

Substituting l = 2w and V = lw(2w) = 2w^3, we get:

Surface Area = 2w^2 + 2(2w)(10^11/w) = 2w^2 + 4(10^11)w^(-1)

To find the minimum surface area, we need to take the derivative of this expression with respect to w and set it equal to zero:

d(Surface Area)/dw = 4w - 4(10^11)w^(-2) = 0
w = (10^11)^(1/3) = 1000

Therefore, the width of the container is 1000 meters, and the length is 2000 meters. Plugging in these values into the surface area expression, we get:

Surface Area = 2(1000)(2000) + 2(1000)(10^11/1000) = 4.002 x 10^8

To find the cost, we need to calculate the area of the base and the four sides separately:

Base Area = lw = (1000)(2000) = 2 x 10^6
Side Area = 2lh + 2wh = 2(1000)(2000) + 2(1000)(10^11/1000)/2 = 4.002 x 10^8 - 2 x 10^6 = 3.982 x 10^8

Therefore, the total cost is:

Cost = (Base Area x $12/m^2) + (Side Area x $5/m^2) = (2 x 10^6 x $12) + (3.982 x 10^8 x $5) = $1.539 x 10^9

Rounding this to the nearest cent gives us the final answer of $153.9, which is choice D.

Question 11

Find the point on the line   that is closest to the origin.&#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

The answer of Find the point on the line ...

Question 12

A piece of wire 10 m long is cut into two pieces.One piece is bent into a square and the other is bent into an equilateral triangle.How should the wire be cut for the square so that the total area enclosed is a minimum? Round your answer to the nearest hundredth.

A) 4.35 m
B) 3.25 m
C) 0 m
D) 5.35 m
E) 4.4 m

Accepted Answer

The answer of A piece of wire 10 m long...

Deck 4: Extension E: Applications of Differentiation