Free Printable optimization using calculus

1.

A farmer wants to construct a rectangular pigpen using 400 ft of fencing. The pen will be built next to an existing stone wall, so only three sides of fencing need to be constructed to enclose the pen. What dimensions should the farmer use to construct the pen with the largest possible area?

a)

100ft x 200ft

b)

102ft x 196 ft

c)

50 ft x 300 ft

d)

50 ft x 175 ft

2.

If y=2x-8, what is the minimum value of the product xy?

a)

-16

b)

-8

c)

-4

d)

2

3.

You want to make a box to contain dirt and your pet earthworm. Using a 7 in by 10 in rectangle of cardboard, you cut congruent squares from the corners and fold up the sides.

Choose the equation would you use in order to do Calculus to find the maximum volume of dirt (including worm) the box can hold?

a)

$V=\left(7-2x\right)\left(10-2x\right)$

b)

$V=x\left(7-2x\right)\left(10-2x\right)$

c)

$V=x\left(7-x\right)\left(10-x\right)$

d)

$V=x\left(7+2x\right)\left(10+2x\right)$

4.

A closed rectangular shipping box with square base is to be made from 120 square inches of cardboard. What dimensions should the box be for maximum volume?

Choose the constraint and optimization equations that represent the problem.

a)

$x^2=120$ and $V=x^3$

b)

$2x+y=120$ and $V=xy$

c)

$x^2y=120$ and $V=2x^2+4xy$

d)

$2x^2+4xy=120$ and $V=x^2y$

5.

A geometry student wants to draw a rectangle inscribed in a semicircle of radius 7. If one side must be on the semicircle's diameter, what is the area of the largest rectangle that the

student can draw?

a)

49

b)

42

c)

14

d)

7√7

6.

Farmer Jo has 32 square feet of land in which to make an enclosure for bunnies, chicks, and penguins. (see picture)

Choose the equation that represents this information.

a)

$2x+4y=32$

b)

$2x+2y=32$

c)

$xy=32$

d)

$A=3xy$

7.

same question, but more ...
There will be fencing put in around the entire enclosure and in the middle (as pictured) to create 3 sections, what dimensions should the overall enclosure be in order to use the least amount of fence?

Which equation represents the fence amount (F) that you want to minimize?

a)

$F=2x+4y$

b)

$F=2x+2y$

c)

$F=xy$

d)

$F=3\left(x+y\right)$

8.

Pick the correct interpretation from the given number line for V', the derivative of V.

a)

V has a minimum at 3

b)

V has a maximum at both 1 and 5

c)

V has a maximum at 0 and a minimum at 6

d)

V has a maximum at 1 and a minimum at 5

9.

$A'=\frac{-2\left(x-12\right)\left(x+12\right)}{x^2}$ and x has to be positive.
What work makes sense to happen next?

a)

plug in 12 and -12, resulting in a maximum

b)

plug in 10 and 14, resulting in a minimum

c)

plug in 10 and 14, resulting in a maximum

d)

take the derivative of the expression

10.

A rectangle is bounded by the x-axis and the parabola $y=12-x^2$ . What length and width should the rectangle have so that its area is a maximum?

Given the constraint equation above and the optimization equation $A=2xy$ , choose the DERIVATIVE of the merged (combined) equation.

a)

$A'=2xy$

b)

$A'=\ \ 24-6x^2$

c)

$A'=2x\left(12-x^2\right)$

d)

$A'=12-2x$

optimization using calculus

Total questions: 10

Worksheet time: 23mins

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