Optimization

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?

• 49

• 42

• 14

• 7√7

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?

• 100ft x 200ft

• 102ft x 196 ft

• 50 ft x 300 ft

• 50 ft x 175 ft

Oil spilling from a ruptured tanker spreads in a circle on the surface of the ocean. The radius of the spill increases at a rate of 5 m/min. How fast is the area of the spill increasing when the radius is 5 m?

• 50π m²/min

• 47π m²/min

• 52π m²/min

• 40π m²/min

We need to enclose a field with a fence. We have 500 feet of fencing material and a building is on one side of the field so we won't need any fencing on that side. Determine the dimensions of the field that will enclose the largest area.

• x=250 ft, y=125 ft
• x=150 ft, y=200 ft
• x=125 ft, y=100 ft
• x=200 ft, y=150 ft

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

• -16
• -8
• -4
• 2

The volume of a cylindrical tin can with a top and a bottom is to be 16π cubic inches. If a minimum amount of tin is to be used to construct the can, what must be the height, in inches, of the can?

• 2 ³√2
• 2√2
• 4
• 2 ³√4

A rectangular  page is to contain 144 sq. inches of print. the margins on each side are 1 inch. Find the dimensions of the page that would use the least paper

• 16, 16
• 15,15
• 14,14
• 13,13

We want to construct a box whose base length is 3 times the base width. If the box must have a volume of 50 ft³, determine the dimensions that will minimize the amount of material used.

• w=2.027ft, h=4.055ft, l=6.082ft
• w=1.488ft, h=3.347ft, l=6.694ft
• w=2.231ft, h=3.347ft, l=6.694ft
• w=0.485ft, h=2.111ft, l=4.222ft

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?

• 49

• 42

• 14

• 7√7