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Math 5 Polynomials Homework #10 -- Applications
1. A farmer has 500 meters of fencing. He would like to enclose a pen that looks like that shown.
a) Write an equation that gives the area of the pens as a function of w.
b) Using your graphing calculator, find the maximum area the farmer can get.
2. A rectangular swimming pool is twice as long as it is wide. A small concrete walkway surrounds the pool. The walkway is a constant 2 feet wide and has an area of 196 square feet. Find the dimensions of the pool.
3. The product of three consecutive integers is 39,270. What are the integers? (think about how you could use a polynomial to find the answer versus just guess and check)
4. A box has a volume of 1,800 cubic cm. The length is 6 times the height. As well, the length is 18 cm longer than the width. If the length is increased by 2 cm, the width by 3 cm and the height by 4 cm, the volume becomes 4,320 cubic cm. What is the original length of the box?
5. Allen cuts half a rectangular lawn, 50 m by 20 m by mowing strips of equal width around the perimeter. Betty cuts the small rectangle left. How wide a strip does Allen cut so that Allen and Betty share the work equally?
6. A ball is shot from a cannon 5 feet off the ground such that it is at its highest point when it has gone 10 meters vertically and 20 meters horizontally from the cannon (see picture). Please give an equation for the path of the ball assuming the origin to be at the bottom of the cannon. (ball travels in a parabolic arch)
7. The equation below gives distance traveled by a falling ball, d in meters, as a function of t, time in seconds. [pic] How many seconds does it take the ball to fall 2 meters?
8. A photo is 4 inches longer than it is wide. A 3-inch border is placed around the photo making the total area of the photo and border 165 square inches. What are the dimensions of the photo?
9. A homeowner is installing a swimming pool in his backyard. He wants its length to be 4 feet longer than its width. Then he wants to surround it with a concrete walkway 3 feet wide. If he can only afford 300 square feet of concrete for the walkway, what should the dimensions for the pool be?
10. A catering company is designing a box. The volume box is to be 54 cubic inches and the bottom of the box to be a square. Suppose the bottom of the box has a width that is 3 inches smaller than the height x of the box. Write a polynomial equation of the box.
11. To carpet my rectangular room (see picture), it costs $25 per square foot plus a flat fee of $100. Write an equation that gives the total cost as a function of x. Find the dimensions of the rectangle (see picture above) that maximize its area if the perimeter is 100 ft.
12. A box with a rectangular base that is twice as long as it is wide has a surface area (including the top) of 3 m2. Give an equation of volume of the box as a function of it’s width, w.
13. A rectangular piece of cardboard will be rolled up to create an open mailing tube used to ship marbles – see picture. Your company’s head of manufacturing tells you that the most economical piece of cardboard she can make has a perimeter of 30 feet. As design leader, your task is to determine the dimensions of the tube that will give the maximum volume.
a) Give the volume of the tube as a function of x.
b) Using your calculator, what are the dimensions of the tube that maximize its volume?
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w
x
x - 20
w
x
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