MATH-2450 HANDOUT 3 (Lesson 2 - Application Problems)

1.A rectangle with length x is inscribed in a semicircle of radius 4 ft. Express the area A of the rectangle as a function of x.

2.A closed rectangular box is to be constructed having a volume of 96 ft 3. The width of the bottom (and hence, the top) of the box is w ft. The length of the bottom (and hence, the top) of the box is four times the width of the bottom (and hence, the top). Express the amount M of material that is needed to make this box as a function of w.

3.Steve wishes to fence a rectangular region of area 500 square yards. Express the amount F of fencing that is required as function of y, which is the length of the rectangle.

4.Suppose that an open rectangular box has a volume of 288 in 3 and a square base with edge length x inches. The material for the bottom of the box costs 8 cents per square inch and the material for the four sides costs 5 cents per square inch. Express the total cost C of the box as a function of x.

5.A rectangle is inscribed in a semicircle of radius 6 cm. Express the area of the rectangle as a function of one variable.

6.An open rectangular box is to be constructed having a volume of 100 cubic inches. The length of the bottom of the box is twice the width of the bottom. Express the amount of material, which is needed to make this box, as a function of one variable.

7.Bill can only afford to buy 100 yards of fencing. If he uses the fencing to enclose his rectangular garden, then express the area of the garden as a function of one variable.

8.Suppose that a closed rectangular box has a volume of 30 ft 3 and a square base. The material for the top and bottom of the box costs 6 cents per square foot and the material for the four sides costs 4 cents per square foot. Express the total cost of the box as a function of one variable.

9.A long rectangular sheet of metal, 16 inches wide, is made into a rain gutter by turning up two sides at right angles to the sheet. Express the cross sectional area of the gutter as a function of one variable.

10.A rectangle is inscribed in an isosceles triangle whose base is 48 feet and sides are 30 feet. If two of the vertices of the rectangle lie on the base of the triangle, then express the area of the rectangle as a function of one variable.

11.A farmer wishes to fence a rectangular pasture with a total area of 1000 yd 2, and he wants to divide it into two parts with a fence across the middle. Fencing around the outside costs $5.00 per yard, but he can use less expensive fence at $2.00 per yard as a divider. Express the cost of this enclosure as a function of one variable.

12.The sum of two numbers is 40. Express the product of these two numbers as a function of one variable.

13.If a box with a square base and open top is to have a volume of 4 ft 3, express the amount of material, that is needed to make the box, as a function of one variable.

14.A rectangle is inscribed in an equilateral triangle of side of a. If two of the vertices of the rectangle lie on the base of the triangle, then express the area of the rectangle as a function of one variable.

15.If (x, y) is a point on the graph of y = x 2 + 1, then express the distance from the point (x, y) and (3, 1) as a function of one variable.

16.If (x, y) is a point on the graph of x 2 + 4y 2 = 9, then express the distance from the point (x, y) and (1, 0) as a function of one variable.

17.An open box with a rectangular base is to be constructed from a rectangular piece of cardboard 16 inches wide and 21 inches long by cutting out a square from each corner and then bending up the sides. Express the volume of the box as a function of one variable.

18.A page of a book is to have an area of 90 square inches, with 1-inch margins at the bottom and sides and a -inch margin at the top. Express the area of printed material as a function of one variable.

19.A veterinarian has 100 feet of fencing and wishes to construct six dog kennels by first building a fence around a rectangular region and then subdividing that region into six smaller rectangles by placing five fences parallel to one of the sides. Express the total area of the kennels as a function of one variable.

ANSWERS:

1. or

2.

3.

4. (in cents) or (in dollars)

5. , where 2x = the length of the rectangle

6. , where x = the width of the bottom of the box

7. or , where x = the length of the rectangular garden

8. (in cents) or (in dollars) , where x = length of one of the sides of the square base of the box

9. or , where y = the width of the rectangular cross sectional region, which is also the depth of the gutter

10. or , where 2x = the length of the rectangle

11.This answer needs to be derived for one of the Computer Problems which is to be graded.

12. or , where x = one of the two numbers

13. , where x = the length of one of the sides of the square base of the box

14. or , where 2x = the length of the rectangle

15. or , where x = the x-coordinate of the point on the graph of

16. , where x = the x-coordinate of the point on the graph of

17. or , where x = the length of one of the sides of the square which is cut from each corner of the piece of cardboard

18. or , where x = the length of the rectangular printed material

19. or , where y = the width of the rectangular region

NOTE: There are other answers that are possible for these problems depending on the set up of the problem and/or the variable which you solve for.