Rose is planting a rectangular herb garden. She has enough seed to cover 432 square inches of ground. Express the length of the garden as x and the width of the garden as y.

Question

. Rose is planting a rectangular herb garden. She has enough seed to cover 432 square inches of ground. Express the length of the garden as x and the width of the garden as y.

a. Write an inequality relating x and y. Then solve the inequality for x.

b. If the width of the garden must be between 4 inches and 8 inches, what are the smallest and largest possible lengths?

c. Is 12 inches a possible length for the garden? If so, what is the maximum possible width?

d. Rose decides to plant exactly 432 square inches of herb garden. She wants to enclose the rectangular garden with rabbit-proof fencing. Rose has only 100 inches of fencing. Write an inequality using x that expresses the 100-inch limit on the length of the fencing.

e. If the length of the garden is 12 inches will Rose be able to enclose the garden with the rabbit-proof fencing? If so, how much fencing will be left over?

Answer 1

A. Since the area is 432 in², we know that the length times the width at most can be 432, but it could be less, so x * y <= 432 in². Solving for x (by dividing y from both sides) gives us x = 432 / y

B. If we use y = 4 in and y = 8 in, we will get x <= 432 / 4 gives us x <= 108 and 432 / 8 give us x <= 54. So the maximum possible length is either 108 in (for a width of 4 in) or a length of 54 in (for a width of 8 in). In both cases the smallest possible length is zero, but then you wouldn't have a garden at all.

C. If x = 12, then we get y <= 36 in (after evaluating 432 / 12), so 36 in is the maximum possible length. However, this is a larger width than part B allows.

D. Since Rose decided to plan 432 sq. in of garden, we know for sure that x * y = 432 sq in now. However, since she only has 100 in of fencing, we know that the perimeter of the garden (2x + 2y) must be less than or equal to the amount of fencing that she has, so 2x + 2y <= 100 in.

If the length of the garden is 12 in, then we know (from part C) that the width is 36 in. Then we evaluate the perimeter 2 * 12 + 2 * 36 = 96 in, so she has used 96 in of her 100 in of fencing, leaving her with 100 - 96 = 4 in left.