The approximate length of the garden is at least _ feet and at most _ feet.

Question

A rectangular garden must have a perimeter of 135 feet and an area of at least 900 square feet. Describe the possible lengths of the garden. Round your answers to the nearest foot.

Answer 1

So the question provides you with some information that is useful for solving the problem. First, it tells you information about the perimeter or the rectangle and then it tells you information regarding the area of the rectangle. Let's assume that the length of the rectangle is x and the width is y. Now, we can use the information the question gives us to write some equations that will help us solve the problem. We can represent the perimeter of the rectangle as 2x + 2y because the opposite sides of the rectangle are equal, so x is counted twice and y is counted twice. Therefore, 2x+2y = 135 feet. Next, we know that x * y has to be greater than or equal to 900. For now, lets assume that x * y = 900. Now we have 2 equations and 2 unknown variables so we can solve for the values of x and y.

2x+2y = 135

x*y = 900

From the second equation, lets divide both sides by y to solve for x. We get x = 900/y. Now we can substitute 900/y in for x in the first equation. We get 2(900/y) + 2y = 135. Now, we can solve for y. If we simplify, we get 1800/y + 2y = 135. Multiplying everything by y, we can get rid of the y in the denominator. And this is equal to 1800 + 2y^2 = 135y. We can treat this as a quadratic and solve for y. If we rearrange the equation, we get 2y^2 - 135y + 1800 = 0. We can now use the quadratic equation to solve for the possible values of y. We get possible values of 18.28 and 49.21. If we calculate values of x using these numbers, we get 49.21 and 18.28. Rounding these numbers, we get 18 and 49 as possible lengths of the garden. However, because we rounded, the answers are not exact.