Suppose you want to enclose a rectangular garden plot against a house using fencing on three sides, as shown at the left. Assume you have 50 ft. of fencing material and want to create a garden with an area of 150 ft².
A. Let w = the width. Write an expression for the length of the plot.
B. Write and solve an equation for the area of the plot. Round to the nearest tenth of a foot.
C. What dimensions should the garden have?
D. Critical Thinking - Find the area of the garden by using the dimensions you found in part (b). Does the area equal 150 ft²? Explain.
I don't even know where to start.
A. To find the length of the plot, we can use the fact that there are three sides being enclosed, and the fencing material is only used on three sides. Since one side is against the house, we need to find the length of the other two sides. Let's call the width of the garden w.
Since the width is w, the length would be the remaining distance between the two sides of the garden. So, the length would be 50 ft - w ft - w ft, or 50 ft - 2w ft.
B. The area of a rectangle is given by the formula A = length × width. In this case, we have the length as 50 ft - 2w ft and the width as w ft. So, the equation for the area of the plot would be:
A = (50 ft - 2w ft) × w ft
To solve this equation, we can distribute and simplify:
A = 50w ft - 2w^2 ft.
C. To find the dimensions of the garden, we can solve the equation for A = 150 ft². So,
150 ft² = 50w ft - 2w^2 ft.
We can rearrange the equation to get a quadratic equation in the form of ax^2 + bx + c = 0:
2w^2 ft - 50w ft + 150 ft² = 0.
We can now solve this quadratic equation to find the values of w. Once we have the value of w, we can substitute it back into the expression for the length (50 ft - 2w ft) to get the dimensions of the garden.
D. To find the area of the garden using the dimensions found in part (C), we can substitute the values of w and the length (50 ft - 2w ft) into the formula for the area of a rectangle:
A = (50 ft - 2w ft) × w ft.
We can simplify and calculate the area using the values of w obtained in part (C). Comparing this calculated area with the given area of 150 ft², we can determine if they are equal or not.