a Use an equation for fence length then rewrite it in terms of y.
B
b Use an equation for area in conjunction with the answer to Part A to solve for x and y.
A
a See solution.
B
b The pastures can have a length and width of 209 ft by 72 ft or a length and width of 54 ft by 278 ft.
Practice makes perfect
a In this exercise, we are given a picture and some information about area and length of fencing. For this part of the exercise, we can concentrate on the length of fencing. Let's redraw the pasture and put values on all the missing pieces of fence length.
From the picture, we can see the total vertical fencing is 3y and the total horizontal fencing is 4x. Let's put those together with the 1050 feet of fencing and solve for y.
b For this part of the exercise, we will use the fact that each pasture area is 15 000 ft^2 to determine possible lengths and widths of the pasture. From the image, we can see that each pasture is a rectangle with length x and width y.
Let's use our formula for area and the solution from Part A to solve for x.
The solutions for this equation are x ≈ 1050 ± 618.478. Let's separate the two cases.
x ≈ 1050 ± 618.47/8
x_1 ≈
x_2 ≈
1050 + 618.47/8
1050 - 618.47/8
1668.47/8
431.53/8
208.56
53.94
From that solution set, we can see that there are two possible values of x, the length of the pasture. Which means we can use each solution to find the corresponding values of y, the width of each pasture. Let's find y using the area equation.
A=x y ⇕ y=A/x
Since we know A=15 000, we can solve for y by dividing it by x. We are starting with approximate values, so all the values in the table are rounded to the nearest whole number.
x
y=15 000/x
y
209
15 000/209
72
54
15 000/54
278
We can conclude from this table that the pastures can either be 209 ft by 72 ft or 54 ft by 278 ft.
