Exercise 56 Page 575

We are given that Ethan and Tariq want to estimate the area of the rectangular field that their team will use for soccer practice. We know that they paced off the width of the field and that they estimate that the angle between the length of the field and the diagonal is about 40^(∘).

First are asked to assume that each of their steps is about 18 inches. With this information, we can evaluate the width of the field in inches. 280steps=280* 18in.=5040in.Since we are asked to give an answer in square feet, we will convert inches to feet. Recall that there are 12 inches in each foot. 5040in.=5040in.*1ft/12in.=420ft The width of the field is 420 feet. Let's add this information to our picture. Since the area of a rectangle is the product of its dimensions, we need to find the length of the field. Let l represents this length.

Now, as the dimensions and the diagonal form a right triangle, we can use one of the trigonometric ratios to evaluate the length. Let's recall the definition of the tangent of an angle. If△ ABCis a right triangle with acute∠ A, then the tangent of∠ Ais the ratio of the length of the leg opposite∠ A to the length of the leg adjacent∠ A.

Let's use this definition to create an equation for l. In our exercise, the length of the leg opposite 40^(∘) is 420 and the length of the leg adjacent to 40^(∘) is l.

The length of the field is approximately 500.54 feet. Finally, we will multiply the width by the length to find the area of the field. Area:420*500.54≈ 210 225 The area of the field is approximately 210 225 square feet.