Question

a frequent problem in surveying city lots and rural lands adjacent to curves of highways and railways is that of finding the area when one or more of the boundary lines is the arc of a circle. find the area of the lot shown in the figure. the area is □ yd². (round to the nearest whole number.)

Explanation:

Step1: Divide the figure into a right - triangle and a sector

The right - triangle has base $b = 120$ yd and height $h = 50$ yd. The sector has radius $r = 50$ yd and central angle $\theta=60^{\circ}=\frac{\pi}{3}$ radians.

Step2: Calculate the area of the right - triangle

The area formula for a triangle is $A_{triangle}=\frac{1}{2}bh$. Substituting $b = 120$ yd and $h = 50$ yd, we get $A_{triangle}=\frac{1}{2}\times120\times50 = 3000$ yd².

Step3: Calculate the area of the sector

The area formula for a sector of a circle is $A_{sector}=\frac{1}{2}r^{2}\theta$ (when $\theta$ is in radians). Substituting $r = 50$ yd and $\theta=\frac{\pi}{3}$, we have $A_{sector}=\frac{1}{2}\times50^{2}\times\frac{\pi}{3}=\frac{2500\pi}{6}\approx\frac{2500\times 3.14}{6}\approx1308.33$ yd².

Step4: Calculate the total area of the figure

$A = A_{triangle}+A_{sector}=3000 + 1308.33=4308.33\approx4308$ yd².

Answer:

4308