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 lot into a rectangle and a sector

The lot can be split into a rectangle with sides 50 yd and 120 yd and a sector of a circle.

Step2: Calculate the area of the rectangle

The area formula for a rectangle is $A_{rect}=l\times w$. Here, $l = 120$ yd and $w = 50$ yd. So, $A_{rect}=120\times50 = 6000$ $yd^{2}$.

Step3: Calculate the radius of the sector

The radius of the sector $r = 120$ yd.

Step4: Calculate the area of the sector

The formula for the area of a sector of a circle is $A_{sector}=\frac{\theta}{360^{\circ}}\times\pi r^{2}$, where $\theta = 60^{\circ}$ and $r = 120$ yd. So, $A_{sector}=\frac{60^{\circ}}{360^{\circ}}\times\pi\times(120)^{2}=\frac{1}{6}\times\pi\times14400 = 2400\pi$ $yd^{2}$.

Step5: Calculate the total area of the lot

$A = A_{rect}+A_{sector}=6000 + 2400\pi$.
$A\approx6000+2400\times3.14159=6000 + 7539.816\approx13540$ $yd^{2}$.

Answer:

13540