Question

a furniture designer builds a trapezoidal desk with a semicircular cutout. what is the area of the desk? a) about 36 540 cm² b) about 30 885 cm² c) about 31 320 cm² d) about 20 010 cm²

Explanation:

Step1: Calculate trapezoid area

The formula for the area of a trapezoid is $A_{t}=\frac{(a + b)h}{2}$, where $a$ and $b$ are the lengths of the parallel - sides and $h$ is the height. Here, assume the parallel sides are $174$ cm and some length related to the semi - circle, and height $h = 300$ cm. First, find the area of the trapezoid without considering the cut - out. Let's assume the trapezoid has parallel sides $a=174$ cm and $b$ (diameter of the semi - circle) $= 120$ cm (since radius $r = 60$ cm), and $h = 300$ cm. Then $A_{t}=\frac{(174 + 120)\times300}{2}=\frac{294\times300}{2}=44100$ $cm^{2}$.

Step2: Calculate semi - circle area

The formula for the area of a circle is $A_{c}=\pi r^{2}$, and for a semi - circle $A_{s}=\frac{1}{2}\pi r^{2}$. Given $r = 60$ cm, then $A_{s}=\frac{1}{2}\times\pi\times60^{2}=\frac{1}{2}\times3.14\times3600 = 5652$ $cm^{2}$.

Step3: Calculate the area of the desk

The area of the desk $A=A_{t}-A_{s}=44100 - 5652=38448$ $cm^{2}$. But if we assume the trapezoid has one base as $174$ cm and the other base is the diameter of the semi - circle ($120$ cm) and height $300$ cm, and calculate the area of the trapezoid $A_{trapezoid}=\frac{(174 + 120)\times300}{2}=44100$ $cm^{2}$, and the area of the semi - circle $A_{semicircle}=\frac{1}{2}\times3.14\times60^{2}=5652$ $cm^{2}$. The area of the desk $A = 44100-5652 = 38448$ $cm^{2}$. If we make a more accurate calculation with $\pi=\frac{22}{7}$, $A_{s}=\frac{1}{2}\times\frac{22}{7}\times60^{2}=\frac{1}{2}\times\frac{22}{7}\times3600=\frac{39600}{7}\approx5657.14$ $cm^{2}$, and $A_{t}=\frac{(174 + 120)\times300}{2}=44100$ $cm^{2}$, then $A = 44100-5657.14 = 38442.86$ $cm^{2}$. Among the given options, the closest one is:

Answer:

a) About 36540 $cm^{2}$