Question

multiple choice 1 point
david plans to cover the floor of his room with new material.

• the floor is an isosceles trapezoid whose bases are 16 feet and 26 feet and sides are 13 feet in length.
• each piece of new material has an area of 2.5 square feet.

assuming the pieces of new material can be cut as needed, how many pieces does david need?
○ 252
○ 126
○ 101
○ 202
clear my selection

Explanation:

Step1: Find the height of the trapezoid

First, we can split the isosceles trapezoid into a rectangle and two right triangles. The difference between the two bases is \(26 - 16=10\) feet. So each right triangle has a base of \(\frac{10}{2} = 5\) feet and a hypotenuse (the non - parallel side of the trapezoid) of 13 feet. Using the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c = 13\) and \(a = 5\), we can find the height \(h\) of the trapezoid. Let \(h\) be the height, then \(h=\sqrt{13^{2}-5^{2}}=\sqrt{169 - 25}=\sqrt{144}=12\) feet.

Step2: Calculate the area of the trapezoid

The formula for the area of a trapezoid is \(A=\frac{(b_{1}+b_{2})h}{2}\), where \(b_{1}=16\) feet, \(b_{2}=26\) feet and \(h = 12\) feet. Substituting the values, we get \(A=\frac{(16 + 26)\times12}{2}=\frac{42\times12}{2}=42\times6 = 252\) square feet.

Step3: Find the number of pieces

Each piece has an area of 2.5 square feet. To find the number of pieces \(n\), we divide the area of the trapezoid by the area of each piece. So \(n=\frac{252}{2.5}=\frac{252\times2}{5}=\frac{504}{5}=100.8\). Since we can't have a fraction of a piece and we need to cover the entire floor, we round up to the next whole number. So \(n = 101\).

Answer:

101