Question

a rectangular sheet of metal has a length of 18 inches and a width of 6 inches. a 45 - 45 - 90 triangle is cut from each corner. the hypotenuse of each triangle is 2 inches. which expression will solve for the area of the shaded portion, in square inches? a. 108 - 2 b. 108 - 4 c. 108 - 8 d. 108 - 16

Explanation:

Step1: Calculate area of rectangle

The area formula for a rectangle is $A = l\times w$. Given $l = 18$ inches and $w = 6$ inches, so $A_{rectangle}=18\times6 = 108$ square - inches.

Step2: Calculate area of one triangle

For a 45 - 45 - 90 triangle, if the hypotenuse is $c$, and the legs are $a$ and $b$ (where $a = b$), by the Pythagorean theorem $c^{2}=a^{2}+b^{2}=2a^{2}$. Given $c = 2$ inches, then $2^{2}=2a^{2}$, $4 = 2a^{2}$, $a^{2}=2$, and the area of a triangle is $A_{\triangle}=\frac{1}{2}a\times b=\frac{1}{2}a^{2}$. Since $a^{2}=2$, $A_{\triangle}=1$ square - inch.

Step3: Calculate area of four triangles

There are 4 triangles cut from the corners. So $A_{total - triangles}=4\times A_{\triangle}=4\times1 = 4$ square - inches.

Step4: Calculate area of shaded region

The area of the shaded region $A_{shaded}=A_{rectangle}-A_{total - triangles}=108 - 4$ square - inches.

Answer:

B. $108 - 4$