Practice Solutions

1. The perimeter of a rectangle is 36 inches. The length is two more than the width. Find the dimensions of the rectangle.

$\begin{array}{rcl}P& =& 2l +2w; P= 36; l= w+2 \\ 36& =& 2(w+2) + 2w \\ 36& =& 2w+4+ 2w \\ 36& =& 4w+4\begin{array}{ccc} & & \mathbf{Check}\end{array}\\ & & –4 –4\begin{array}{ccc} & & 36=2\left(10\right)+2\left(8\right) \end{array}✓\\ \left(\frac{1}{4}\right)32& =& 4w\left(\frac{1}{4}\right) \\ w& =& 8\\ l& =& \left(8\right)+2=10\end{array}$

The dimensions of the rectangle are 10 inches by 8 inches.

2. The perimeter of a triangle is 19 cm. The largest side is twice the smallest side plus one. The middle side is two more than the smallest side. Find the dimensions of the triangle.

$\begin{array}{ccccc}P=a+b+c& & & & \\ a: \mathrm{shortest}& b: \mathrm{middle}& & c: \mathrm{longest}& \\ P=19;& b=a+2;& & c=2a+1& \\ 19=a+\left(a+2\right)+\left(2a+1\right)& & & & \\ 19=4a+3& & & & \mathbf{Check}\\ –3 –3& & & & 19=4+6+9 ✓\\ \left(\frac{1}{4}\right)16=4a\left(\frac{1}{4}\right)& b=\left(4\right)+2& & c=2\left(4\right)+1& \\ a=4& b=6& & c=9& \end{array}$

The dimensions of the triangle are 4 cm, 6 cm, and 9 cm.

3. Find the height of a pyramid if the volume is $35 {\mathrm{cm}}^3$, the length of the base is 3 cm, and the width of the base is 5 cm.

$$\begin{gathered} V=\frac{1}{3}lwh \\ V=35; l=3; w=5 \\ \begin{array}{ccc}35=\frac{1}{\cancel{3}}\left(\cancel{3}\right)\left(5\right)h& & \\ \left(\frac{1}{5}\right)35=5h\left(\frac{1}{5}\right)& & \mathbf{Check}\\ h=7& & 35=\frac{1}{3}\left(3\right)\left(5\right)\left(7\right) ✓\end{array} \end{gathered}$$

The height of the pyramid is 7 cm.

4. Find the length of a rectangular prism if the height is 12 inches, the width is 4 inches, and the volume is 72 cubic inches.

$\begin{array}{rcl}V& =& lwh\\ V& =& 72; h=12; w=4\\ 72& =& l\left(4\right)\left(12\right)\\ \left(\frac{1}{48}\right)72& =& 48l\left(\frac{1}{48}\right)\\ l& =& \frac{72}{48}=\frac{3}{2}\begin{array}{cc}& \mathbf{Check}\end{array}\\ l& =& 1.5\begin{array}{cc} & 72=\left(1.5\right)\left(4\right)\left(12\right)\end{array}✓\end{array}$

The length of the rectangular prism is 1.5 inches.