### 6.8 Problem Solving: Using Equations

Problem #

Problem Statement

Problems Similar To

1.

The sum of a number and its square is 72. Find the number?

\begin{equation}x^{2}+x=72\end{equation}

2.

A rectangular counter-top has a hole drilled in it to hold a cylindrical container (a utensil holder). The area of the entire counter top is given by (3x+4)*(x+6), while the area of the hole is given by $(x+6)^{2}$. Write an expression for the area (in factored form) of the counter-top that is left after the hole is drilled.

\begin{equation}(3x+4)(x+6)-(x+6)^{2}\end{equation}

3.

Daphne and Stephanie have competing refreshment stand businesses. Daphne's profit can be modeled by the polynomial $x^{2} + 10x + 7$, where 'x' is the number of items sold. Stephanie's profit can be modeled by the polynomial $9x^{2} + 7x + 7$, where 'x' has the same meaning. How much more is Stephanie's profit than that of Daphne's?

\begin{equation}9x^{2}+7x+7-(x^{2}+10x+7)\end{equation}

4.

Area of a circle is 144*$\pi$ sq inches. If the radius is (x - 7), what is the value of 'x'?

\begin{equation}(x-7)^{2}=144\end{equation}

5.

Area of a square is 169. If a side of the square is (7*x - 4) inches long, what is the value of 'x'?

\begin{equation}(7x-4)^{2}=169\end{equation}

6.

A rectangle has one side 8 ft longer than the other, and its area is 153 $ft^{2}$. Find the length of the shorter side of the rectangle.

\begin{equation}(x+8)x=153\end{equation}

7.

A triangular banner has an area of 70 $ft^{2}$. The height of the banner is 4 ft longer than its base. Find the base of the triangle.

\begin{equation}\frac{1}{2} \times x(x+4)=\frac{140}{2}\end{equation}