Read and Understand: We need to find a way to describe the shaded region of this shape using polynomials.  We know the formula for the area of a circle is [latex]A=\pi{r^2}[/latex].  The figure we are working with is a circle with a smaller circle cut out.

Define and Translate: The larger circle has radius = r, and the smaller circle has radius= 3.  If we find the area of the larger circle, then subtract the area of the smaller circle, the remaining area will be the shaded region. First define the area of the larger circle:

[latex]A_{1}=\pi{r^2}[/latex]

Then define the area of the smaller circle.

[latex]A_{2}=\pi{3^2}=9\pi[/latex]

Write and Solve: The shaded region can be found by subtracting the smaller area from the larger area.

[latex]\begin{array}{c}A_{1}-A_{2}\\=\pi{r^2}-9\pi\end{array}[/latex]

The area of the shaded region is [latex]\pi{r^2}-9\pi[/latex]

Answer

[latex]\pi{r^2}-9\pi[/latex]

First, we will define the polynomial that describes the area of the rectangle as one figure.

Read and Understand: We are tasked with writing an expressions for the area of the figure above. The area of a rectangle is given as [latex]A=lw[/latex].  We need to consider the whole figure in our dimensions.

Define and Translate: Use the formula for area:  [latex]A=lw[/latex],the sides of the figure are the sum of the defined sides.  [latex]\begin{array}{c}l=\left(y+7\right)\\w=\left(y+9\right)\end{array}[/latex]

You could define [latex]\begin{array}{c}w=\left(y+7\right)\\l=\left(y+9\right)\end{array}[/latex] because it doesn’t matter in which order you multiply.

Write and Solve: 

[latex]\begin{array}{c}A=lw\\l=\left(y+7\right)\\w=\left(y+9\right)\end{array}[/latex],

We can use either method we learned to multiply binomials to simplify this expression, we will use a table.

Sum the terms from the table, and simplify:

[latex]\begin{array}{c}\left(y+7\right)\left(y+9\right)\\=y^2+7y+9y+63\\=y^2+16y+63\end{array}[/latex]

Answer 1

[latex]A=y^2+16y+63[/latex]

Now we will find an expression for the area of the whole figure as comprised by the areas of the four rectangles added together.

Read and Understand:  The area of a rectangle is given as [latex]A=lw[/latex].  We need to first define the areas of each rectangle, then sum them all together to get the area of the whole figure. It helps to label the four rectangle in the figure so you can keep the dimensions organized.

Define and Translate: Use the formula for area:  [latex]A=lw[/latex] for each rectangle:

[latex]A_{1}=7\cdot{y}[/latex]

[latex]A_{2}=7\cdot{9}=63[/latex]

[latex]A_{3}=y\cdot{y}=y^2[/latex]

[latex]A_{4}=y\cdot{9}[/latex]

Write and Solve: 

[latex]\begin{array}{c}A=A_{1}+A_{2}+A_{3}+A_{4}\\=7y+63+y^2+9y\\\text{ reorganize and simplify }\\=y^2+16y+63\end{array}[/latex]

Answer 2

[latex]A=y^2+16y+63[/latex]

Hopefully it isn;t surprising that both expressions simplify to the same thing.

Read and Understand:  We are tasked with writing an expressions for the volume of the cylinder in the figure above. The volume of a cylinder is given as[latex]V=\pi{r^2}h[/latex], where [latex]\pi[/latex] is a constant, and r is the radius and h is the height of the cylinder.

Define and Translate: Use the formula for volume:[latex]V=\pi{r^2}h[/latex],  we need to define r and h.

[latex]\begin{array}{c}r=\left(t-2\right)\\h=7\end{array}[/latex]

Write and Solve: Substitute r and h into the formula for volume.

[latex]\begin{array}{c}V=\pi{r^2}h\\=\pi\left(t-2\right)^2\cdot{7}\\=7\pi\left({t^2}-2t-2t+4\right)\\=7\pi\left({t^2}-4t+4\right)\end{array}[/latex]

Note that we usually write other constants that are multiplied by [latex]\pi[/latex] in front of it.  We can now distribute [latex]7\pi[/latex] to each term in the polynomial.

[latex]\begin{array}{c}7\pi\left({t^2}-4t+4\right)\\=\left(7\pi\right){t^2}-\left(7\pi\right)\cdot{4t}+\left(7\pi\right)\cdot{4}\\=7\pi{t^2}-28\pi{t}+28\pi\end{array}[/latex].

Note again how we left [latex]\pi[/latex] as a greek letter.  If we needed to use this calculation for measurement of materials, we would round pi, or a computer would round for us.

Answer

[latex]V=\pi{r^2}h=7\pi{t^2}-28\pi{t}+28\pi[/latex]