Question

A new chocolate company is estimating how many candy bars per week college students will consume of their line of products. The graph shows the probable number of candy bars students (age 18-22) will consume from year 0 to year 10. B(x) gives the number of candy bars for boys, G(x) gives the number of candy bars for girls, and T(x) gives the total number for both groups. Use the graph to answer the question.

Answer 1

Solve for the slope and equation for each of the function B(x), G(x) and T(x)

Solving for B(x)

We have the following points for B(x)

$\begin{gathered} (x_1,y_1)=(0,1) \\ (x_2,y_2)=(5,2) \\ \\ m = \dfrac{y_2 - y_1}{x_2 - x_1} \\ m = \dfrac{2 - 1}{5 - 0} \\ m = \dfrac{1}{5} \end{gathered}$

Use the point (0,1) to solve for the equation of B(x)

$\begin{gathered} y-y_1=m(x-x_1) \\ y-1=\frac{1}{5}(x-0) \\ y-1=\frac{1}{5}x \\ y=\frac{1}{5}x+1 \\ B(x)=\frac{1}{5}+1 \end{gathered}$

Solving for G(x)

$\begin{gathered} (x_1,y_1)=(0,1) \\ (x_2,y_2)=(5,3) \\ \\ m = \dfrac{y_2 - y_1}{x_2 - x_1} \\ m = \dfrac{3 - 1}{5 - 0} \\ m = \dfrac{2}{5} \end{gathered}$

Use the point (0,1) to solve for the equation G(x)

$\begin{gathered} y-y_1=m\left(x-x_1\right) \\ y - 1 = \dfrac{2}{5}\left(x - 0\right) \\ y-1=\frac{2}{5}x \\ y=\frac{2}{5}x+1 \\ G(x)=\frac{2}{5}x+1 \end{gathered}$

Solving for T(x)

$\begin{gathered} (x_1,y_1)=(0,2) \\ (x_2,y_2)=(5,5) \\ \\ m = \dfrac{y_2 - y_1}{x_2 - x_1} \\ m = \dfrac{5 - 2}{5 - 0} \\ m = \dfrac{3}{5} \end{gathered}$

Use the point (0,2) to solve for the equation T(x)

$\begin{gathered} y-y_1=m\left(x-x_1\right) \\ y - 2 = \dfrac{3}{5}\left(x - 0\right) \\ y-2=\frac{3}{5}x \\ y=\frac{3}{5}x+2 \\ T(x)=\frac{3}{5}x+2 \end{gathered}$

Substituting x = 4 to B(x), G(x), and T(x)

$\begin{gathered} B(x)=\frac{1}{5}x+1 \\ B(4)=\frac{1}{5}(4)+1 \\ B(4)=\frac{4}{5}+1 \\ B(4)=\frac{9}{5} \\ B(4)=1.8 \\ \\ G(x)=\frac{2}{5}x+1 \\ G(4)=\frac{2}{5}(4)+1 \\ G(4)=\frac{8}{5}+1 \\ G(4)=\frac{13}{5} \\ G(4)=2.6 \\ \\ T(x)=\frac{3}{5}x+2 \\ T(4)=\frac{3}{5}(4)+2 \\ T(4)=\frac{12}{5}+2 \\ T(4)=\frac{22}{5} \\ T(4)=4.4 \end{gathered}$

Therefore, B(4) = 1.8, G(4) = 2.6, and T(4) =4.4