Q. 4.119 - Intermediate Algebra

Question

Tension needs to eat at least an extra $1,000$ calories a day to prepare for running a marathon. He has only $\backslash (25$ to

spend on the extra food he needs and will spend it on $\backslash )0.75$ donuts which have $360$ calories each and $\$2$ energy

drinks which have $110$ calories.

ⓐ Write a system of inequalities that models this situation.

ⓑ Graph the system.

ⓒ Can he buy $8$  donuts and $4$ energy drinks and satisfy his caloric needs?

ⓓ Can he buy $1$ donut and  $3$ energy drinks and satisfy his caloric needs?

Step-by-Step Solution

Answer

Part(a). The obtained system of inequalities is:

$\left\{\begin{array}{l}360x+110y\ge 1000\\ 0.75x+2y\le 25\\ x\ge 0\\ y\ge 0\end{array}\right.$

Part (b). The graph of the system of inequalities is

Part (c). Yes

Part (b). No.

Part (a) Step 1. Form the inequalities

Let 

$$\begin{gathered} x=number of donuts \\ y=number of energy drinks \end{gathered}$$

Tension has to take $1000$ calories where each donut has $360 calories$ and each energy drink has $110$ calories. So,

$360x+110y\ge 1000$

$\$0.75$ for donut and $\$2$ for energy drink must not greater than $\$25$. So,

$0.75x+2y\le 25$

The number of donut is equal or more than zero. So,

$x\ge 0$

The number of energy drinks is equal or more than zero. So,

$y\ge 0$

The obtained system of inequalities is:

$\left\{\begin{array}{l}360x+110y\ge 1000\\ 0.75x+2y\le 25\\ x\ge 0\\ y\ge 0\end{array}\right.$

part (b) Step 1. graph the system of inequalities

The graph for the obtained system of inequalities is: 

Part (c) Step 1.Find 8 , 4 in the solution region.

To determine if he can $8$ donuts and $4$ energy drinks and satisfy his caloric needs. we look at the graph to see if the point $\left(8,4\right)$ is in the solution region (dark blue).

Yes, $\left(8,4\right)$ is in the solution region. So Tension can buy $8$ donuts and $4$ energy drinks.

Part (d) Step 1. part (d) step 4. Find 1 , 3 in the solution region.

To determine if he can $1$ donuts and $3$energy drinks and satisfy his caloric needs. we look at the graph to see if the point $\left(1,3\right)$  is in the solution region (dark blue).

No, $\left(1,3\right)$ is not in the solution region. So Tension cannot buy $1$donut and $3$energy drinks.