Q 325 - Intermediate Algebra

Question

Mark is increasing his exercise routine by running and walking at least $4$ miles each day. His goal is to burn a minimum of $1500$ calories from this exercise. Walking burns $270$ calories/mile and running burns $650$ calories.

ⓐ Write a system of inequalities to model this situation. 

ⓑ Graph the system. 

ⓒ Could he meet his goal by walking $3$ miles and running $1$ mile?

ⓓ Could he meet his goal by walking $2$ miles and running $2$ mile?

Step-by-Step Solution

Answer

Part a. A system of inequalities to model the situation is $\left\{\begin{array}{l}x+y\ge 4\\ 270x+650y\ge 1500\\ x\ge 0\\ y\ge 0\end{array}\right.$

Part b. The graph of the system is

Part c. He cannot meet his goal by walking $3$ miles and running $1$ mile.

Part d. He can meet his goal by walking $2$ miles and running $2$ mile

Part (a) Step 1. Form the system of inequalities

Let $x$ represents number of miles of walking and $y$ represents number of miles of running.

Mark wants to run or walk at least $4$ miles, so the inequality is $x+y\ge 4$

Walking burns $270$ calories per mile and running burns $650$ calories per mile. Mark wants to burn at least $1500$ calories, so the inequality is $270x+650y\ge 1500$

Number of miles is always a non-negative quantity, so the system of inequalities is

$\left\{\begin{array}{l}x+y\ge 4\\ 270x+650y\ge 1500\\ x\ge 0\\ y\ge 0\end{array}\right.$

Part (b) Step 1. Graph the first inequality

Draw a solid line $x+y=4$ and the point $(0,0)$ does not satisfy the inequality $x+y\ge 4$. So shade the region black that does not contain the point $(0,0)$ to get the graph.  

Part (b) Step 2. Graph the second inequality on the same coordinate plane

Draw a solid line $270x+650y=1500$ and the point $(0,0)$ does not satisfy the inequality $270x+650y\ge 1500$. So shade the region red that does not contain the point $(0,0)$ to get the graph.  

The common shaded region is the solution of the system of inequality. 

Part (c) Step 1. Check the point on the graph

$3$ miles of walking and $1$ mile of running represents the point $(3,1)$ on the coordinate plane. On plotting the point we get

The point does not lie in the common shaded region, so he cannot meet his goal by walking three miles and running one mile.

Part (d) Step 1. Check the point on the graph

$2$ miles of walking and $2$ miles of running represents the point $(2,2)$ on the coordinate plane. On plotting the point we get

The point lies in the common shaded region, so he can meet his goal by walking two miles and running two miles.