Question

A truck that can carry no more than 6600 lb is being used to transport refrigerators and upright pianos. Each refrigerator weighs 200 lb and each piano weighs 475 lb. Write and graph an inequality to show how many refrigerators and how many pianos the truck could carry. Will 17 refrigerators and 8 pianos overload the​ truck? Explain. Let x be the number of refrigerators in the truck and y be the number of pianos in the truck. Write an inequality to show how many refrigerators and how many pianos the truck could carry. y equals 15 ​(Use integers or simplified fractions for any numbers in the inequality. Do not​ factor.)

Answer 1

The inequality to represent this problem is:

`y\leq -(8)/(19) x+(264)/(19)`

And its graph is shown in the attached image.

17 refrigerators and 8 pianos would certainly overload the truck.

Explanation:

The total weight of refrigerators and pianos (if there are x number of refrigerators and y number of pianos) is given by the following algebraic expression:

`200\,x+475\,y`

since each piano weighs 200 pounds and each refrigerator wirghs 475 pounds.

We are limited by the maximum load of 6600 pounds that the truck can carry, therefore, the inequality we are looking for becomes:

`200\,x+475\,y\leq 6600`

This inequality can be re-written by isolating the term in y on one side of the inequality symbol:

`200\,x+475\,y\leq 6600\\475\,y\leq -200\,x+6600\\y\leq -(200)/(475) \,x+(6600)/(475) \\y\leq -(8)/(19) \,x+(264)/(19)`

In order to graph this inequality, we start by graphing the boundary line:

`y= -(8)/(19) \,x+(264)/(19)`

and then by selecting the half plane that satisfies the "smaller than" part of the inequality (see the highlighted region in blue in the attached image). Notice that we have only considered the regions of the plane that contain x and y values larger than or equal zero, since there is no meaning of considering negative x or y values given that they represent the number of refrigerators and pianos to be loaded in the truck.

Using this algebraic representation, we notice that 17 refrigerators (x=17) would render a maximum value for y of 6.7368 (which is smaller than 8) therefore yes, 17 refrigerators and 8 pianos would certainly overload the truck.