A truck that can carry no more than 5900 lb is being used to transport refrigerators and upright pianos. Each refrigerator weighs 250 lb and each piano weighs 525 lb. Write and graph an inequality to show how many refrigerators and how many pianos the truck could carry. Will 12 refrigerators and 7 pianos overload the truck? Explain.
Let x be the number of refrigerators and y be the number of pianos. Then we can write the inequality:
250x + 525y ≤ 5900
To graph this inequality, we can plot the boundary line 250x + 525y = 5900 (which represents the maximum weight the truck can carry) as a dashed line, since it does not include the points on the line. Then we can shade the region below the line, since any point in this region will satisfy the inequality.
To determine if 12 refrigerators and 7 pianos will overload the truck, we can substitute x = 12 and y = 7 into the inequality:
250(12) + 525(7) = 5900
This shows that the weight of the refrigerators and pianos combined is exactly equal to the maximum weight the truck can carry. Therefore, the truck would not be overloaded by carrying 12 refrigerators and 7 pianos.
A truck that can carry no more than 5900 lb is being used to transport refrigerators and upright pianos. Each refrigerator weighs 250 lb and each piano weighs 525 lb. Write and graph an inequality to show how many refrigerators and how many pianos the truck could carry. Will 12 refrigerators and 7 pianos overload the truck? Explain.
Write an equation in slope-intercept form.
The inequality is 250x + 525y ≤ 5900, where x is the number of refrigerators and y is the number of pianos.
To graph this inequality in slope-intercept form, we need to solve for y:
250x + 525y ≤ 5900
525y ≤ -250x + 5900
y ≤ (-250/525)x + (5900/525)
y ≤ (-10/21)x + (112/21)
The slope of the line is -10/21 and the y-intercept is 112/21. We can plot the y-intercept on the y-axis and then use the slope to find additional points and draw a line. The inequality is satisfied by all points below this line.
To answer the second part of the question:
Substituting x = 12 and y = 7 into the inequality, we get:
250(12) + 525(7) = 5900
Since the weight of the refrigerators and pianos combined is exactly equal to the maximum weight the truck can carry, carrying 12 refrigerators and 7 pianos will not overload the truck.
Let's assume the number of refrigerators as "x" and the number of pianos as "y".
According to the given information:
- Each refrigerator weighs 250 lb, so the total weight of refrigerators would be 250x.
- Each piano weighs 525 lb, so the total weight of pianos would be 525y.
The weight of both the refrigerators and pianos should be less than or equal to the maximum carrying capacity of the truck, which is 5900 lb.
Therefore, the inequality representing the weight restriction would be:
250x + 525y ≤ 5900
To determine if 12 refrigerators and 7 pianos would overload the truck, we substitute x = 12 and y = 7 in the inequality:
250(12) + 525(7) ≤ 5900
Simplifying:
3000 + 3675 ≤ 5900
6675 ≤ 5900
This statement is false, as 6675 is not less than or equal to 5900.
Therefore, 12 refrigerators and 7 pianos would overload the truck.