A truck that can carry no more than 8,000 Ib is being used to transport refrigerators and upright pianos. Each refrigerator weighs 250 Ib and each piano weighs 475 Ib. Write and graph an inquality to shower how many refigerators and how many pianos that truck could carry. Will 14 refrigerators and 10 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.
(Use integers or simplified fractions for any numbers in the inequality. Do not factor.)
250x + 475y ≤ 8000
To graph this inequality, first rewrite it in slope-intercept form:
475y ≤ -250x + 8000
y ≤ (-250/475)x + 16.84
Then, graph the line y = (-250/475)x + 16.84 (note that the slope is negative because as x increases, y decreases).
To determine which side of the line represents the valid solutions, choose a test point not on the line, such as (0,0). Substitute the x and y values into the inequality:
250(0) + 475(0) ≤ 8000
0 ≤ 8000
Since this is true, the valid solutions are below the line. Shade in the region below the line on the graph.
To determine if 14 refrigerators and 10 pianos will overload the truck, substitute x = 14 and y = 10 into the inequality:
250(14) + 475(10) ≤ 8000
3550 + 4750 ≤ 8000
8300 ≤ 8000
This is false, so the truck can carry 14 refrigerators and 10 pianos without overloading.
To determine how many refrigerators and pianos the truck can carry, we need to consider the weight capacity of the truck.
Let x be the number of refrigerators in the truck and y be the number of pianos in the truck.
Each refrigerator weighs 250 Ib, so the total weight of refrigerators is 250x Ib.
Each piano weighs 475 Ib, so the total weight of pianos is 475y Ib.
The total weight of refrigerators and pianos combined should not exceed the truck's weight capacity of 8,000 Ib.
Therefore, the inequality representing this situation is:
250x + 475y ≤ 8000
To graph this inequality, we will first rewrite it in slope-intercept form:
475y ≤ -250x + 8000
y ≤ (-250/475)x + (8000/475)
y ≤ (-10/19)x + 16.842
Now, we can graph this inequality on a coordinate plane to visually represent the possible combinations of refrigerators and pianos.
Regarding 14 refrigerators and 10 pianos, we can substitute these values into the inequality to check if they overload the truck:
250(14) + 475(10) ≤ 8000
3500 + 4750 ≤ 8000
8250 ≤ 8000
Since 8250 is greater than 8000, this means that 14 refrigerators and 10 pianos would overload the truck.