To find the length of the spring when no weights are attached, we can use the rate of change between the two scenarios where weights are attached.
Let x be the length of the spring when no weights are attached.
We know that the rate of change is the change in length of the spring divided by the change in weight.
So, the rate of change when going from 2 pounds to 5 pounds is:
(18 inches - 12 inches) / (5 pounds - 2 pounds) = 6 inches / 3 pounds = 2 inches/pound
Using this rate of change, we can calculate the length of the spring when no weights are attached:
(x - 12 inches) / (0 pounds - 2 pounds) = 2 inches/pound
(x - 12 inches) / -2 pounds = 2 inches/pound
x - 12 inches = -4 inches
x = 8 inches
Therefore, the length of the spring when no weights are attached is 8 inches.
Use the image to answer the question.
An illustration shows three coiled wire springs stretched to varying lengths. The first spring does not have any weights pulling it downward, and the length of the spring is marked as a question mark inches. The second spring has a 2-pound weight attached to the bottom, and its length is marked as 12 inches. The third spring has a 5-pound weight attached to the bottom and its length is marked as 18 inches.
A spring has a length of 12 inches when a 2-pound weight is attached, and a length of 18 inches when a 5-pound weight is attached. Use rate of change to find the length of the spring when no weights are attached.
7 answers
The Kims are hosting a catered dinner. The cost for 3 servings is $18. The cost for 10 servings is $60. What is the cost per serving?
To find the cost per serving, we need to determine the rate of change in cost as the number of servings increases.
Let x be the cost per serving.
From the given information, the rate of change in cost as the number of servings increases is:
(cost for 10 servings - cost for 3 servings) / (10 servings - 3 servings) = 42 / 7 = 6
Therefore, the cost per serving is $6.
Let x be the cost per serving.
From the given information, the rate of change in cost as the number of servings increases is:
(cost for 10 servings - cost for 3 servings) / (10 servings - 3 servings) = 42 / 7 = 6
Therefore, the cost per serving is $6.
A graph of the cost of gas purchased depending on the number of gallons pumped has the points (4,15) and (8,30). What is the rate of change in the cost of the gas as each gallon is pumped?
To find the rate of change in the cost of gas as each gallon is pumped, we can use the given points (4, 15) and (8, 30) on the graph of the cost of gas versus the number of gallons pumped.
Let x represent the number of gallons pumped and y represent the cost of the gas.
Using the two points, we can calculate the rate of change (slope) of the cost of gas with respect to the number of gallons pumped:
Slope = (change in y) / (change in x)
Slope = (30 - 15) / (8 - 4)
Slope = 15 / 4
Slope = 3.75
Therefore, the rate of change in the cost of gas as each gallon is pumped is $3.75.
Let x represent the number of gallons pumped and y represent the cost of the gas.
Using the two points, we can calculate the rate of change (slope) of the cost of gas with respect to the number of gallons pumped:
Slope = (change in y) / (change in x)
Slope = (30 - 15) / (8 - 4)
Slope = 15 / 4
Slope = 3.75
Therefore, the rate of change in the cost of gas as each gallon is pumped is $3.75.
A linear graph of the cost of broccoli in dollars per pound has the points (3,8.25) and (6,16.50). What is the rate of change in the purchase price in dollars per pound?
To find the rate of change in the purchase price in dollars per pound, we can use the given points (3, 8.25) and (6, 16.50) on the graph of the cost of broccoli versus the number of pounds purchased.
Let x represent the number of pounds purchased and y represent the cost of broccoli.
Using the two points, we can calculate the rate of change (slope) of the cost of broccoli with respect to the number of pounds purchased:
Slope = (change in y) / (change in x)
Slope = (16.50 - 8.25) / (6 - 3)
Slope = 8.25 / 3
Slope = 2.75
Therefore, the rate of change in the purchase price of broccoli in dollars per pound is $2.75.
Let x represent the number of pounds purchased and y represent the cost of broccoli.
Using the two points, we can calculate the rate of change (slope) of the cost of broccoli with respect to the number of pounds purchased:
Slope = (change in y) / (change in x)
Slope = (16.50 - 8.25) / (6 - 3)
Slope = 8.25 / 3
Slope = 2.75
Therefore, the rate of change in the purchase price of broccoli in dollars per pound is $2.75.