At the bowling alley, Terry paid $22.50 and bowled 3 games while Tommy bowled 2 games and paid $15. Interpret the rate of change by identifying the cost of each game
We can use a system of equations to solve for the cost of each game:
Let x be the cost per game for Terry, and let y be the cost per game for Tommy.
From the problem, we know that:
3x = 22.50 (Terry paid $22.50 for 3 games)
2y = 15 (Tommy paid $15 for 2 games)
Simplifying these equations, we get:
x = 7.50 (dividing both sides of the first equation by 3)
y = 7.50 (dividing both sides of the second equation by 2)
Therefore, the cost of each game is $7.50 for both Terry and Tommy.
The rate of change can be interpreted as the cost per game. In this case, the rate of change is $7.50 per game for both Terry and Tommy.
Let's use the formula for the equation of a straight line to find the equation that relates the weight and length of the spring:
y = mx + b
where y is the length of the spring, x is the weight attached to the spring, m is the rate of change (the increase in length of the spring per unit increase in weight), and b is the length of the spring when no weight is attached.
We can use the information from the problem to find the slope (rate of change):
m = (y2 - y1) / (x2 - x1)
m = (18 - 12) / (5 - 2)
m = 2
Therefore, the equation that relates the weight and length of the spring is:
y = 2x + b
To find the length of the spring when no weights are attached, we can use the fact that the length is b when x = 0:
y = 2x + b
b = y - 2x
b = 12 - 2(2)
b = 8
Therefore, the length of the spring when no weights are attached is 8 inches.
The slope of the graph represents the rate of change of parking fees in dollars per hour parked. The slope indicates how much the cost of parking increases or decreases for each additional hour parked.
To calculate the slope, we can use the formula:
slope = (change in y) / (change in x)
where (2, 20) and (6, 44) are two points on the line.
slope = (44 - 20) / (6 - 2)
slope = 24 / 4
slope = 6
This means that for every additional hour parked, the fee increases by $6.
Therefore, we can interpret the slope of the graph as the rate of change in the parking fee for each additional hour of parking, or the increase in cost per hour.
Let's use the formula for the slope of a line to find the rate of change of the cost of gas per gallon:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) = (4, 15) and (x2, y2) = (8, 30) are the two points we have.
m = (30 - 15) / (8 - 4)
m = 15 / 4
Therefore, the rate of change of the cost of gas per gallon is $15/4 or $3.75 per gallon pumped.
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.
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?(1 point)
Let's use the formula for the equation of a straight line to find the equation that relates the number of servings and the cost:
y = mx + b
where y is the cost, x is the number of servings, m is the rate of change (the increase in cost per unit increase in servings), and b is the fixed cost (the cost for 0 servings).
We can use the information from the problem to find the slope (rate of change):
m = (y2 - y1) / (x2 - x1)
m = (60 - 18) / (10 - 3)
m = 6
Therefore, the equation that relates the number of servings and the cost is:
y = 6x + b
To find the fixed cost, we can use the fact that the cost is $18 when x = 3:
y = 6x + b
18 = 6(3) + b
b = 0
Therefore, the equation that relates the number of servings and the cost is:
y = 6x
To find the cost per serving, we can substitute x = 1 (1 serving) into the equation:
y = 6x
y = 6(1)
y = 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?(1 point)
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?(1 point)
Let's use the formula for the slope of a line to find the rate of change of the cost of broccoli per pound:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) = (3, 8.25) and (x2, y2) = (6, 16.50) are the two points we have.
m = (16.50 - 8.25) / (6 - 3)
m = 8.25 / 3
Therefore, the rate of change of the cost of broccoli per pound is $8.25/3 or $2.75 per pound.