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 $2.75 per pound.
Height Above Canyon Floor Time (minutes)
360 0
280 3
160 10
80 14
Which statement best interprets the rate of change of the linear model shown in the table?
The table represents a linear relationship between the height above the canyon floor and the time in minutes. As time increases, the height decreases. The rate of change of the linear model can be interpreted as the amount that the height changes for each minute increase in time.
To find the rate of change of the linear model, we can use the formula for the slope of a line:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are any two points on the line. Let's choose the points (0, 360) and (14, 80) from the table:
m = (80 - 360) / (14 - 0)
m = -280 / 14
m = -20
Therefore, the rate of change of the linear model is -20. This means that for each minute increase in time, the height decreases by 20 feet.
Use the table to answer the question.
Number of Months Cost ($)
1 74
2 99
3 124
4 149
The table below shows the cost for a gym membership at the local fitness center with an initial start-up fee charged in the first month. Find the slope, or monthly cost.
To find the monthly cost, we need to first determine the rate of change of the cost over time, which is the slope of the line connecting the points in the table. We can use the formula for the slope of a line:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are any two points on the line. Let's use the points (1, 74) and (4, 149) from the table:
m = (149 - 74) / (4 - 1)
m = 75 / 3
m = 25
Therefore, the slope, or monthly cost, is $25.
A linear graph of parking fees in dollars based on hours parked has the points (2,20) and (6,44). How would you interpret the slope of the graph as the rate of change in the parking fee for each additional hour of parking?(1 point)
The graph shows the number of gallons in a wading pool that is being drained. Using the points (1,160) and (3,80) as (x1,y1) and (x2,y2), what is the rate of change in the number of gallons in the pool after each hour has passed?
To find the rate of change in the number of gallons in the pool after each hour has passed, we need to calculate the slope of the line connecting the points (1,160) and (3,80). We can use the slope formula:
slope = (y2 - y1) / (x2 - x1)
where (x1, y1) = (1,160) and (x2, y2) = (3,80)
slope = (80 - 160) / (3 - 1)
slope = -80 / 2
slope = -40
Therefore, the rate of change in the number of gallons of water in the pool is -40 gallons per hour. This means that the number of gallons in the pool decreases by 40 for every hour that passes, on average.
Which formula can be used to find the slope of a line from two points (x1,y1) and (x2,y2)?
The formula to find the slope of a line from two points (x1,y1) and (x2,y2) is:
slope = (y2 - y1) / (x2 - x1)
The number of gallons g in a swimming pool being filled with water is g=50x+1,000, where x is the number of hours since sunrise. It has been 10 hours since sunrise. What is the initial value?
The number of gallons g in the swimming pool being filled with water is given by g = 50x + 1,000, where x is the number of hours since sunrise. Since it has been 10 hours since sunrise, we can find the number of gallons in the pool by replacing x with 10:
g = 50(10) + 1,000
g = 500 + 1,000
g = 1,500
Therefore, there are 1,500 gallons in the pool after 10 hours since sunrise.
The initial value in this context is the number of gallons in the pool before any time has passed since sunrise. Notice that when x = 0, the formula reduces to:
g = 50(0) + 1,000
g = 0 + 1,000
g = 1,000
Therefore, the initial value is 1,000 gallons.
Days Fine
2 30 cents
3 45 cents
4 60 cents
The table shows the fine owed for an overdue library book according to how many days it is overdue. What is the initial value?
In this problem, the initial value represents the fine owed for a book that is returned on time or before the due date. Since there is no fine for a book that is returned on time, the initial value, in this case, is zero.
Therefore, the initial value is zero cents or $0.
Years Amount
1 $1,080
2 $1,160
3 $1,240
The table shows the amount of money in a bank account after a certain number of years. What is the initial value?
In this problem, the initial value represents the amount of money in the bank account at the beginning, or when no time has passed. In other words, it represents the starting balance of the account.
From the table, we can see that after 1 year, the amount in the account is $1,080. Therefore, the initial value is $1,080.
Therefore, the initial value is $1,080.
Victor will start paying off his credit card in January. He uses the function f(x)=−35x+2,500 to model the amount of money he owes x months after January. How much money does he owe at the beginning of January?(1 point)
To find how much money Victor owes at the beginning of January, we need to find the value of f(x) when x = 0 (since January is the starting month).
f(x) = -35x + 2,500
f(0) = -35(0) + 2,500
f(0) = 2,500
Therefore, at the beginning of January, Victor owes $2,500.
Cesar rents a bicycle for a flat rate plus $10 per hour. He models the cost of the rental with the function f(x)=10x+15, where x is the number of hours he rents the bicycle, and f(x) is the total cost of the bicycle rental, including the flat rate and the hourly fee. What is the flat rate for the bicycle rental?(1 point)
In this problem, the flat rate represents the cost of renting the bicycle for 0 hours, which means the cost would not include the hourly fee of $10 per hour.
To find the flat rate, we need to find the y-intercept of the function, or the value of f(x) when x = 0.
f(x) = 10x + 15
f(0) = 10(0) + 15
f(0) = 15
Therefore, the flat rate for the bicycle rental is $15.
The current temperature in degrees Celsius, where x equals the number of hours after noon, is 2x+10. Interpreting this equation, what is the initial value?
In this equation, the current temperature in degrees Celsius is represented as 2x + 10, where x is the number of hours after noon.
The initial value in this context represents the temperature at noon, or when x equals zero. This means that we need to find the value of the function when x = 0.
2x + 10 = 2(0) + 10
2x + 10 = 10
Therefore, the initial value of the temperature is 10 degrees Celsius.
From a height of 3,000 feet, a falcon descends at a rate of 250 ft./sec. What is the rate of change in the falcon’s elevation, and what is the initial value?
We can use the formula for the slope of a line to find the rate of change of the falcon's elevation:
slope = (y2 - y1) / (x2 - x1)
where y is the height in feet and x is the time in seconds. Let's choose two points on the line: (0, 3000) and (t, 3000 - 250t), where t is the time in seconds.
slope = (3000 - (3000 - 250t)) / (t - 0)
slope = 250t / t
slope = 250
Therefore, the rate of change of the falcon's elevation is -250 feet per second (negative because the falcon is descending). This means that the falcon's elevation decreases by 250 feet for every second that passes, on average.
The initial value in this context is the elevation from which the falcon descended. Since the falcon descended from a height of 3000 feet, the initial value is 3000 feet.
Hector would like to join a gym that has a one-time membership fee plus a monthly fee. He can use the function f(x)=35x+50 to model the cost of gym membership after x months. Identify and interpret the initial value of the function
In this function, the monthly fee is represented by the coefficient of x, which is 35, and the one-time membership fee is represented by the constant term, which is 50.
The initial value in this context represents the total cost of gym membership when x equals zero, or when no time has passed (i.e., the first month).
f(x) = 35x + 50
f(0) = 35(0) + 50
f(0) = 50
Therefore, the initial value of the function is $50, which represents the one-time membership fee.
Interpretation: This means that Hector will have to pay a one-time fee of $50 to join the gym, and this fee is separate from the monthly membership fee of $35.
The function f(x)=−75x+1,200 represents the value of your cell phone x months after you purchase it. Identify and interpret the initial value of the function.(1 point)
In this function, the value of your cell phone is represented by f(x), where x is the number of months since you purchased it.
The initial value in this context represents the value of your cell phone when x equals zero, or when you first purchased it.
f(x) = -75x + 1,200
f(0) = -75(0) + 1,200
f(0) = 1,200
Therefore, the initial value of the function is $1,200, which represents the value of your cell phone when you first purchased it.
Interpretation: This means that when you first purchased your cell phone, it was worth $1,200, but as time passes, the value decreases at a rate of $75 per month.