Given the function with f(3)=127
and f(1)=95
, determine the rate of change over the interval 1≤x≤3
(1 point)
Responses
116
1 over 16
16
16
32
32
132
We can find the rate of change over the interval 1≤x≤3 by finding the slope of the line passing through the points (1, f(1)) and (3, f(3)).
The coordinates are (1, 95) and (3, 127).
Let's calculate the slope:
slope = (change in y-coordinates)/(change in x-coordinates)
slope = (127 - 95)/(3 - 1)
slope = 32/2
slope = 16
Therefore, the rate of change over the interval 1≤x≤3 is 16.
The correct response is 16.
What is the value of f(5)
if f(3)=10
and the rate of change over the interval 3≤x≤5
is 15?(1 point)
Responses
f(5)=40
f of 5 is equal to 40
f(5)=10
f of 5 is equal to 10
f(5)=20
f of 5 is equal to 20
f(5)=5
To find the value of f(5) given that f(3) = 10 and the rate of change over the interval 3≤x≤5 is 15, we can use the rate of change to determine how much f(x) increases from x=3 to x=5.
The rate of change is given by the slope of the line passing through the points (3, f(3)) and (5, f(5)). The slope is equal to the change in y-coordinates divided by the change in x-coordinates.
We are given that the rate of change is 15. This means that for every unit increase in x, f(x) increases by 15.
So, for the interval 3≤x≤5, the change in x is 2 (since x increases from 3 to 5), and the change in y (or f(x)) is 15 * 2 = 30 (since f(x) increases by 15 for each unit increase in x).
Since f(3) = 10, we can find f(5) by adding the change in y to f(3):
f(5) = f(3) + change in y = 10 + 30 = 40.
Therefore, the value of f(5) is 40.
The correct response is f(5) = 40 or "f of 5 is equal to 40."
The snow starts at a depth of 10 inches and melts to 2 inches over the span of 4 hours. Determine the rate of change over the time interval 0≤x≤4
(1 point)
Responses
2 inches per hour
2 inches per hour
4 inches per hour
4 inches per hour
8 inches per hour
8 inches per hour
0.5 inches per hour
To determine the rate of change over the time interval 0≤x≤4, we need to find the change in depth (y) divided by the change in time (x).
Given that the snow starts at a depth of 10 inches and melts to 2 inches over the span of 4 hours, the change in depth is 10 - 2 = 8 inches, and the change in time is 4 - 0 = 4 hours.
Hence, the rate of change is:
rate of change = (change in depth)/(change in time)
rate of change = 8 inches / 4 hours
rate of change = 2 inches per hour
Therefore, the rate of change over the time interval 0≤x≤4 is 2 inches per hour.
The correct response is 2 inches per hour.
The population in a neighborhood increased from 120 to 156 people from 1990 to 1994. Find the rate of change over the time interval 1990≤x≤1994
(1 point)
Responses
19 of a person per year
19 of a person per year
9 people per year
9 people per year
4 people per year
4 people per year
36 people per year
To find the rate of change over the time interval 1990≤x≤1994, we need to determine the change in population (y) divided by the change in time (x).
Given that the population increased from 120 to 156 people from 1990 to 1994, the change in population is 156 - 120 = 36 people, and the change in time is 1994 - 1990 = 4 years.
Thus, the rate of change is:
rate of change = (change in population)/(change in time)
rate of change = 36 people / 4 years
rate of change = 9 people per year
Therefore, the rate of change over the time interval 1990≤x≤1994 is 9 people per year.
The correct response is 9 people per year.
The temperature increases from 60 degrees Fahrenheit to 84 degrees Fahrenheit from 8 in the morning to 12 in the afternoon. Find the rate of change over the time interval 8≤x≤12
(1 point)
Responses
6°F per hour
6°F per hour
4°F per hour
4°F per hour
24°F per hour
24°F per hour
16°F per hour
16°F per hour
To find the rate of change over the time interval 8≤x≤12, we need to determine the change in temperature (y) divided by the change in time (x).
Given that the temperature increases from 60°F to 84°F from 8 in the morning to 12 in the afternoon, the change in temperature is 84°F - 60°F = 24°F, and the change in time is 12 - 8 = 4 hours.
Thus, the rate of change is:
rate of change = (change in temperature)/(change in time)
rate of change = 24°F / 4 hours
rate of change = 6°F per hour
Therefore, the rate of change over the time interval 8≤x≤12 is 6°F per hour.
The correct response is 6°F per hour.
To determine the rate of change over the interval 1 ≤ x ≤ 3, we need to find the difference in the values of the function at those points and divide it by the difference in their respective x-values.
Given:
f(3) = 127
f(1) = 95
The difference in the function values is:
127 - 95 = 32
The difference in the x-values is:
3 - 1 = 2
To find the rate of change, we divide the difference in the function values by the difference in the x-values:
Rate of change = (difference in the function values) / (difference in the x-values)
Rate of change = 32 / 2 = 16
Therefore, the rate of change over the interval 1 ≤ x ≤ 3 is 16.