The total stopping distance T of a vehicle is shown below, where T is in feet and x is the speed in miles per hour.
T = 2.5x + 0.5x^2
Approximate the change and percent change in total stopping distance as speed changes from x = 15 to x = 18 miles per hour. (Round your answers to one decimal place.)
To find the change in total stopping distance, we need to subtract the value of T at x = 15 from the value of T at x = 18.
Step 1: Find T at x = 15:
T = 2.5x + 0.5x^2
T = 2.5(15) + 0.5(15)^2
T = 37.5 + 0.5(225)
T = 37.5 + 112.5
T = 150
Step 2: Find T at x = 18:
T = 2.5x + 0.5x^2
T = 2.5(18) + 0.5(18)^2
T = 45 + 0.5(324)
T = 45 + 162
T = 207
Step 3: Find the change in total stopping distance:
Change in T = T at x = 18 - T at x = 15
Change in T = 207 - 150
Change in T = 57
The change in total stopping distance as speed changes from x = 15 to x= 18 miles per hour is 57 feet.
To find the percent change, we need to calculate the percent change relative to the initial value (T at x = 15).
Step 4: Calculate the percent change:
Percent Change = (Change in T / T at x = 15) * 100
Percent Change = (57 / 150) * 100
Percent Change = 0.38 * 100
Percent Change = 38%
The percent change in total stopping distance as speed changes from x = 15 to x = 18 miles per hour is 38%.
To approximate the change and percent change in total stopping distance as speed changes from x = 15 to x = 18 miles per hour using the given equation T = 2.5x + 0.5x^2, we need to calculate the stopping distances at each speed and then find the difference and percent change.
First, let's substitute x = 15 into the equation to find the total stopping distance at a speed of 15 miles per hour:
T(15) = 2.5(15) + 0.5(15^2)
T(15) = 37.5 + 0.5(225)
T(15) = 37.5 + 112.5
T(15) = 150
Next, let's substitute x = 18 into the equation to find the total stopping distance at a speed of 18 miles per hour:
T(18) = 2.5(18) + 0.5(18^2)
T(18) = 45 + 0.5(324)
T(18) = 45 + 162
T(18) = 207
Now, to find the change in total stopping distance, we subtract the stopping distance at x = 15 from the stopping distance at x = 18:
Change in total stopping distance = T(18) - T(15)
Change in total stopping distance = 207 - 150
Change in total stopping distance = 57 feet
To calculate the percent change, we use the following formula:
Percent change = (Change in total stopping distance / T(15)) * 100
Percent change = (57 / 150) * 100
Percent change = 0.38 * 100
Percent change = 38%
Therefore, the approximate change in total stopping distance is 57 feet and the percent change is 38%.
since this is calculus, and you want to approximate the change ∆T, you must want to use
∆T ≈ T' ∆x
So, plug in your numbers to get ∆T. Then of course, the % change is
∆T/T(15)