The total stopping distance T of a vehicle is T=2.5x+0.5x^2 where T is in feet and x is the speed in miles per hour. Approximate the change and percent change in total stopping distance as sped changes from x=25 to x=26 miles per hour.
I started taking the derivative, but that just left me with no where to go....
dt= 2.5+x(dx)
2.5+25=27.5
dx=1 --- because change on x (26-25=1)
so dt=27.5(1)= 27.5ft
To approximate the change and percent change in total stopping distance as the speed changes from x = 25 to x = 26 miles per hour, we can use the concept of the derivative.
First, let's find the derivative of the total stopping distance function T(x) = 2.5x + 0.5x^2 with respect to x.
dT/dx = 2.5 + 1x
Now, we can evaluate the derivative at x = 25 to find the change in total stopping distance:
dT/dx = 2.5 + 1(25) = 2.5 + 25 = 27.5 ft/mph
This means that for a 1 mile per hour increase in speed, the total stopping distance is expected to increase by approximately 27.5 feet.
To find the percent change in total stopping distance, we can use the following formula:
Percent Change = (Change / Initial Value) * 100
Initial Value (total stopping distance at x = 25 mph) is given by:
T(25) = 2.5(25) + 0.5(25)^2 = 62.5 + 312.5 = 375 ft
Therefore, the percent change in total stopping distance is:
Percent Change = (27.5 / 375) * 100 = 7.3%
Approximately, the percent change in total stopping distance as speed changes from x = 25 to x = 26 miles per hour is 7.3%.
To approximate the change and percent change in total stopping distance as the speed changes from x = 25 to x = 26 miles per hour, we can use calculus.
First, let's start by differentiating the equation for the total stopping distance T with respect to x (the speed in miles per hour):
dT/dx = 2.5 + (2)(0.5x) = 2.5 + x
The derivative tells us the rate at which the total stopping distance is changing with respect to speed.
Now, let's find the change in total stopping distance by substituting x = 25 and x = 26 into the derivative:
Change in T = dT/dx * change in x
= (2.5 + x) * (26 - 25)
= 2.5 + 26
= 28.5 feet
Next, to find the percent change, we can use the formula:
Percent change = (Change in T / Initial T) * 100
The initial total stopping distance (at x = 25 mph) can be calculated by substituting x = 25 into the original equation: T = 2.5x + 0.5x^2.
Initial T = 2.5(25) + 0.5(25^2)
= 62.5 + 312.5
= 375 feet
Now we can calculate the percent change:
Percent change = (28.5 / 375) * 100
= 7.6%
Therefore, the approximate change in total stopping distance is 28.5 feet and the percent change is approximately 7.6% as the speed changes from 25 to 26 miles per hour.
dT/dx=2.5+x
dT=2.5dx+xdx x=25, dx=1 solve for dT