The height, h, in metres, of a ski ramp over a horizonatal distance, x, in metres, is given by h(x)=0.01x^3-0.3x^2+60 for the interval 0 less than or equal to x less than or equal to 22.
a) Find the vertical drop from the top of the ramp to the lowest point on the ramp.
b) Find the vertical rise from the lowest point of the ramp to the end of the ramp.
The lowest point i found was at (0,60) using the first derivative, but i do not know how to calculate the above two questions.
h'=0=.03x^2-.6x
x=0; x=20
vertical drop= 60-h(20)
vertical rise:h(22)-h(20)
check my thinking.
To find the vertical drop from the top of the ramp to the lowest point on the ramp, you need to subtract the height at the top of the ramp from the height at the lowest point on the ramp.
First, let's find the height at the top of the ramp. You mentioned that the lowest point is at (0,60), so we need to substitute x=0 into the equation h(x)=0.01x^3-0.3x^2+60:
h(0) = 0.01(0)^3 - 0.3(0)^2 + 60
= 0 + 0 + 60
= 60
The height at the top of the ramp is 60 meters.
Now, let's find the height at the lowest point of the ramp. Since you mentioned that the lowest point is at (0,60), we already know that the height at that point is 60 meters.
To find the vertical drop, we subtract the height at the top from the height at the lowest point:
Vertical Drop = Height at Lowest Point - Height at Top
= 60 - 60
= 0 meters
So, the vertical drop from the top of the ramp to the lowest point on the ramp is 0 meters.
Next, let's find the vertical rise from the lowest point of the ramp to the end of the ramp. To do this, we need to subtract the lowest point's height from the height at the end of the ramp.
Substitute x=22 into the equation h(x) = 0.01x^3 - 0.3x^2 + 60:
h(22) = 0.01(22)^3 - 0.3(22)^2 + 60
Use a calculator or simplify the expression:
h(22) ≈ 10.648
The height at the end of the ramp is approximately 10.648 meters.
Now, let's find the vertical rise by subtracting the lowest point's height from the height at the end of the ramp:
Vertical Rise = Height at End of Ramp - Height at Lowest Point
= 10.648 - 60
= -49.352 meters
Notice that the vertical rise is negative, which indicates that the height decreases from the lowest point to the end of the ramp. In this case, it means that there is a downward slope from the lowest point to the end of the ramp.
Thus, the vertical rise from the lowest point of the ramp to the end of the ramp is approximately -49.352 meters.