The complete question is;
A snowboarder leaves an 8-foot-tall ramp with an upward velocity of 28 feet per second. The function h = -16t² + 28t + 8 gives the height h (in feet) of the snowboarder after t seconds. The snowboarder earns 1 point per foot of the maximum height reached, 5 points per second in the air, and 25 points for a perfect landing. With a perfect landing, how many total points does the snowboarder receive?
Answer:
Total points earned with a perfect landing = 111 points
Step-by-step explanation:
First of all, let's find the maximum height of the given parabolic function h = -16t² + 28t + 8
h_max = c - (b²/4a)
where: a = -16, b = 28, c = 8
h_max = 8² - 28²/(4*(-16))
h_max = 64 + 784/64
h max = 76.25 ft ≈ 76 ft
The question says that the snowboarder earns 1 point per foot of the maximum height reached.
Thus, the points from the maximum height reached are 76 points
Now, let's find the maximum time in air by solving the equation for h = 0 Thus;
-16t² + 28t +8 = 0
Using quadratic formula,
t = [-28 ± √(28² - (4*-16*8))]/(2*-16)
t = 2 or -0.25
We'll use 2 as we are looking for maximum time.
The question says at maximum time, it's 5 points for each second in the air,
thus, points for seconds in air = 5 × 2 = 10 points
We are told 25 points for perfect landing.
Thus,
Total points earned with a perfect landing = 76 + 10 + 25 = 111 points
![\bf A=\cfrac{h(a+b)}{2}\quad \begin{cases} A=20\\ a=6z-1\\ b=2z+3\\ h=z \end{cases}\implies 20=\cfrac{z[(6z-1)~+~(2z+3)]}{2} \\\\\\ 20=\cfrac{z(8z+2)}{2}\implies 20=\cfrac{2z(4z+1)}{2}\implies 20=z(4z+1) \\\\\\ 20=4z^2+z\implies 0=4z^2+z-20](https://tex.z-dn.net/?f=%5Cbf%20A%3D%5Ccfrac%7Bh%28a%2Bb%29%7D%7B2%7D%5Cquad%20%0A%5Cbegin%7Bcases%7D%0AA%3D20%5C%5C%0Aa%3D6z-1%5C%5C%0Ab%3D2z%2B3%5C%5C%0Ah%3Dz%0A%5Cend%7Bcases%7D%5Cimplies%2020%3D%5Ccfrac%7Bz%5B%286z-1%29~%2B~%282z%2B3%29%5D%7D%7B2%7D%0A%5C%5C%5C%5C%5C%5C%0A20%3D%5Ccfrac%7Bz%288z%2B2%29%7D%7B2%7D%5Cimplies%2020%3D%5Ccfrac%7B2z%284z%2B1%29%7D%7B2%7D%5Cimplies%2020%3Dz%284z%2B1%29%0A%5C%5C%5C%5C%5C%5C%0A20%3D4z%5E2%2Bz%5Cimplies%200%3D4z%5E2%2Bz-20)
