Question

A skier leaves an 8-foot-tall ramp with an initial vertical velocity of 28 feet per second. the function h = −16t^2 28t 8 represents the height h (in feet) of the skier after t seconds. the skier has a perfect landing. how many points does the skier earn? 1 point per foot in the air, 5 points per second in the air, and a perfect landing is 25 points.

Answer 1

The skier earns approximately 27 points.

Step-by-step explanation:

To find the total points earned by the skier, we need to calculate the time it takes for the skier to reach the ground. In this case, the height function h = -16t^2 + 28t + 8 represents the height of the skier after t seconds. To find the time it takes for the skier to reach the ground, we need to find the value of t when h = 0.

Setting h = 0, we get:

-16t^2 + 28t + 8 = 0

Using the quadratic formula, t = (-b ± sqrt(b^2 - 4ac)) / (2a), where a = -16, b = 28, and c = 8:

t = (-28 ± sqrt(28^2 - 4(-16)(8))) / (2(-16))

Simplifying the equation, we get:

t = (-28 ± sqrt(672)) / -32

t = (-28 ± 26) / -32

Since the skier has a perfect landing, we can disregard the negative value and take the positive value of t:

t = (-28 + 26) / -32

t = -2 / -32

t = 1/16

This means the skier takes 1/16 seconds to reach the ground. To calculate the total points earned, we need to find the total time the skier is in the air. Since the skier starts with an initial vertical velocity of 28 ft/s, we can use this velocity to find the time it takes to reach the ground:

t = h / v

t = 8 / 28

t = 2/7 seconds

Therefore, the skier is in the air for 2/7 seconds. To find the total points earned:

Points = (h * 1) + (t * 5) + 25

Points = (8 * 1) + ((2/7) * 5) + 25

Points = 8 + (10/7) + 25

Points = 189/7

So, the skier earns approximately 27 points.

Answer 2

The skier earns 35.875 points.

We can find the height in the air by using -b/2a:
-28/2(-16) = -28/-32 = 0.875

This will give the skier 0.875 points.

To find the amount of time in the air, we solve the related equation:
0=-16t²+28t+8

We will first factor out the GCF, -4:
0=-4(4t²-7t-2)

Now we will factor the trinomial in parentheses using grouping. We want factors of 4(-2)=-8 that sum to -7; -8(1) = -8 and -8+1=-7. This is how we will "split up" bx:
0=-4(4t²-8t+1t-2)

Now we will group the first two and last two terms:
0=-4[(4t²-8t)+(1t-2)]

We will factor out the GCF of each group:
0=-4[4t(t-2)+1(t-2)]

This gives us the factored form:
0=-4(4t+1)(t-2)

Using the zero product property, we know that either t-2=0 or 4t+1=0:
t-2=0
t-2+2=0+2
t=2

4t+1=0
4t+1-1=0-1
4t=-1
4t/4 = -1/4
t=-1/4

Negative time makes no sense, so t=2. This gives the skier 5(2) = 10 points.

Counting the perfect landing, we have 25+10+0.875 = 35.875 points.