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Question

A snowboarder is preparing for the 2024 Olympics by practicing his tricks. His height relative to the top of the halfpipe on one of his tricks can be represented by the equation 

      h(t)=0.1t^2 - 3.4t +10

Where h represents the height in metres from the top of the halfpipe, and t represents the time in seconds.

1. How long will it be until he reaches the exact bottom of the halfpipe? Explain how you know this answer is true.

b) Using the quadratic formula, determine how long his snowboard is touching the halfpipe between arial tricks (to 2 decimal places). Explain how you know.

Answer 1

a) To find the time it takes for the snowboarder to reach the exact bottom of the halfpipe, we need to find the value of t when h(t) = 0. The equation for the height of the snowboarder is:

h(t) = 0.1t^2 - 3.4t + 10

Setting h(t) = 0, we get:

0 = 0.1t^2 - 3.4t + 10

This is a quadratic equation that can be solved using the quadratic formula:

t = (-b ± sqrt(b^2 - 4ac)) / 2a

where a = 0.1, b = -3.4, and c = 10. Plugging in these values, we get:

t = (-(-3.4) ± sqrt((-3.4)^2 - 4(0.1)(10))) / 2(0.1) t = (3.4 ± sqrt(11.24)) / 0.2

Simplifying, we get:

t ≈ 7.47 seconds or t ≈ 2.53 seconds

The snowboarder reaches the exact bottom of the halfpipe twice during each jump - once on the way down and once on the way up. The first solution t ≈ 7.47 seconds is the time when the snowboarder reaches the bottom on the way down, while the second solution t ≈ 2.53 seconds is the time when the snowboarder reaches the bottom on the way up. Therefore, it will take the snowboarder 7.47 seconds to reach the exact bottom of the halfpipe.

b) To find how long his snowboard is touching the halfpipe between aerial tricks, we need to find the roots of the equation h(t) = 0. The time when the snowboarder takes off and lands will be when h(t) = 0, so the length of time his snowboard is touching the halfpipe is the time between these two roots.

Using the quadratic formula, we find the roots of the equation:

t = (-b ± sqrt(b^2 - 4ac)) / 2a

where a = 0.1, b = -3.4, and c = 10. Plugging in these values, we get:

t = (-(-3.4) ± sqrt((-3.4)^2 - 4(0.1)(10))) / 2(0.1) t = (3.4 ± sqrt(11.24)) / 0.2

Simplifying, we get:

t ≈ 7.47 seconds or t ≈ 2.53 seconds

Since the snowboarder takes off and lands at the bottom of the halfpipe, the length of time his snowboard is touching the halfpipe is the difference between these two roots:

t = 7.47 seconds - 2.53 seconds t ≈ 4.94 seconds



Therefore, the snowboarder's snowboard is touching the halfpipe for approximately 4.94 seconds between aerial tricks.