Welcome to the article The function shown represents the height h in feet of a firework t seconds after it is launched The firework explodes at its highest point. On this page, you will learn the essential and logical steps to better understand the topic being discussed. We hope the information provided helps you gain valuable insights and is easy to follow. Let’s begin the discussion!
Answer :
Final answer:
The firework explodes at t = 0 seconds and t = 4 seconds. The height at which the firework explodes is 768 feet.
Explanation:
To determine when the firework explodes, we need to find the time when the height reaches its maximum. The highest point is reached when the vertical velocity (Vy) is equal to zero. By using the equation v² = V₀y - 2g(y - y₀), where v is the vertical velocity, V₀y is the initial vertical velocity, g is the acceleration due to gravity, y is the height, and y₀ is the initial height, we can solve for y.
By substituting the given equation h = 16t² + 128t for y and simplifying, we get 0 = -32t² + 128t. We can solve for t by factoring and setting each factor to zero. Therefore, the firework explodes at t = 0 seconds and t = 4 seconds.
To find the height at which the firework explodes, we can substitute the time t into the given equation h = 16t² + 128t. Thus, the height at which the firework explodes is h = 16(4)² + 128(4) = 256 + 512 = 768 feet.
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Rewritten by : Brahmana
Final answer:
The firework explodes 4 seconds after it is launched at a height of 768 feet.
Explanation:
The problem given is a quadratic function [tex]h = 16t2 + 128t[/tex]which represents the height in feet (h) of a firework t seconds after it is launched. The firework explodes at its maximum height which corresponds to the maximum point of this quadratic function.
To find the time when the firework explodes, we set the derivative of h with respect to t to zero. The derivative of h with respect to t is 32t + 128. Setting this to zero and solving for t, we have [tex]t = -128/32 = -4.[/tex] Since time can't be negative, we discard this solution. Hence, the firework explodes at 4 seconds.
To find the height at which the firework explodes, we substitute t = 4 into the function h. So [tex]h = 16*(4)2 + 128*4 = 256 + 512 = 768[/tex] feet. Therefore, the firework explodes at a height of 768 feet.
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