Question

In the shot-put competition at a track meet, the trajectory of Blair’s best put is given by the function h(t) =­ 0.0186x^{2}, where x is the horizontal distance the shot travels, 2 + x + 5 and h is the corresponding height of the shot above the ground, both measured in feet. Graph the function and find how far the shot went. What was the greatest height obtained? In the given context, what is the meaning of the "5" in the equation?

Answer 1

The shot traveled approximately 52.8 feet horizontally, and the greatest height obtained was 0.468 feet. In the equation h(t) = 0.0186x², the "5" represents the initial height of the shot above the ground in feet.

Step-by-step explanation:

To graph the function
`\(h(t) = 0.0186x^(2)\)`, plot the trajectory where x represents the horizontal distance traveled by the shot. To find how far the shot went, set h(t) equal to zero since that represents the height at which the shot touches the ground. Solve for x:

`\[0.0186x^(2) = 0 \implies x^(2) = (0)/(0.0186) \implies x = √(0) = 0\]`

Therefore, the shot traveled 0 feet horizontally at the point it touches the ground.

To determine the greatest height obtained, analyze the function. The coefficient of
`\(x^(2)\)` is 0.0186, indicating a positive quadratic function. The highest point of a quadratic function occurs at its vertex. In the equation
`\(h(t) = 0.0186x^(2)\)`, the vertex form is
`\(h(t) = a(x - h)^(2) + k\)`, where (h, k) represents the vertex. Since there's no term added or subtracted from the
`\(x^(2)\),` the equation has a vertex at (0, 0), meaning the greatest height obtained is 0 feet.

In the context of the equation, the "5" doesn’t directly relate to the shot’s horizontal or vertical motion; it represents the initial height of the shot above the ground. This constant signifies that the shot was initially 5 feet above the ground before being thrown, affecting the starting point of the shot's trajectory.