Question

g. the height in meters of a bullet fired into the air after t seconds its given by h(t) = 2000t - 200t². find the maximum height reached by the bullet.
h. the curve traced by a point on a circle as it rolls on a straight line has a parametric equations x = θ - sinθ, y = 1 - cosθ. find \\( \frac { d ^ { 2 } y } { d x ^ { 2 } } \\) in terms of θ.

Explanation:

Step1: Find the first - derivative of \(h(t)\)

The function is \(h(t)=2000t - 200t^{2}\).
Using the power rule \((x^{n})^\prime=nx^{n - 1}\), we have \(h^\prime(t)=\frac{d}{dt}(2000t)-\frac{d}{dt}(200t^{2})\).
\(h^\prime(t)=2000-400t\).

Step2: Set the first - derivative equal to zero to find the critical points

Set \(h^\prime(t) = 0\), so \(2000-400t=0\).
Solve for \(t\):
\(400t=2000\), then \(t = 5\).

Step3: Find the second - derivative of \(h(t)\)

Differentiate \(h^\prime(t)=2000 - 400t\) with respect to \(t\).
Using the power rule, \(h^{\prime\prime}(t)=\frac{d}{dt}(2000)-\frac{d}{dt}(400t)=-400\lt0\). Since \(h^{\prime\prime}(t)\lt0\) when \(t = 5\), the function \(h(t)\) has a maximum at \(t = 5\).

Step4: Substitute \(t = 5\) into the original function \(h(t)\)

\(h(5)=2000\times5-200\times5^{2}\).
First, calculate \(200\times5^{2}=200\times25 = 5000\) and \(2000\times5=10000\).
Then \(h(5)=10000 - 5000=5000\).

Answer:

The maximum height reached by the bullet is \(5000\) meters.