g. the height in meters of a bullet fired into the air after t seconds its given by h(t) = 2000t - 200t²…

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.\nh. 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.

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.\nh. 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.

Answer

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.