You need to evaluate the minimum value of the given function `h(t), ` hence, you need to solve for t the equation `h'(t) = 0` .

You need to differentiate the given function with respect to t, using the chain rule, such that:

`h'(t) = (65 sin(12(t- 7.5)) + 70)'`

`h'(t)...

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You need to evaluate the minimum value of the given function `h(t), ` hence, you need to solve for t the equation `h'(t) = 0` .

You need to differentiate the given function with respect to t, using the chain rule, such that:

`h'(t) = (65 sin(12(t- 7.5)) + 70)'`

`h'(t) = 65*cos(12(t - 7.5))*(12(t - 7.5))' + 0`

`h'(t) = 65*12*cos(12(t - 7.5))`

You need to solve for t the equation `h'(t) = 0` , such that:

`65*12*cos(12(t - 7.5)) = 0`

Dividing by `65*12` yields:

`cos(12(t - 7.5)) = 0 => 12(t - 7.5) = cos^(-1)(0) + 2n*pi`

Substituting `pi/2` for `cos^(-1)(0)` yields:

`12(t - 7.5) = pi/2 + 2n*pi => t - 7.5 = pi/24 + n*pi/6`

`t = pi/24 + n*pi/6 + 7.5`

Substituting `(3pi)/2` for `cos^(-1)(0)` yields:

`12(t - 7.5) = (3pi)/2 + 2n*pi => t - 7.5 = pi/8 + n*pi/6`

`t = pi/8 + n*pi/6 + 7.5`

**Hence, evaluating the points where the function `h(t) ` reaches its minimum point yields that `t = pi/8 + n*pi/6 + 7.5.` **