To find the rate of change \( \frac{dP}{dt} \) of \( P \) with respect to \( t \), you'll need to differentiate the expression for \( P \) with respect to \( t \). Given \( P(t) = \frac{t + 1750}{30(t + 2)} \) To differentiate the function, we'll use the quotient rule: \[ \frac{d}{dx} \left( \frac{u(x)}{v(x)} ight) = \frac{u'v - uv'}{v^2} \] Where \( u(x) \) is the numerator and \( v(x) \) is the denominator. Let \( u(t) = t + 1750 \) and \( v(t) = 30(t + 2) \). Differentiating: \[ u'(t) = 1 \] \[ v'(t) = 30 \] Using the quotient rule: \[ \frac{dP}{dt} = \frac{u'v - uv'}{v^2} \] \[ = \frac{(1)(30(t + 2)) - (t + 1750)(30)}{[30(t + 2)]^2} \] \[ = \frac{30t + 60 - 30t - 52500}{900(t^2 + 4t + 4)} \] \[ = \frac{-52440}{900(t^2 + 4t + 4)} \] \[ = \frac{-58.27}{t^2 + 4t + 4} \] (Rounded to two decimal places) Now, for: (a) \( t = 1 \): \[ \frac{dP}{dt} = \frac{-58.27}{1^2 + 4(1) + 4} \] \[ = \frac{-58.27}{9} \] \[ \approx -6.47 \] (Rounded to two decimal places) So, the rate of change of \( P \) with respect to \( t \) when \( t = 1 \) is \( -6.47 \)