Let's solve the problem step by step from the image you provided. **Given Formula:** \[ H = 28.1 + 3.86t - 0.207t^2 \] where: - \( H \) = head circumference (cm) - \( t \) = age (in weeks) --- ### (a) Calculate the rate of head growth (dH/dt). The rate of head growth is the derivative of \( H \) with respect to \( t \): \[ \frac{dH}{dt} = \frac{d}{dt}[28.1 + 3.86t - 0.207t^2] \] - The derivative of 28.1 is 0 (since it's a constant). - The derivative of \( 3.86t \) is \( 3.86 \). - The derivative of \( -0.207t^2 \) is \( -2 \times 0.207t = -0.414t \). So, \[ \frac{dH}{dt} = 3.86 - 0.414t \] --- ### (b) How many cm per week is the head growing at \( t = 8 \) weeks and \( t = 16 \) weeks? Plug in the values for \( t \): #### At \( t = 8 \) weeks: \[ \frac{dH}{dt} = 3.86 - 0.414 \times 8 = 3.86 - 3.312 = 0.548\, \text{cm/week} \] #### At \( t = 16 \) weeks: \[ \frac{dH}{dt} = 3.86 - 0.414 \times 16 = 3.86 - 6.624 = -2.764\, \text{cm/week} \] --- ### (c) Repeat part (b) but for instantaneous rate of growth at \( t = 24 \) weeks. \[ \frac{dH}{dt} = 3.86 - 0.414 \times 24 = 3.86 - 9.936 = -6.076\, \text{cm/week} \] --- ## **Summary of Answers** **(a)** \[ \frac{dH}{dt} = 3.86 - 0.414t \] **(b)** - At \( t = 8 \) weeks: \( 0.548\, \text{cm/week} \) - At \( t = 16 \) weeks: \( -2.764\, \text{cm/week} \) **(c)** - At \( t = 24 \) weeks: \( -6.076\, \text{cm/week} \) --- **(Negative values indicate a decrease in head circumference, which may not be physiologically plausible for this context, but mathematically that's the answer!)** Let me know if you need further clarification!