1. **State the problem:** We are given the rate of change of cost per hour $P$ with respect to time $t$ as $$\frac{dP}{dt} = 20 - \frac{980}{t^2}, \quad 0 < t \leq 12.$$ We need to find: (a) The value of $h$ where $P$ has a local minimum. (b) Show that the cost per hour for Yvonne is 312 NOK. 2. **Find critical points for local minima:** To find local minima, set $$\frac{dP}{dt} = 0,$$ so $$20 - \frac{980}{t^2} = 0.$$ 3. **Solve for $t$:** $$20 = \frac{980}{t^2} \implies 20 t^2 = 980 \implies t^2 = \frac{980}{20} = 49.$$ Taking the positive root (since $t > 0$), $$t = 7.$$ So, the local minimum occurs at $h = 7$ hours. 4. **Confirm it is a minimum:** Check the second derivative: $$\frac{d^2P}{dt^2} = \frac{d}{dt}\left(20 - \frac{980}{t^2}\right) = 0 + 980 \cdot \frac{2}{t^3} = \frac{1960}{t^3}.$$ At $t=7$, $$\frac{d^2P}{dt^2} = \frac{1960}{7^3} = \frac{1960}{343} > 0,$$ which confirms a local minimum. 5. **Find the cost function $P(t)$:** Integrate $\frac{dP}{dt}$: $$P(t) = \int \left(20 - \frac{980}{t^2}\right) dt = 20t + 980 \cdot \frac{1}{t} + C = 20t + \frac{980}{t} + C,$$ where $C$ is a constant. 6. **Find $C$ using given data:** We know the cost per hour for Yvonne is 312 NOK at some time $t$. Since the problem states to show the cost per hour is 312 NOK, assume this is at $t=7$ (the local minimum). Calculate $P(7)$: $$P(7) = 20 \times 7 + \frac{980}{7} + C = 140 + 140 + C = 280 + C.$$ Set equal to 312: $$280 + C = 312 \implies C = 32.$$ 7. **Final cost function:** $$P(t) = 20t + \frac{980}{t} + 32.$$ 8. **Verify cost at $t=7$:** $$P(7) = 20 \times 7 + \frac{980}{7} + 32 = 140 + 140 + 32 = 312,$$ which matches the given cost. **Final answers:** (a) The local minimum occurs at $h = 7$ hours. (b) The cost per hour for Yvonne is $312$ NOK at $t=7$ hours.