A company produces and sells electric cars. The company's profit, P, in thousands of dollars, changes based on the number of cars, x, they produce per month. The rate of change of their profit from producing x electric cars is modelled by $\frac{dP}{dx} = -1.6x + 48, x ≥ 0$ The company makes a profit of 260 (thousand dollars) when they produce 15 electric cars.

recognition of need to integrate (eg reverse power rule or integral symbol) (M1) $P(x)=-0.8x^2+48x(+c)$ (A1) $260=-0.8\times(15)^2+48\times(15)+c$ (M1) Note: Award $\boldsymbol{M1}$ for correct substitution of $x=15$ and $P=260$. A constant of integration must be seen (can be implied by a correct answer). $c=-280$ (A1) $P(x)=-0.8x^2+48x-280$ (A1)

profit will decrease (with each new car produced) (A1)

EITHER

because the profit function is decreasing / the gradient is negative / the rate of change of $P$ is negative (R1)

OR

$\int_{30}^{50}-1.6x+48(\mathrm{d}x)=-320$ (R1)

OR

evidence of finding $P(30)=440$ and $P(50)=120$ (R1)

Note: Award at most R1AO if $P(30)$ or $P(50)$ or both have incorrect values.