Calculate total change by integrating instantaneous or marginal rates of change, with and without technology.

A factory produces ceramic tiles. The rate of production, in tiles per hour, varies throughout an 8-hour shift according to the function \( r(t) = 45 + 12t - 0.8t^2 \), where \( t \) is the time in hours since the shift began, \( 0 \le t \le 8 \). Refer to the table below, which shows the rate of production at selected times during the shift. Calculate the total number of tiles produced during the 8-hour shift.

A manufacturer's marginal cost function is given by $\frac{dC}{dx} = 8x + 15$, where $x$ is the number of units produced and cost is in dollars. The fixed cost is $\$500$. a) Find the total cost function $C(x)$. (1 mark) b) Calculate the total cost of producing 20 units. (1 mark) c) Hence, find the additional cost of producing units 21 to 30 inclusive. (1 mark)

A manufacturer's production facility has a marginal cost function (in dollars per unit) given by $\(MC(x) = 0.06x^2 - 2x + 50\)$, where $x$ is the number of units produced. Calculate the total additional cost of increasing production from $100$ units to $150$ units. Give your answer to the nearest dollar.