$1.$ Use the graph below to answer the questions:

a$.$ State the equation for the demand curve $($inverse demand function$)$ shown in the graph above using the format Px $=$ a $$ bQx: $1$ point

b$.$ State the equation for the demand function implied in the graph using the format Qx $=$ c $$ dPx:

$1$ point

c$.$ Find the equation for Total Revenue, where TR is a function of output $($Qx$)$: $1$ point

d$.$ Find the equation for Marginal Revenue, where MR is a function of output $($Qx$)$: $1$ point

e$.$ If MC $=$ $$4,$ find the output level that maximizes Total Revenue: $1$ point

f$.$ If MC $=4,$ find the output level that maximizes Total Profit: $1$ point

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$2.$ Suppose that you have been hired as a consultant to a firm that has estimated its demand curve to be P $=600-3$Q$,$ where P denotes price in dollars and Q denotes total unit sales. The firm$$s cost equation is TC $=3,200+50$Q $+2$Q$2.$

a$.$ Derive the equations for marginal revenue and marginal cost. $2$ points

b$.$ Determine the dollar values for Marginal Revenue $($MR$),$ and Marginal Cost $($MC$)$ if the firm is currently producing at an output level $($Q$)$ of $75$ units. $2$ points

c$.$ Provide a recommendation to this firm if they should increase, decrease, or keep the quantity at Q $=75$ to maximize its profit $($you must motivate your answer$)?1$ point

d$.$ How should this firm change $($increase$,$ decrease, or keep the same$)$ its profit maximizing quantity compared to your finding in part c if the firm$$s fixed cost changes from $3,200$ to $3,800$ so TC $=3,800+50$Q $+2$Q$2($you must motivate your answer$)?1$ point

e$.$ How should this firm change $($increase$,$ decrease, or keep the $)$ its profit maximizing quantity compared to your finding in part c if the firm$$s variable cost changes so TC $=3,200+80$Q $+2$Q$2($you must motivate your answer$)?1$ point

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$3.$ Swedish Automobiles Are the Best $($SAAB$)$ sells two types of vehicles: a truck and a car. The monthly inverse demand or price equation for trucks is estimated as Pt $=65-0\mathrm{.}05$Qt $($where Qt is the number of trucks sold and Pt is the price per truck in thousands$).$ For cars is: Pc $=55-0\mathrm{.}1$Qc$.$ It costs SAAB $$27($thousand$)$ to produce each additional truck and $$21($thousand$)$ to produce each additional car, so MCt $=27$ and MCc $=21.$ Assume no fixed cost so TC $=$ MC $\backslash$times Q$.$

a$.$ State the total profit equation AND marginal profit for trucks: $2$ points

b$.$ Determine the profit$-$maximizing output for trucks $($Qt$)$ and compute the total profit that SAAB will earn from selling this quantity of trucks: $2$ points

c$.$ State the total profit equation AND marginal profit for cars: $2$ points

d$.$ Determine the profit$-$maximizing output for cars $($Qc$)$ and compute the total profit that SAAB will earn from selling this quantity of cars: $2$ points

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Swedish Automobiles Are the Best $($SAAB$)$ sells two types of vehicles: a truck and a car. The monthly inverse demand or price equation for trucks is estimated as Pt $=65-0\mathrm{.}05$Qt $($where Qt is the number of trucks sold and Pt is the price per truck in thousands$).$ For cars is: Pc $=55-0\mathrm{.}1$Qc$.$ It costs SAAB $$27($thousand$)$ to produce each additional truck and $$21($thousand$)$ to produce each additional car, so MCt $=27$ and MCc $=21.$ Assume no fixed cost so TC $=$ MC $\backslash$times Q$.$

e$.$ Suppose now that SAAB$$s production capacity is limited to $400$ vehicles per month. Determine the profit$-$maximizing quantities of trucks $($Qt$)$ and cars $($Qc$)$ and the total profit for each vehicle type with the new optimal quantities caused by the constraint. $4$ points