# The manager of a small furniture manufacturing company has agreed to deliver an order of up to...

## Question:

The manager of a small furniture manufacturing company has agreed to deliver an order of up to 600 chairs to a customer. If the size of the order is 400 chairs or less, the price per chair will be $60. If more than 400 chairs are ordered, then the price per chair will be discounted by $0.20 for each chair over 400 ordered (the discounted price applies to the entire order). Let {eq}R(x){/eq} denote the revenue from an order of {eq}x{/eq} chairs.

a) Calculate {eq}R(320){/eq} and {eq}R'(320){/eq}. Label each appropriately.

b) Calculate {eq}R(450){/eq} and {eq}R'(450){/eq}. Label each appropriately.

## Rate of Change:

Rate of change of a mathematical function is computed by taking the derivative of the function. This gives the slope of a tangent line to the curve, which defines the rate of change.

## Answer and Explanation:

a) Calculate {eq}R(320) {/eq} and {eq}R'(320) {/eq}.

If the order is 400 chairs or less, then the price per chair will be $60 and the revenue in dollars is

{eq}R(x) = 60x {/eq}.

So {eq}R(320) = 60(320) = 19200 {/eq}

and {eq}R'(x) = 60 {/eq}

Answer: {eq}\boxed{R(320) = 19200} {/eq} dollars and {eq}\boxed{R'(320) = 60} {/eq} dollars/chair.

b) Calculate {eq}R(450) {/eq} and {eq}R'(450) {/eq}.

If the order is 400 chairs or more, then the price per chair will be $60 minus $0.20 for each chair over 400, and the revenue in dollars is

{eq}R(x) = (60-(0.20(x-400)))x = 60x - 0.2x^2 +80x =140x - 0.2x^2 {/eq}.

So {eq}R(450) = 140(450) - 0.2(450)^2 = 22500 {/eq}

and {eq}R'(x) = 140 - 0.4x {/eq}

{eq}R'(450) = 140 - 0.4(450) = -40 {/eq}

Answer: {eq}\boxed{R(450) = 22500} {/eq} dollars and {eq}\boxed{R'(450) = -40} {/eq} dollars/chair.

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