The monthly sales of Sunny Electronics\' new sound system are given by q(t) = 6,

Question

The monthly sales of Sunny Electronics' new sound system are given by q(t) = 6,000t 90t2 units per month, t months after its introduction. The price Sunny charges is p(t) = 3,000 t2 dollars per sound system, t months after introduction. Find the rate of change of monthly sales. HINT [See Example 4(a).]

q'(t) =_______

Find the rate of change of the price. p'(t) = ______

Find the rate of change of monthly revenue 5 months after the introduction of the sound system. $________ per month Interpret your answer.

When t =______ , the revenue is increasing at a rate of $______ per month.

Explanation / Answer

q'(t) = d/dt(6,000t 90t2) ==> 6,000 - 2t (90) ==> 6,000 - 180t

p'(t) = d/dt( 3,000 t2) ==> 0-2t ==> -2t

q'(5) ==>  6,000 - 180(5) ==> 5100 units/month.Therefore, the rate of change of sales 5 months after the introduction of the system is 5100 units /month i.e. the sale of the system is increased by 5100 units after 5 months.

p'(5) = -2(5) = -10 /month.The price of the sound system is dropping at a rate of $10 per month

Therefore, the rate of change of price 5 months after the introduction of the system is

-$10/month i.e. the price per system is decreased by 10 dollars after 5 months.

Revenue = quantity x price per system

Revenue R(t)  = q(t) x p(t) t months after the introduction of the system.

Therefore R(t) = (6000t – 90t2)( 3000 – t2). To find the rate of change of the revenue we need to find R’(t). We will use product rule to calculate that.

R'(t) =  d/dt((6000t – 90t2)( 3000 – t2))

==>(6000t – 90t2) d/dt( 3000 – t2) + ( 3000 – t2) d/dt(6000t – 90t2)

R'(t)====> (6000t – 90t2)(-2t) +( 3000 – t2) (6,000 - 180t)

R'(5)==> (6000(5) – 90(52))(-2(5)) +( 3000 – 52) (6,000 - 180(5))

R'(5) ==> 14781600

Therefore, the rate of change of the revenue 5 months after the introduction is $14781600.00.

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