Question

Consumers are spending more on music from digital sources than they do on CDs. Purchases of streamed music from services such as Pandora and Spotify are projected to grow in accordance with the model f(t) = 3.7 + 0.84 ln(t + 1) (0 ≤ t ≤ 5) where f(t) is measured in billions of dollars and t is measured in years, with t = 0 corresponding to 2012.
(ai) What was the amount spent by consumers on streamed music in 2012?



$__________ billion



(aii) What was the amount spent by consumers on streamed music in 2014? (Round your answer to one decimal place.)



$__________ billion



(b) How fast was the amount spent by consumers on streamed music growing in 2014? (Round your answer to one decimal place.)



$__________ billion/year

Answer 1

Answer:

(a) (i) 3.7 billion

(ii) $ 4.6 billion

(b) $ 0.3 billion/year

Step-by-step explanation:

Given equation that shows the amount( in billions of dollar ) spent by consumers on streamed music after t years since 2012,

f(t) = 3.7 + 0.84 ln(t + 1) ,

(a)

(i) for 2012, t = 0,

So, the amount spent by consumers on streamed music in 2012,

f(0) = 3.7 + 0.84 ln (1)

= 3.7 + 0.84(0)

= $ 3.7 billion

(ii) for 2014, t = 2 years,

So, the amount spent by consumers on streamed music in 2014,

f(0) = 3.7 + 0.84 ln (1+2)

= 3.7 + 0.84 ln(3)

= 3.7 + 0.9228

= 4.6228 billion

≈ $ 4.6 billion

(b) ∵ f(t) = 3.7 + 0.84 ln(t + 1),

Differentiating with respect to t,

$f'(t) = 0.84 \frac{1}{t+1}$

For t = 2 years,

$f'(2) = 0.84 \frac{1}{2+1}=\frac{0.84}{3}=0.28\approx 0.3$

Hence, the amount spent by consumers on streamed music is growing with the rate of $ 0.3 billion/year in 2014