Suppose the rate of oil imports to the United States from a certain country can

Question

Suppose the rate of oil imports to the United States from a certain country can be approximated by

r(t) = 5t2 + 40t + 900 million barrels per year (0 t 8)

where t is time in years since the start of 2000. During that time, the price of oil was approximately

p(t) = 25e^(0.1t) dollars per barrel

Obtain an expression for the total oil revenue R(x) the country earned from the United States since the start of 2000 to the start of year x as a function of x. (Do not simplify the answer.) HINT [Rate of revenue = p(t)r(t).]

Explanation / Answer

rate of oil imports to the United States from a certain country r(t) = (5t2 + 40t + 900)*106 barrels per year

price of oil was approximately p(t) = 25e(0.1t)

Rate of revenue R'(t)= p(t)r(t)

R'(t)= 25e(0.1t) (5t2 + 40t + 900)*106

R'(t)=  25*106(5t2 + 40t + 900)e(0.1t)

total oil revenue R(x) =[0 to x] 25*106(5t2 + 40t + 900)e(0.1t)dt

integration by parts : u =(5t2 + 40t + 900), dv=e(0.1t)dt, v=10e(0.1t),du=(10t + 40)dt

u dv=uv-v du

R(x) =25*106[[0 to x] (5t2 + 40t + 900)10e(0.1t)-[0 to x] 10e(0.1t)(10t + 40)dt]

R(x) =25*107[[0 to x] (5t2 + 40t + 900)e(0.1t)+[0 to x] (10t - 40)e(0.1t)dt]

R(x) =25*107[[0 to x] (5t2 + 40t + 900)e(0.1t)+[[0 to x] (10t - 40)10e(0.1t)-[0 to x]10e(0.1t)10dt]]

R(x) =25*107[[0 to x] (5t2 + 40t + 900)e(0.1t)+[0 to x] (100t - 400)e(0.1t)-[0 to x]1000e(0.1t)]

R(x) =25*107[0 to x]e(0.1t)[5t2 + 40t + 900+100t - 400-1000]

R(x) =25*107[0 to x]e(0.1t)(5t2 + 140t - 500)

R(x) =25*107[e(0.1x)(5x2 + 140x - 500) +500] dollars

R(x) =250[e(0.1x)(5x2 + 140x - 500) +500] milliondollars

Related Questions

Navigate

• Browse (All)
• Browse S
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.