Question

The price of crude oil during the period 2000-2010 can be approximated by P(t) = 6t + 18 dollars per barrel (0 ≤ t ≤ 10) in year t, where t = 0 represents 2000. Saudi Arabia's crude oil production over the same period can be approximated by Q(t) = −0.036t^2 + 0.62t + 8 million barrels per day (0 ≤ t ≤ 10). Use these models to estimate Saudi Arabia's daily oil revenue and also its rate of change in 2007. (Round your answers to the nearest $1 million.)

A) Daily oil revenue in 2007: $306 million, Rate of change in 2007: $24 million/yr
B) Daily oil revenue in 2007: $240 million, Rate of change in 2007: $18 million/yr
C) Daily oil revenue in 2007: $312 million, Rate of change in 2007: $26 million/yr
D) Daily oil revenue in 2007: $360 million, Rate of change in 2007: $28 million/yr

Answer 1

To estimate Saudi Arabia's daily oil revenue in 2007, we need to calculate the production (Q) for that year and then multiply it by the price per barrel (P). The production (Q) for 2007 can be determined by substituting t = 7 into the equation Q(t) = -0.036t^2 + 0.62t + 8. The rate of change in 2007 can be found by taking the derivative of Q(t) with respect to t and then evaluating it at t = 7.

Step-by-step explanation:

To estimate Saudi Arabia's daily oil revenue in 2007, we need to calculate the production (Q) for that year and then multiply it by the price per barrel (P). The production (Q) for 2007 can be determined by substituting t = 7 into the equation Q(t) = -0.036t^2 + 0.62t + 8. Once we have the production for 2007, we can multiply it by the price per barrel (P) for 2007, which can be calculated by substituting t = 7 into the equation P(t) = 6t + 18. The rate of change in 2007 can be found by taking the derivative of Q(t) with respect to t and then evaluating it at t = 7.

Substituting t = 7 into the equation Q(t) = -0.036t^2 + 0.62t + 8, we get:

Q(7) = -0.036(7)^2 + 0.62(7) + 8

Q(7) = -0.176

To calculate the price per barrel (P) for 2007, we substitute t = 7 into the equation P(t) = 6t + 18:

P(7) = 6(7) + 18

P(7) = 60

Now we can calculate the daily oil revenue in 2007 by multiplying the production (Q) and the price per barrel (P):

Daily oil revenue in 2007 = Q(7) * P(7) = -0.176 * 60 = $10.56 million

To find the rate of change in 2007, we take the derivative of Q(t) with respect to t, which gives us:

Q'(t) = -0.072t + 0.62

Substituting t = 7 into the derivative equation, we get:

Q'(7) = -0.072(7) + 0.62

Q'(7) = -0.504 + 0.62

Q'(7) = 0.116

Therefore, the daily oil revenue in 2007 is $10.56 million and the rate of change in 2007 is $0.116 million/year.