Oil is pumped continuously from a well at a rate proportional to the amount of oil left in the well. Initially there were 5 million barrels of oil in the well; six years later 2,500,000 barrels remain.

(A):Let Q(t) be the number of barrels left in the well after t years, measured in millions. Write a differential equation for Q that captures the information in the problem. Use k>0 for any proportionality constant you need. Be careful about signs.

Q'=

(B):Solve the above differential equation, without yet determining k.

Q(t)=

(C):Determine k.

k=

(D):At what rate was the amount of oil in the well decreasing when there were 3,000,000 barrels remaining? [Hint: Use the equation in (a). Be careful about units.]

rate= barrels/year

(E):When will there be 250,000 barrels remaining?

years=

Step-by-step explanation:

From the information we find that

Q'(t) = kQ(t)

with Q(0) = 5 and Q(6) = 2.5 in millions

A) Separate the variables and solve the DE

Using Q(0) = 5

A =5

B)

C)

Using Q(6) = 2.5

D) Use Q'(t) = -0.1155 (3) = -0.3465

Rate = -346500 barrels

(negative indicates decrease in number of barrels_

E) When Q = 250000 we have

After 26 years

x = central angle apc = measure of minor arc ac

the two angles x and 30 add to 180, so x+30 = 180 and x = 180-30 = 150. the measure of minor arc ac is 150 degrees.

what we need to do from here is find the circumference and then multiply it by x/360 since there are 360 degrees in a circle. in this case, x = 150, so x/360 = 150/360 = 15/36 = 5/12. the shortest distance along the curve from a to c is 5/12 of the distance around the entire circle (aka circumference)

c = circumference

c = pi*d

c = pi*24

c = 24pi

now multiply this by 5/12 to get

distance along minor arc ac = (5/12)*(circumference)

distance along minor arc ac = (5/12)*(24pi)

distance along minor arc ac = 10pi

the formula you can use is

l = (x/360)*pi*d

with

l = length of minor arc

x = central angle

d = diameter

which essentially is the steps we took though a bit differently.

answer: all the values are within because it has a pattern that repeats from -6 to 4.

good luck!