Question

[HELP, VERY IMPORTANT]The Baker family started a farm in 1900, and the farm has been passed down from generation to generation. The Bakers primarily produce corn and potatoes on their farm. The following equation models the number of pounds of corn they produced each year t years after 1900.

C(t) = 25,000 + 500 ln(t + 1)

The graph below models the number of pounds of potatoes they produced each year t years after 1900.

Which of the following statements best describes the quantity of corn and potatoes produced as time passed?

A.) The Baker family has always and will always produce more potatoes than corn.

B.) The Baker family has always and will always produce more corn than potatoes.

C.) The Baker family started off producing more corn than potatoes, but as time passed, they eventually produced more potatoes than corn.

D.) The Baker family started off producing more potatoes than corn, but as time passed, they eventually produced more corn than potatoes.

Answer 1

Looking first at the graph, we see that the intial potato production was 25000 lb, and that potato production tapered off to below 20000 lb over time.

Analyzing the equation for corn production: C(t) = 25,000 + 500 ln(t + 1), we see

that the initial production was 25000 lb per year and that this increases steadily with time.

Thus, answer A is false.

Answer B is false because at the outset the amounts of corn and potatoes was equal (25000 lb).

Answer D makes sense. Productions were both 25000 lb at the beginning, but the corn production rose steadily over time, exceeding the potato production.

Answer 2

D) The Baker family started off producing more potatoes than corn, but as time passed, they eventually produced more corn than potatoes.

Explanation:

Since, the function that represents the quantity ( in pounds ) of corn produced after t years since 1900,

C(t) = 25,000 + 500 ln(t + 1)

Since, initially,

t = 0,

So, the quantity of corn produced,

C(0) = 25000,

After 10 year, the quantity is,

C(1) = 25000 + 500 ln(10+1) ≈ 26198.95 pounds,

After 2 years, the quantity is,

C(2) = 25000 + 500 ln(20+1) ≈ 26522.26 pounds,

After 3 year, the quantity is,

C(3) = 25000 + 500 ln (3+1) ≈ 26716.99 pounds,

..........., so, on.....

Now, by the given graph,

The initial quantity of the potato is,

P(0) = 30,000 pounds,

After 10 years, the quantity of the potato is,

P(1) ≈ 27,000,

After 20 years, the quantity of the potato is,

P(2) ≈ 25,500,

After 30 years, the quantity of the potato is,

P(3) ≈ 24,750,

...... so, on,......

Hence, from the above explanation it is clear that,

The Baker family started off producing more potatoes than corn, but as time passed, they eventually produced more corn than potatoes.

Option D is correct.