Math / Data & Statistics

QuestionGuidelines for the amount of supplementary nitrogen needed to grow winter wheat depend on the amount of nitrogen in the soil (as determined by a soil test), the price of fertilizer, and the price of wheat at harvest. Suppose the soil on a particular farm has a nitrogen content of 2 ppm (parts per million) and 50 acres will be planted in winter wheat. Consider two pricing scenarios.
Case A: The price of fertilizer is $\$ 0.20 / \mathrm{lb}$ , the price of wheat is $\$ 3.50 / \mathrm{bushel}$ , and the expected yield is 60 bushels/acre. Case B: The price of fertilizer is $\$ 0.45 / \mathrm{lb}$ , the price of wheat is $\$ 5.50 /$ bushel, and the expected yield is 40 bushels/acre. In Case A, the guidelines recommend adding 90 pounds of nitrogen per acre, and in Case B, 70 pounds of nitrogen per acre. Assuming all other factors are equal, comput and compare the net profits (income minus expenses) for the two scenarios.
Select the correct choice below and fill in the answer box(es) to complete your choice. (Round to the nearest whole number as needed.) A. The net profit of Case $A$ is $\$$ $\square$ which is less than the net profit of $\$$ $\square$ in Case B. B. The net profit of Case A is $\$$ $\qquad$ which is the same as the net profit in Case B. C. The net profit of Case A is $\$$ $\square$ which is greater than the net profit of $\$$ $\square$ in Case B.

Studdy Solution

STEP 1

What is this asking? We need to calculate the profit from growing wheat in two different scenarios, where the cost of fertilizer and the price of wheat are different, and then compare the profits. Watch out! Don't mix up the units – we're dealing with pounds of nitrogen, bushels of wheat, and acres of land.
Also, remember profit is income minus expenses!

STEP 2

1. Calculate Case A profit
2. Calculate Case B profit
3. Compare profits

STEP 3

Alright, in Case A, we expect to yield 60 bushels/acre and we're planting 50 acres.
So, we'll have a total of $60 \cdot 50 = \textbf{3000}$ bushels of wheat.
Each bushel sells for $\$\textbf{3.50}$ , so our total income is $3000 \cdot \$3.50 = \$\textbf{10,500}$ .
Cha-ching!

STEP 4

We're adding 90 pounds of nitrogen per acre to our 50 acres, which means we need $90 \cdot 50 = \textbf{4500}$ pounds of nitrogen.
With the fertilizer costing $\$\textbf{0.20}$ per pound, our total expense is $4500 \cdot \$0.20 = \$\textbf{900}$ .

STEP 5

Time for the good stuff!
Profit is income minus expenses.
So, for Case A, it's $\$10,500 - \$900 = \$\textbf{9,600}$ .
Not bad!

STEP 6

In Case B, we're expecting $\textbf{40}$ bushels per acre on our $\textbf{50}$ acres, giving us a total of $40 \cdot 50 = \textbf{2000}$ bushels.
The price per bushel is $\$\textbf{5.50}$ , making our total income $2000 \cdot \$5.50 = \$\textbf{11,000}$ .
Even more cha-ching!

STEP 7

We need $\textbf{70}$ pounds of nitrogen per acre on our $\textbf{50}$ acres, totaling $70 \cdot 50 = \textbf{3500}$ pounds.
The fertilizer now costs $\$\textbf{0.45}$ per pound, so our total expense is $3500 \cdot \$0.45 = \$\textbf{1,575}$ .

STEP 8

Let's see that profit!
For Case B, it's $\$11,000 - \$1,575 = \$\textbf{9,425}$ .
Still pretty good!

STEP 9

Now, let's compare.
Case A gave us a profit of $\$\textbf{9,600}$ , and Case B gave us $\$\textbf{9,425}$ .
Clearly, $\$9,600$ is greater than $\$9,425$ .

STEP 10

The net profit of Case A is $\$\textbf{9,600}$ which is greater than the net profit of $\$\textbf{9,425}$ in Case B.
So, the answer is C.

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Studdy Solution

STEP 1

STEP 2

1. Calculate Case A profit
2. Calculate Case B profit
3. Compare profits

STEP 3

Alright, in Case A, we expect to yield 60 bushels/acre and we're planting 50 acres.
So, we'll have a total of $60 \cdot 50 = \textbf{3000}$ bushels of wheat.
Each bushel sells for $\$\textbf{3.50}$ , so our total income is $3000 \cdot \$3.50 = \$\textbf{10,500}$ .
Cha-ching!

STEP 4

We're adding 90 pounds of nitrogen per acre to our 50 acres, which means we need $90 \cdot 50 = \textbf{4500}$ pounds of nitrogen.
With the fertilizer costing $\$\textbf{0.20}$ per pound, our total expense is $4500 \cdot \$0.20 = \$\textbf{900}$ .

STEP 5

Time for the good stuff!
Profit is income minus expenses.
So, for Case A, it's $\$10,500 - \$900 = \$\textbf{9,600}$ .
Not bad!

STEP 6

In Case B, we're expecting $\textbf{40}$ bushels per acre on our $\textbf{50}$ acres, giving us a total of $40 \cdot 50 = \textbf{2000}$ bushels.
The price per bushel is $\$\textbf{5.50}$ , making our total income $2000 \cdot \$5.50 = \$\textbf{11,000}$ .
Even more cha-ching!

STEP 7

We need $\textbf{70}$ pounds of nitrogen per acre on our $\textbf{50}$ acres, totaling $70 \cdot 50 = \textbf{3500}$ pounds.
The fertilizer now costs $\$\textbf{0.45}$ per pound, so our total expense is $3500 \cdot \$0.45 = \$\textbf{1,575}$ .

STEP 8

Let's see that profit!
For Case B, it's $\$11,000 - \$1,575 = \$\textbf{9,425}$ .
Still pretty good!

STEP 9

Now, let's compare.
Case A gave us a profit of $\$\textbf{9,600}$ , and Case B gave us $\$\textbf{9,425}$ .
Clearly, $\$9,600$ is greater than $\$9,425$ .

STEP 10

The net profit of Case A is $\$\textbf{9,600}$ which is greater than the net profit of $\$\textbf{9,425}$ in Case B.
So, the answer is C.

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Studdy Solution

STEP 1

STEP 2

1. Calculate Case A profit2. Calculate Case B profit3. Compare profits

STEP 3

STEP 4

STEP 5

Time for the good stuff!Profit is income minus expenses.So, for Case A, it's $10,500−$900=$9,600\$10,500 - \$900 = \$\textbf{9,600}$10,500−$900=$9,600.Not bad!

STEP 6

STEP 7

STEP 8

Let's see that profit!For Case B, it's $11,000−$1,575=$9,425\$11,000 - \$1,575 = \$\textbf{9,425}$11,000−$1,575=$9,425.Still pretty good!

STEP 9

Now, let's compare.Case A gave us a profit of $9,600\$\textbf{9,600}$9,600, and Case B gave us $9,425\$\textbf{9,425}$9,425.Clearly, $9,600\$9,600$9,600 is greater than $9,425\$9,425$9,425.

STEP 10

The net profit of Case A is $9,600\$\textbf{9,600}$9,600 which is greater than the net profit of $9,425\$\textbf{9,425}$9,425 in Case B.So, the answer is C.

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1. Calculate Case A profit
2. Calculate Case B profit
3. Compare profits

Time for the good stuff!
Profit is income minus expenses.
So, for Case A, it's $\$10,500 - \$900 = \$\textbf{9,600}$ .
Not bad!

Let's see that profit!
For Case B, it's $\$11,000 - \$1,575 = \$\textbf{9,425}$ .
Still pretty good!

Now, let's compare.
Case A gave us a profit of $\$\textbf{9,600}$ , and Case B gave us $\$\textbf{9,425}$ .
Clearly, $\$9,600$ is greater than $\$9,425$ .

The net profit of Case A is $\$\textbf{9,600}$ which is greater than the net profit of $\$\textbf{9,425}$ in Case B.
So, the answer is C.