The cost and revenue functions of a product are given by c(x) = 20x + 4000 and R(x) = 60x + 2000 respectively where x is the number of items produced and sold. The value of x to earn profit is
- >50
- >60
- >80
- >40
The Correct Option is A
Approach Solution - 1
We are given the following:
- Cost function: \( C(x) = 20x + 4000 \)
- Revenue function: \( R(x) = 60x + 2000 \)
To find the value of \( x \) (the number of items) that will give a profit, we need to calculate the profit function. The profit function is the difference between revenue and cost:
\[ \text{Profit} = R(x) - C(x) \]
Substituting the given functions:
\[ \text{Profit} = (60x + 2000) - (20x + 4000) \]
Simplifying:
\[ \text{Profit} = 60x + 2000 - 20x - 4000 \]
\[ \text{Profit} = 40x - 2000 \]
To earn a profit, the profit must be greater than 0:
\[ 40x - 2000 > 0 \]
Solving for \( x \):
\[ 40x > 2000 \]
\[ x > \frac{2000}{40} \]
\[ x > 50 \]
Therefore, the value of \( x \) to earn a profit is greater than 50.
The correct answer is (A): \( x > 50 \)
Approach Solution -2
We are given the cost function \( C(x) \) and the revenue function \( R(x) \) for a product, where x is the number of items produced and sold:
- Cost function: \( C(x) = 20x + 4000 \)
- Revenue function: \( R(x) = 60x + 2000 \)
The Profit function \( P(x) \) is calculated as the difference between the revenue and the cost:
\[ P(x) = R(x) - C(x) \]
Substitute the given functions into the profit formula:
\[ P(x) = (60x + 2000) - (20x + 4000) \]
Simplify the expression:
\[ P(x) = 60x + 2000 - 20x - 4000 \]
\[ P(x) = (60x - 20x) + (2000 - 4000) \]
\[ P(x) = 40x - 2000 \]
To earn a profit, the profit \( P(x) \) must be greater than zero:
\[ P(x) > 0 \]
Substitute the expression for \( P(x) \):
\[ 40x - 2000 > 0 \]
Now, solve this inequality for x:
Add 2000 to both sides:
\[ 40x > 2000 \]
Divide both sides by 40:
\[ x > \frac{2000}{40} \]
\[ x > \frac{200}{4} \]
\[ x > 50 \]
Therefore, a profit is earned when the number of items produced and sold (x) is greater than 50.
Comparing this result with the given options:
- >50
- >60
- >80
- >40
The correct option is >50.
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