Answer :
To solve this problem, let's break down the information and solution step-by-step.
1. Define the Variables and Given Information:
- Let [tex]\( x \)[/tex] be the number of accessories produced in millions.
- The price per accessory is given by the equation [tex]\( 100 - 10x^2 \)[/tex].
- The cost to make each accessory is [tex]\( \$10 \)[/tex].
- The company currently produces [tex]\( 2 \)[/tex] million accessories and makes a profit of [tex]\( \$100 \)[/tex] million.
2. Calculate Current Revenue and Profit:
- Current Price per Accessory:
[tex]\[ \text{Price per accessory when } x = 2: 100 - 10(2^2) = 100 - 40 = 60 \text{ dollars} \][/tex]
- Revenue:
[tex]\[ \text{Revenue} = \text{Price per accessory} \times \text{number of accessories} = 60 \times 2 = 120 \text{ million dollars} \][/tex]
- Total Cost:
[tex]\[ \text{Cost} = \text{Cost per accessory} \times \text{number of accessories} = 10 \times 2 = 20 \text{ million dollars} \][/tex]
- Profit:
[tex]\[ \text{Profit} = \text{Revenue} - \text{Total Cost} = 120 - 20 = 100 \text{ million dollars} \][/tex]
This confirms the given information.
3. Form the Profit Equation:
- The profit equation in general form:
[tex]\[ \text{Profit} = (\text{Price per accessory} \times \text{number of accessories}) - (\text{Cost per accessory} \times \text{number of accessories}) \][/tex]
- Replace the price per accessory and cost per accessory in the equation:
[tex]\[ 100 = (100 - 10x^2)x - 10x \][/tex]
- Simplify and solve for [tex]\( x \)[/tex]:
[tex]\[ 100 = (100x - 10x^3) - 10x \][/tex]
[tex]\[ 100 = 90x - 10x^3 \][/tex]
[tex]\[ 10x^3 - 90x + 100 = 0 \][/tex]
4. Solve the Polynomial Equation:
- The solutions are obtained by solving the equation [tex]\( 10x^3 - 90x + 100 = 0 \)[/tex]. By analyzing the roots, we find the following values for [tex]\( x \)[/tex]:
[tex]\[ x \approx 2.000 \text{ and } x \approx 1.45 \][/tex]
5. Filter the Solutions:
- The positive, real number of accessories produced that yields the same profit of [tex]\( \$100 \)[/tex] million are:
[tex]\[ x = 2 \text{ million (already given)} \][/tex]
[tex]\[ x \approx 1.45 \text{ million} \][/tex]
Therefore, the other number of accessories that yields the same profit is 1.45 million.
Thus, the correct answer is:
[tex]\[ \boxed{1.45 \text{ million}} \][/tex]
1. Define the Variables and Given Information:
- Let [tex]\( x \)[/tex] be the number of accessories produced in millions.
- The price per accessory is given by the equation [tex]\( 100 - 10x^2 \)[/tex].
- The cost to make each accessory is [tex]\( \$10 \)[/tex].
- The company currently produces [tex]\( 2 \)[/tex] million accessories and makes a profit of [tex]\( \$100 \)[/tex] million.
2. Calculate Current Revenue and Profit:
- Current Price per Accessory:
[tex]\[ \text{Price per accessory when } x = 2: 100 - 10(2^2) = 100 - 40 = 60 \text{ dollars} \][/tex]
- Revenue:
[tex]\[ \text{Revenue} = \text{Price per accessory} \times \text{number of accessories} = 60 \times 2 = 120 \text{ million dollars} \][/tex]
- Total Cost:
[tex]\[ \text{Cost} = \text{Cost per accessory} \times \text{number of accessories} = 10 \times 2 = 20 \text{ million dollars} \][/tex]
- Profit:
[tex]\[ \text{Profit} = \text{Revenue} - \text{Total Cost} = 120 - 20 = 100 \text{ million dollars} \][/tex]
This confirms the given information.
3. Form the Profit Equation:
- The profit equation in general form:
[tex]\[ \text{Profit} = (\text{Price per accessory} \times \text{number of accessories}) - (\text{Cost per accessory} \times \text{number of accessories}) \][/tex]
- Replace the price per accessory and cost per accessory in the equation:
[tex]\[ 100 = (100 - 10x^2)x - 10x \][/tex]
- Simplify and solve for [tex]\( x \)[/tex]:
[tex]\[ 100 = (100x - 10x^3) - 10x \][/tex]
[tex]\[ 100 = 90x - 10x^3 \][/tex]
[tex]\[ 10x^3 - 90x + 100 = 0 \][/tex]
4. Solve the Polynomial Equation:
- The solutions are obtained by solving the equation [tex]\( 10x^3 - 90x + 100 = 0 \)[/tex]. By analyzing the roots, we find the following values for [tex]\( x \)[/tex]:
[tex]\[ x \approx 2.000 \text{ and } x \approx 1.45 \][/tex]
5. Filter the Solutions:
- The positive, real number of accessories produced that yields the same profit of [tex]\( \$100 \)[/tex] million are:
[tex]\[ x = 2 \text{ million (already given)} \][/tex]
[tex]\[ x \approx 1.45 \text{ million} \][/tex]
Therefore, the other number of accessories that yields the same profit is 1.45 million.
Thus, the correct answer is:
[tex]\[ \boxed{1.45 \text{ million}} \][/tex]
