Math / Algebra

QuestionA musician charges $C(x)=64 x+30,000$ where $x$ is the total number of attendees at the concert. The venue charges $\$ 84$ per ticket. After how many people buy tickets does the venue break even, and what is the value of the total tickets sold at that point?
The venue breaks even after $\square$ tickets are sold for a total value of $\$$ $\square$

Studdy Solution

STEP 1

1. The cost function for the musician is given by $C(x) = 64x + 30,000$ .
2. The venue charges $84 per ticket.
3. We need to find the number of tickets sold for the venue to break even.
4. We need to calculate the total value of tickets sold at the break-even point.

STEP 2

1. Define the revenue function.
2. Set up the break-even equation.
3. Solve for the number of tickets.
4. Calculate the total value of tickets sold.

STEP 3

Define the revenue function. The revenue $R(x)$ from selling $x$ tickets at $84 each is:
$R(x) = 84x$

STEP 4

Set up the break-even equation. The break-even point occurs when the revenue equals the cost:
$R(x) = C(x)$
Substitute the expressions for $R(x)$ and $C(x)$ :
$84x = 64x + 30,000$

STEP 5

Solve for the number of tickets. Subtract $64x$ from both sides to isolate the variable term:
$84x - 64x = 30,000$ $20x = 30,000$
Divide both sides by 20:
$x = \frac{30,000}{20}$ $x = 1,500$
The venue breaks even after $\boxed{1,500}$ tickets are sold.

STEP 6

Calculate the total value of tickets sold at the break-even point. Multiply the number of tickets by the price per ticket:
$\text{Total value} = 1,500 \times 84$ $\text{Total value} = 126,000$
The total value of tickets sold at the break-even point is $\$\boxed{126,000}$ .

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Studdy Solution

STEP 1

1. The cost function for the musician is given by $C(x) = 64x + 30,000$ .
2. The venue charges $84 per ticket.
3. We need to find the number of tickets sold for the venue to break even.
4. We need to calculate the total value of tickets sold at the break-even point.

STEP 2

1. Define the revenue function.
2. Set up the break-even equation.
3. Solve for the number of tickets.
4. Calculate the total value of tickets sold.

STEP 3

Define the revenue function. The revenue $R(x)$ from selling $x$ tickets at $84 each is:
$R(x) = 84x$

STEP 4

Set up the break-even equation. The break-even point occurs when the revenue equals the cost:
$R(x) = C(x)$
Substitute the expressions for $R(x)$ and $C(x)$ :
$84x = 64x + 30,000$

STEP 5

Solve for the number of tickets. Subtract $64x$ from both sides to isolate the variable term:
$84x - 64x = 30,000$ $20x = 30,000$
Divide both sides by 20:
$x = \frac{30,000}{20}$ $x = 1,500$
The venue breaks even after $\boxed{1,500}$ tickets are sold.

STEP 6

Calculate the total value of tickets sold at the break-even point. Multiply the number of tickets by the price per ticket:
$\text{Total value} = 1,500 \times 84$ $\text{Total value} = 126,000$
The total value of tickets sold at the break-even point is $\$\boxed{126,000}$ .

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Studdy Solution

STEP 1

1. The cost function for the musician is given by C(x)=64x+30,000 C(x) = 64x + 30,000 C(x)=64x+30,000.2. The venue charges $84 per ticket.3. We need to find the number of tickets sold for the venue to break even.4. We need to calculate the total value of tickets sold at the break-even point.

STEP 2

1. Define the revenue function.2. Set up the break-even equation.3. Solve for the number of tickets.4. Calculate the total value of tickets sold.

STEP 3

Define the revenue function. The revenue R(x) R(x) R(x) from selling x x x tickets at $84 each is:R(x)=84x R(x) = 84x R(x)=84x

STEP 4

Set up the break-even equation. The break-even point occurs when the revenue equals the cost:R(x)=C(x) R(x) = C(x) R(x)=C(x)Substitute the expressions for R(x) R(x) R(x) and C(x) C(x) C(x):84x=64x+30,000 84x = 64x + 30,000 84x=64x+30,000

STEP 5

STEP 6

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1. The cost function for the musician is given by $C(x) = 64x + 30,000$ .
2. The venue charges $84 per ticket.
3. We need to find the number of tickets sold for the venue to break even.
4. We need to calculate the total value of tickets sold at the break-even point.

1. Define the revenue function.
2. Set up the break-even equation.
3. Solve for the number of tickets.
4. Calculate the total value of tickets sold.

Define the revenue function. The revenue $R(x)$ from selling $x$ tickets at $84 each is:
$R(x) = 84x$

Set up the break-even equation. The break-even point occurs when the revenue equals the cost:
$R(x) = C(x)$
Substitute the expressions for $R(x)$ and $C(x)$ :
$84x = 64x + 30,000$