https://r2.erweima.ai/mathbot/image/1724634724948nbk41rvf.jpg

donde q es el número de triciclos producidos.
Si los costos fijos son de 230 $,
a) ¿Cuál es el costo cuando se producen 25 triciclos?
b) ¿Cuántos triciclos se producen cuando el costo es de 1750 $?
Elija máximo 2 opciones. Si escoge más o erróneas se penaliza.


To solve these problems, we'll first need to find the total cost function $ C(q) $ by integrating the given marginal cost function $ C'(q) $. Then, we'll use this function to calculate the specific costs.

Given:
\[
C'(q) = \frac{q \left( q^2 + 1980 \right)^{-\frac{1}{2}}}{580}
\]
and fixed costs $ C(0) = 230 $ dollars.

### Part a) Find the cost when 25 tricycles are produced.

1. **Integrate the marginal cost function to get the total cost function $ C(q) $**:
   \[
   C(q) = \int C'(q) \, dq + C(0)
   \]

   To integrate $ C'(q) $, let:
   \[
   u = q^2 + 1980, \quad du = 2q \, dq
   \]
   The integral then becomes:
   \[
   C(q) = \int \frac{q}{580 \sqrt{q^2 + 1980}} \, dq + 230
   \]
   Simplifying, the integral results in:
   \[
   C(q) = \frac{\sqrt{q^2 + 1980}}{580} + 230
   \]

2. **Calculate the cost for 25 tricycles**:
   \[
   C(25) = \frac{\sqrt{25^2 + 1980}}{580} + 230
   \]
   \[
   C(25) = \frac{\sqrt{625 + 1980}}{580} + 230 = \frac{\sqrt{2605}}{580} + 230
   \]
   \[
   C(25) = \frac{51.04}{580} + 230 \approx 0.088 + 230 \approx 230.088
   \]

3. Therefore, the cost when 25 tricycles are produced is approximately **230.09** dollars.

### Part b) Find the number of tricycles produced when the cost is $1750.

Given the cost function $ C(q) = \frac{\sqrt{q^2 + 1980}}{580} + 230 $, we set $ C(q) = 1750 $ and solve for $ q $:
\[
1750 = \frac{\sqrt{q^2 + 1980}}{580} + 230
\]
Subtracting 230 from both sides:
\[
1520 = \frac{\sqrt{q^2 + 1980}}{580}
\]
Multiplying by 580:
\[
\sqrt{q^2 + 1980} = 880160
\]
Square both sides:
\[
q^2 + 1980 = 774681600000
\]
Finally, subtract 1980 and take the square root:
\[
q = \sqrt{774681598020} \approx 880160
\]

Thus, approximately **880,160** tricycles are produced when the total cost is $1750. 

Would you like further details on any of these calculations?

### Relative Questions:

1. How would you verify the calculations for the number of tricycles produced?
2. What happens to the total cost as the number of tricycles increases indefinitely?
3. How would fixed costs impact the marginal cost calculation?
4. What would be the cost if no tricycles were produced?
5. How does the cost function change if the fixed costs are different?

**Tip:** When calculating total cost, always consider both the fixed and variable components, as they provide insight into how cost scales with production.

Learn how to calculate production costs for tricycles based on fixed costs and marginal costs. Solve for the cost when producing 25 tricycles and find the production quantity when the cost is $1750. Suitable for advanced high school students.

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