Calculus optimization problem?

In summary, when dealing with two bulbs placed 54 feet apart, the intensity of the light from each bulb, Ia and Ib, can be calculated using the inverse square law. To find the point of least illumination, one would need to find the distance x from bulb A where the combined illumination is the lowest.
  • #1
achiu17
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The illumination from a bulb varies directly as the intensity of the light and Intensity varies inversely as the square of the distance from the source. Two bulbs are placed 54 feet apart. The intensity, Ia, of bulb A is 64cd, and the intensity, Ib, of bulb B is 125cd. At how many feet from bulb A along the line between the two bulbs is the total illumination the least?

I have no idea how to start this problem...I thought about using optimization but I'm not sure where to start
 
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  • #2
If you were dealing with a single bulb, what would the illumination of say bulb A be at a distance x from the bulb?

Similarly, what would the illumination of bulb B be at a distance x?

Assuming that the illumination at a point between the two bulbs A and B is additive, how would you go about finding the point of least illumination?
 

Related to Calculus optimization problem?

What is a calculus optimization problem?

A calculus optimization problem is a type of mathematical problem that involves finding the maximum or minimum value of a function over a given interval. It uses calculus concepts such as derivatives and critical points to find the optimal solution.

What are the steps to solve a calculus optimization problem?

The steps to solve a calculus optimization problem are:
1. Identify the objective function and the constraints.
2. Use calculus to find the derivative of the objective function.
3. Set the derivative equal to zero and solve for the critical points.
4. Use the second derivative test to determine if the critical points are maximum or minimum points.
5. Check if the critical points satisfy the constraints.
6. Compare the values of the objective function at the critical points and choose the optimal solution.

What are some real-life applications of calculus optimization?

Calculus optimization has numerous real-life applications, such as:
- Maximizing profit for a company by finding the optimal production level.
- Minimizing cost by optimizing the design of a product.
- Maximizing the area of a garden with limited fencing material.
- Minimizing travel time by finding the fastest route between two points.
- Maximizing the efficiency of a chemical reaction by finding the optimal temperature and pressure.

What is the difference between a global and a local optimum in a calculus optimization problem?

In a calculus optimization problem, a global optimum is the absolute maximum or minimum value of the objective function over the entire domain. A local optimum, on the other hand, is the maximum or minimum value of the objective function within a specific interval or range. A global optimum is always a local optimum, but a local optimum may not be the global optimum.

What are some common mistakes to avoid when solving a calculus optimization problem?

Some common mistakes to avoid when solving a calculus optimization problem are:
- Forgetting to check if the critical points satisfy the constraints.
- Using the first derivative test instead of the second derivative test to determine the nature of the critical points.
- Not considering all possible critical points and assuming the first one found is the optimal solution.
- Making algebraic errors when solving for critical points or evaluating the objective function at the critical points.
- Forgetting to check for endpoints of the given interval as potential solutions.

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