How much money did the soda machines generate during the second semester?

In summary, the school year in a West Texas school district runs from August 1 to May 31. A Soft Drink company installed soda machines in the cafeteria on August 1 and after t months, the machines generated income at a rate of 500t/(5t^2+2) dollars per month. To find the total income for the second semester starting on January 1, the integral of the function was used with the correct limits of 1 and 5, resulting in an income of $50 ln(27) or approximately $90.03.
  • #1
Momentum09
71
0

Homework Statement


In a West Texas school district the school
year began on August 1 and lasted until May
31. On August 1 a Soft Drink company in-
stalled soda machines in the school cafeteria.
It found that after t months the machines
generated income at a rate of
f(t) =
500t/ (5t^2 + 2)
dollars per month. Find the total income,
$Tsec, produced during the second semester
beginning on January 1.


Homework Equations



I know to find the income we first have to find the integral for the function t. I did this and got 50 ln [5x^2 +2].
I set u = 5t^2 +2, so du = 10t dt

The Attempt at a Solution



I used 6 as my upper limit and 1 as my lower limit but I can't get the right answer. Can someone please tell me what is the right limit of integration to use in this case?
Thank you so much for your help!
 
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  • #2
Your limits are a little off. Why did you use 6? The machine was functioning for 5 months from Aug 1 to Jan 1 the next year. And it's not supposed to be 0 either for the lower limits. That ignores the revenue for the first month.
 

Related to How much money did the soda machines generate during the second semester?

1. What is an integration word problem?

An integration word problem is a type of mathematical problem that involves finding the integral of a function in order to solve a real-world scenario. It requires understanding of both calculus and the given situation in order to set up and solve the appropriate integral.

2. What are some common types of integration word problems?

Some common types of integration word problems include finding the area under a curve, determining the volume of a solid, and calculating the work done by a variable force. Other examples include related rates problems, optimization problems, and motion problems.

3. How do I solve an integration word problem?

To solve an integration word problem, you must first carefully read and understand the given scenario. Next, you will need to set up the appropriate integral based on the information given. Finally, you can use calculus techniques to evaluate the integral and find the solution to the problem.

4. What are some tips for solving integration word problems?

Some tips for solving integration word problems include carefully reading and understanding the given scenario, identifying the relevant quantities and variables, choosing the appropriate integral to represent the problem, and using calculus techniques such as substitution and integration by parts to solve the integral.

5. Are there any common mistakes to avoid when solving integration word problems?

Yes, some common mistakes to avoid when solving integration word problems include forgetting to include the constant of integration, making errors in setting up the integral, and not carefully checking your work for mistakes. It is also important to make sure that your answer makes sense in the context of the given scenario.

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