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Find the interest earned on an interest-bearing account

Every month, a particular interest-bearing account earns 0.2 percent interest on the average balance for that month. The function B(t) =7.5t^2 -300t +5000 represents one investor's balance in this account during the month of November. B(t) gives the number of dollars in the account on day t, with t=0 being the beginning of the first day. How much interest will the account earn this month?

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Andre W. | Friendly tutor for ALL math and physics coursesFriendly tutor for ALL math and physics ...
5.0 5.0 (3 lesson ratings) (3)
The average value of a function f(t) on an interval [a,b] is given by
ab f(t) dt /(b-a).
In your case, the average of B(t)=7.5t2-300t+500 on [0,30] is given by
030 B(t) dt /30 = 2750.
0.2% interest on this balance is 2750*0.002 = $5.5.
Ryan Y. | Harvey Mudd/Cornell grad; multi-subject, empathetic tutorHarvey Mudd/Cornell grad; multi-subject,...
5.0 5.0 (427 lesson ratings) (427)
Hi Bella,

Can I assume that you have calculus for this? In order to calculate an average, I would integrate B(t) with respect to t, then divide it by the total time (31 days for July  30 days for November).

Int(5.5t2-300t+5000)dt = (5.5/3)t3-150t2+5000t |310 = $65,466.83

Int(7.5t2-300t+5000)dt = (7.5/3)t3-150t2+5000t |300 = $82,500
Average daily balance = $82,500/30 = $2,750.

The interest is 0.2% on the average balance, or

(0.002)($2,750) = $5.50


Answer updated to account for new information.
Kirill Z. | Physics, math tutor with great knowledge and teaching skillsPhysics, math tutor with great knowledge...
4.9 4.9 (174 lesson ratings) (174)
The interest will be 0.003*<B(t)>, where <> means average.
Interest compounded will be 0.003*2111.83≈6.34
Answer: $6.34.
William S. | Experienced scientist, mathematician and instructor - WilliamExperienced scientist, mathematician and...
4.4 4.4 (10 lesson ratings) (10)
Dear Bella,
July has 31 days, so, if I'm reading the problem correctly
B(t) = (5.5)t2 - 300t + 5000
Bella, I'm afraid there might be something wrong with this equation.  As written, it predicts that our poor account holder will be losing money.  For 0 > t ≤ 54.5455, 300*t will be greater than (5.5)*t2 and the $5000 in the account will be diminished.  Not until day 55 arrives will the (5.5)t2 term be larger than the (300)*t term.
Something doesn't flush her, Bella.  Have a look at that original equation again and see if you entered it correctly.