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Hi there.
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So for this problem, we are told that when a professional puzzler attempts a 1 ,000 piece g -saw puzzle, the rate at which they can place pieces into the puzzle is given by the following equation.
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That is the differential of the pieces with respect to time is equal to 2 times 1 .004 elevated to the times t, where t is the amount of times since they started the puzzle in minutes.
00:30
Now for part a of this problem, we need to find an equation for the function p of t that predicts the number of pieces placed after working on the puzzle for t minutes.
00:43
So what we need to do to solve this differential equation is to separate the variables and then do the integral.
00:51
So we are going to have that p of t is equal to the integral of this expression right here.
01:01
And we know that an integral that has the following form, a, which is a constant, elevated to the s, the integrated over x is equal to a to the x divided by the neferial logarithm of a.
01:19
So from this, immediately we obtained that this is equal to two times this divided by the neferian logarithm of 1 .1.
01:35
In zero four.
01:37
And this, because this is an indefinite integral, we need to add a constant of integration c.
01:43
Now, we know that the initial amount of pieces that we have, that we have put, is equal to zero.
01:53
So with this, we obtained that the constant c should be equal to when we substitute the time equals to zero.
02:00
In here, we know that every number, any number that is limited to zero is one.
02:05
So, immediately we obtained 2 divided by the neparian logarithm of 1 .004.
02:10
So substituting this into the equation, we finally obtained that the function, the solution for this problem, is equal to 2 times 1 .004, elevated to the times t, divided by the neparian logarithm of 1 .004.
02:27
And this plus 2 divided by the neparian logarithm of 1 .004.
02:39
So that's a solution for part a of this problem.
02:44
Now for part b, we are asked about the time at which the puzzle, the puzzler has completed the puzzle.
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So in this case, we need to find the time of which the number of pieces is 1 ,000.
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So we just simply substitute this in here and then start sort for, start...