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ioda
3 years ago
11

Mrs. Carlson wrote the following question for her class: "Ricardo's basketball team is selling raffle tickets, Each ticket costs

$0.75, They want to sell $285 worth of raffle tickets. So far, they have sold $75 worth of tickets. How many tickets do they still need to sell?" Some of the student responses are shown in the table below: Michelle's respone was 285 - 75 = 210 and 210 + 0.75 = 28, Kaley's respone was 285 - 75 = 210 and 210 + 0.75 = 210.75, Eva's response was 285 - 75 is 210 and 210 + 0.75 = 280, Rebecca's response was 285 - 75 = 210 and 210 + 0.75 = 118. Which students answered Incorrectly? Why are they wrong?​
Mathematics
1 answer:
IceJOKER [234]3 years ago
7 0

Answer:

michelle, kaley, and Rebecca.

Step-by-step explanation:

They have sold 100 tickets so far ($75), they need to sell 380 ($285) tickets in total and need to sell 280 more tickets.

Michelle was wrong because she should have divided 210 (how much more money they need to sell on tickets) into .75 (which is how much each ticket costs).

Kaley was wrong because she should have also divided 210 into .75 like Michelle.

Rebecca should have also divided 210 into .75.

Eva was the only correct one.

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Step-by-step explanation:

It's a bit hard to tell what constitutes an "iteration" when using the bisection method to approximate a polynomial root. For the purpose here, we'll say one iteration consists of ...

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Thus, the result of the iteration consists of a bracketing interval and the choice of one of the interval's ends as the solution approximation.

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(a) We observe that the graphs intersect in the interval (-1, 0). For the first iteration, we evaluate f(x)-g(x) at x=-1/2. This tells us the solution is in the interval (-1, -1/2). The x-value closest to the root is x=-1/2.

For the second iteration, we evaluate the function f(x)-g(x) at x=-3/4. This tells us the solution is in the interval (-1, -3/4). The x-value closest to the root is x=-3/4.

For the third iteration, we evaluate the function f(x)-g(x) at x=-7/8. This tells us the solution is in the interval (-7/8, -3/4). The x-value closest to the root is x=-3/4.

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(b) The graph tells us the solution is approximately 0.7549. Rounded to 2 decimal places, the solution is approximately 0.75.

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(c) The above solution found after 3 iterations rounded to 2 decimal places is exactly 0.75.

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See the attached table for function values.

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<em>Comment on bisection iteration</em>

Since you cut the interval containing the root in half with each iteration, you gain approximately one decimal place for each 3 iterations. When the function value is very nearly zero at one of the interval endpoints, it can take many more iterations to achieve a better result.

Here, it takes 4 more iterations before an x-value becomes closer to the solution (x≈-97/128). And it takes one more iteration to move the end of the interval away from -3/4. After these 5 more iterations (8 total), the solution is known to lie in the interval (-97/128, -193/256). The corresponding solution approximation is -193/256. It is still only correct to 2 decimal places.

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