Video: AP Calculus AB Exam 1 • Section I • Part B • Question 82

The given graph shows the number of people who visited a museum over a 10-hour period and number of people per hour. Which of the following is the closest approximation to the total number of people who visited the museum over the 10-hour period? [A] 3000 [B] 1050 [C] 1450 [D] 2500

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Video Transcript

The given graph shows the number of people who visited a museum over a 10-hour period and number of people per hour. Which of the following is the closest approximation to the total number of people who visited the museum over the 10-hour period? a) 3000, b) 1050, c) 1450, and d) 2500.

In this question, we are asked to approximate the total number of people who visited the museum in this 10-hour period. Now, on the 𝑦-axis of our graph, we have the number of people who visited per hour. And on the 𝑥-axis, we have hours. Now, if we multiply the number of people per hour by the number of hours, then this will give us the number of people who visited the museum. Therefore, we can say that the total number of people will be equal to the area under the curve.

Let’s now work out how many people each large square on our graph represents. Each square has a height of 50 people per hour and a length of two hours. Therefore, its area represents 50 multiplied by two which is also equal to 100 people. Okay, so now we know that one large square represents 100 people. We can work out how many large squares are represented by each of our options. We do this by dividing each option by 100 since there are 100 people per large square.

So 3000 people will be represented by 30 squares. 1050 people will be represented by 10 and a half squares. 1450 people will be represented by 14 and a half squares. And finally, 2500 people will be represented by 25 squares. Now, let’s estimate the number of large squares which are under our graph. And we can then compare this to our options to see which one is closest and therefore gives the best approximation.

Along the bottom of our graph, we can see that we have five squares. We then have three more full squares on the row above, taking us up to eight squares. We have another two full squares above that, taking us up to 10. And now, we’ve run out of full squares to count. However, we can add together pieces of the remaining area in order to get a rough estimate of the number of squares there are under this graph. The reason why we don’t have to be quite so accurate here is since we’re finding the closest approximation to the total number of people.

On the second row, we have two very nearly full squares. And if we add on this small area here, this will give us two more squares. This will take our total up to 12. On the next row up, we have another two nearly full squares. And if we add this area in here, we can say that we have roughly another two full squares. This takes our total up to 14. If we combine the area right at the top of the graph with the two other remaining areas, then we will see that we have roughly 15 full squares.

Of our four options, 14 and a half squares or 1450 people is the closest approximation to the number which we found here. Therefore, we can say that the closest approximation to the total number of people who visited the museum over the 10-hour period is option c, 1450 people.

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