Question

Select all the correct locations on the image.
Two hourglasses are turned over at the exact same time. The first hourglass contains 300 grams of sand in the upper bulb and sand flows down into the lower bulb at a rate of 5 grams per minute. The second hourglass contains 200 grams of sand in the upper bulb and sand flows down into the lower bulb at a rate of 2 grams per minute.

The system of equations below model the amount of sand remaining, y, in the upper bulbs of the hourglasses after x minutes.
y = 300 - 5x
y = 200 - 2x
First, select the point on the graph that represents the solution to the system of equations. Notice that one of the equations in the system has already been graphed.

Then, determine the approximate number of minutes it takes for the amount of sand in the top bulbs of the hourglasses to be equal.

Answer 1

To solve the problem, we first need to analyze the two equations provided in the system:

1. For the first hourglass: \( y = 300 - 5x \)
2. For the second hourglass: \( y = 200 - 2x \)

We are tasked with finding the point where these two equations intersect, which would represent the time at which both hourglasses have the same amount of sand left in the upper bulbs.

To find this point, we can set the two equations equal to each other:

\[
300 - 5x = 200 - 2x
\]

Now, we solve for \( x \):

1. Rearranging gives us:
\[
300 - 200 = 5x - 2x
\]
\[
100 = 3x
\]

2. Dividing by 3:
\[
x = \frac{100}{3} \approx 33.33 \ \text{minutes}
\]

Now, to find the corresponding \( y \)-value (the amount of sand left in the upper bulbs) at this point, we can substitute \( x = \frac{100}{3} \) back into either equation. Let's use the first equation:

\[
y = 300 - 5\left(\frac{100}{3}\right)
\]
\[
y = 300 - \frac{500}{3}
\]
\[
y = \frac{900}{3} - \frac{500}{3} = \frac{400}{3} \approx 133.33 \ \text{grams}
\]

Therefore, the point where the two hourglasses have the same amount of sand in the upper bulb is approximately \( \left(33.33, 133.33\right) \).

In summary:

- The approximate number of minutes it takes for the amount of sand in the top bulbs of the hourglasses to be equal is **33.33 minutes**.
- The solution point is approximately \( (33.33, 133.33) \), and should be selected on the graph.

If you have an image or the graph available, you would select the corresponding point that aligns with these coordinates.