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.
[tex]
\begin{array}{l}
y=300-5 x \\
y=200-2 x
\end{array}
[/tex]
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)
Set the two equations equal to each other: 300 − 5 x = 200 − 2 x .
Solve for x : 3 x = 100 .
Find the value of x : x = 3 100 ≈ 33.33 .
The amount of sand in the top bulbs of the hourglasses will be equal after approximately 33.33 minutes.
Explanation
Understanding the Problem We are given a system of equations that model the amount of sand remaining in the upper bulbs of two hourglasses. The equations are:
y = 300 − 5 x y = 200 − 2 x
where y is the amount of sand remaining in grams and x is the time in minutes. We want to find the time x when the amount of sand remaining in both hourglasses is equal.
Setting the Equations Equal To find the time when the amount of sand is equal, we set the two equations equal to each other:
300 − 5 x = 200 − 2 x
Solving for x Now, we solve for x :
Add 5 x to both sides:
300 = 200 + 3 x
Subtract 200 from both sides:
100 = 3 x
Divide by 3:
x = 3 100 = 33.333...
Finding the Time Therefore, it takes approximately 33.33 minutes for the amount of sand in the top bulbs of the hourglasses to be equal.
Examples
Imagine you're managing two water tanks that are being drained at different rates. One tank starts with 300 liters and drains 5 liters per minute, while the other starts with 200 liters and drains 2 liters per minute. This problem helps you determine when both tanks will have the same amount of water left. Understanding how to solve such systems of equations is crucial for managing resources efficiently, whether it's water in tanks, money in accounts, or any other depleting asset.
