To determine the point at which the amount of sand in the top bulbs of both hourglasses is equal, we need to find the intersection of the two given equations:
[tex]\[ y = 300 - 5x \][/tex]
[tex]\[ y = 200 - 2x \][/tex]
Let's solve these equations step-by-step to find the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
1. Set the equations equal to each other: Since [tex]\(y\)[/tex] is the same in both equations at the point of intersection,
[tex]\[ 300 - 5x = 200 - 2x \][/tex]
2. Isolate the variable [tex]\(x\)[/tex]:
- First, subtract 200 from both sides of the equation:
[tex]\[ 300 - 200 - 5x = -2x \][/tex]
[tex]\[ 100 - 5x = -2x \][/tex]
- Next, add [tex]\(5x\)[/tex] to both sides to combine the [tex]\(x\)[/tex] terms:
[tex]\[ 100 = 3x \][/tex]
3. Solve for [tex]\(x\)[/tex]:
- Divide both sides by 3 to isolate [tex]\(x\)[/tex]:
[tex]\[ x = \frac{100}{3} \][/tex]
[tex]\[ x \approx 33.33 \][/tex]
So, it takes approximately 33.33 minutes for the amount of sand in the top bulbs of the hourglasses to be equal.
4. Calculate the amount of sand remaining in the top bulbs at this time:
- Substitute [tex]\(x = 33.33\)[/tex] back into one of the original equations (we can use either equation since they are equal at this point):
[tex]\[ y = 300 - 5 \left(\frac{100}{3}\right) \][/tex]
[tex]\[ y = 300 - 5 \times 33.33 \][/tex]
[tex]\[ y = 300 - 166.67 \][/tex]
[tex]\[ y \approx 133.33 \][/tex]
Therefore, the coordinates [tex]\( (x, y) \)[/tex] of the intersection point are approximately [tex]\( (33.33, 133.33) \)[/tex].
In conclusion, the solution to the system of equations, representing the point at which the amount of sand in the top bulbs of both hourglasses is equal, is approximately [tex]\( (33.33, 133.33) \)[/tex]. This means it takes about 33.33 minutes for the sand amounts to be equal, and at that time, there will be approximately 133.33 grams of sand remaining in the upper bulbs of both hourglasses.
