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Water is poured into a rectangular container 10 centimeters (cm) long, 12cm wide, and 9cm high, until it is (1)/(20) full. All this water is then poured into an empty cylindrical container with a circular base of radius 3cm. The height of the water in the cylindrical container is (y)/( pi) centimeters, where y is a constant. What is the value of y ?

Water is poured into a rectangular container 1010 centimeters (cm) (\mathrm{cm}) long, 12 cm 12 \mathrm{~cm} wide, and 9 cm 9 \mathrm{~cm} high, until it is 120 \frac{1}{20} full. All this water is then poured into an empty cylindrical container with a circular base of radius 3 cm 3 \mathrm{~cm} . The height of the water in the cylindrical container is yπ \frac{y}{\pi} centimeters, where y y is a constant. What is the value of y y ?

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Q. Water is poured into a rectangular container 1010 centimeters (cm) (\mathrm{cm}) long, 12 cm 12 \mathrm{~cm} wide, and 9 cm 9 \mathrm{~cm} high, until it is 120 \frac{1}{20} full. All this water is then poured into an empty cylindrical container with a circular base of radius 3 cm 3 \mathrm{~cm} . The height of the water in the cylindrical container is yπ \frac{y}{\pi} centimeters, where y y is a constant. What is the value of y y ?
  1. Calculate volume of rectangular container: First, we need to calculate the volume of water in the rectangular container when it is (1)/(20)(1)/(20) full.\newlineThe volume of the rectangular container is length ×\times width ×\times height.\newlineSo, the full volume is 1010 cm ×\times 1212 cm ×\times 99 cm.
  2. Calculate full volume: Now, we calculate the full volume: 10 cm×12 cm×9 cm=1080 cm310\,\text{cm} \times 12\,\text{cm} \times 9\,\text{cm} = 1080\,\text{cm}^3.
  3. Calculate volume of water poured: Since the rectangular container is only (1)/(20)(1)/(20) full, we need to find (1)/(20)(1)/(20) of the full volume.(1)/(20)(1)/(20) of 10801080 cm³ is 10801080 cm³ ÷ 2020.
  4. Find volume of cylindrical container: Calculating 120\frac{1}{20} of the full volume gives us 1080 cm3÷20=54 cm31080 \, \text{cm}^3 \div 20 = 54 \, \text{cm}^3. This is the volume of water that is poured into the cylindrical container.
  5. Set up equation for volume of water: Next, we need to find the volume of the cylindrical container that the water will occupy.\newlineThe volume of a cylinder is given by the formula V=πr2hV = \pi r^2 h, where rr is the radius and hh is the height.\newlineWe know the radius r=3r = 3 cm, and we need to find the height hh, which is expressed as yπ\frac{y}{\pi} cm.
  6. Simplify equation: The volume of water 54 cm354 \, \text{cm}^3 must be equal to the volume of the cylindrical container that it occupies.\newlineSo, π×(3 cm)2×yπ cm=54 cm3\pi \times (3 \, \text{cm})^2 \times \frac{y}{\pi} \, \text{cm} = 54 \, \text{cm}^3.
  7. Solve for y: We simplify the equation: π×9 cm2×yπ cm=54 cm3\pi \times 9 \, \text{cm}^2 \times \frac{y}{\pi} \, \text{cm} = 54 \, \text{cm}^3. The π\pi on both sides of the equation cancels out.
  8. Calculate y: Now we have: 9 cm2×yπ cm=54 cm39 \, \text{cm}^2 \times \frac{y}{\pi} \, \text{cm} = 54 \, \text{cm}^3. To find yy, we divide both sides of the equation by 9 cm29 \, \text{cm}^2.
  9. Calculate y: Now we have: 9 cm2×yπ cm=54 cm39 \, \text{cm}^2 \times \frac{y}{\pi} \, \text{cm} = 54 \, \text{cm}^3. To find yy, we divide both sides of the equation by 9 cm29 \, \text{cm}^2. Dividing by 9 cm29 \, \text{cm}^2 gives us yπ cm=54 cm39 cm2\frac{y}{\pi} \, \text{cm} = \frac{54 \, \text{cm}^3}{9 \, \text{cm}^2}.
  10. Calculate y: Now we have: 9 cm2×yπ cm=54 cm39 \, \text{cm}^2 \times \frac{y}{\pi} \, \text{cm} = 54 \, \text{cm}^3. To find yy, we divide both sides of the equation by 9 cm29 \, \text{cm}^2. Dividing by 9 cm29 \, \text{cm}^2 gives us yπ cm=54 cm39 cm2\frac{y}{\pi} \, \text{cm} = \frac{54 \, \text{cm}^3}{9 \, \text{cm}^2}. Calculating the division gives us yπ cm=6 cm\frac{y}{\pi} \, \text{cm} = 6 \, \text{cm}. Therefore, y=6π cmy = 6\pi \, \text{cm}.

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