A parabolic trough is 3 meters long, one foot across the top and one foot deep. Find the volume of water in the trough in liters if it is filled to a depth of 6inches.

To find the volume of water in the trough, we first need to calculate the cross-sectional area of the trough.

The shape of the cross-section of a parabolic trough is a parabola. The depth of the trough is given as 1 foot, which is equivalent to 12 inches. The width of the trough at the top is also given as 1 foot, or 12 inches.

The equation of a parabola in standard form is y^2 = 4ax, where "a" is the distance between the vertex and the focus of the parabola.

In this case, the parabolic trough has a depth of 12 inches, which acts as the "a" value. So the equation becomes y^2 = 48x.

To find the value of x at a given depth (y), we can rearrange the equation and solve for x:

x = y^2 / 48

Now, with the depth of 6 inches, we can calculate the corresponding value of x:

x = (6^2) / 48
x = 1/8

This means that at a depth of 6 inches, the width of the trough at that point is 1/8 of a foot.

To calculate the cross-sectional area of the trough, we multiply the width (1/8 ft) by the depth (6 inches) and length (3 meters):

Cross-sectional area = (1/8 ft) * (6 inches) * (3 meters)
Note: To continue the calculation, we need to convert the length from meters to feet.

1 meter is equal to approximately 3.28 feet. So, the length of the trough in feet is:

Length in feet = 3 meters * 3.28 feet/meter
Length in feet = 9.84 feet

Therefore, the cross-sectional area is:

Cross-sectional area = (1/8 ft) * (6 inches) * (9.84 feet)
Cross-sectional area = 0.75 square feet

Finally, to find the volume of water, we multiply the cross-sectional area by the depth of water:

Volume = Cross-sectional area * Depth
Note: we need to convert the depth from inches to feet.

Depth in feet = 6 inches / 12 inches/foot
Depth in feet = 0.5 feet

Now, we can calculate the volume:

Volume = 0.75 square feet * 0.5 feet
Volume = 0.375 cubic feet

Since there are approximately 28.3168466 liters in a cubic foot, we can convert the volume to liters:

Volume in liters = 0.375 cubic feet * 28.3168466 liters/cubic foot
Volume in liters ≈ 10.628 liters

Therefore, the volume of water in the trough, when filled to a depth of 6 inches, is approximately 10.628 liters.

To calculate the volume of water in the parabolic trough, we need to find the area of the cross-section and multiply it by the length of the trough.

First, let's convert the given measurements to a consistent unit. Since we are calculating the volume in liters, we will convert all the dimensions to meters:

Length of the trough = 3 meters
Width of the trough (across the top) = 1 foot = 0.3048 meters
Depth of the trough = 1 foot = 0.3048 meters
Filled depth of water = 6 inches = 0.1524 meters

Now, let's find the area of the cross-section of the trough:

The cross-section can be approximated as a rectangle with the width and depth of the trough.

Area of cross-section = width * depth
Area of cross-section = 0.3048 meters * 0.3048 meters
Area of cross-section = 0.093 m^2

Finally, we can calculate the volume of water in the trough by multiplying the area of the cross-section by the length of the trough:

Volume of water = area of cross-section * length of trough
Volume of water = 0.093 m^2 * 3 meters
Volume of water = 0.279 cubic meters

To convert the volume to liters, we need to multiply by 1000 since there are 1000 liters in a cubic meter:

Volume of water = 0.279 cubic meters * 1000 = 279 liters

Therefore, the volume of water in the parabolic trough, when filled to a depth of 6 inches, is 279 liters.