Question

Water Depth \(\mathrm{A}\) trough is 12 feet long, 3 feet deep, and 3 feet wide (see figure). Find the depth of the water when the trough contains 70 gallons of water. (1 gallon \(\approx 0.13368\) cubic foot.)

Short answer

The depth of the water when the trough contains 70 gallons of water is approximately \(0.26\) feet.

Step-by-step solution

Convert Water Volume from Gallons to Cubic Feet

Firstly, the volume of the water needs to be converted from gallons to cubic feet. This can be done by multiplying the given volume of the water in gallons (70 gallons) by the conversion factor: 1 gallon \(\approx 0.13368\) cubic foot. So, the volume in cubic feet will be \(70 * 0.13368\) cubic feet.

Calculate the Full Volume of the Trough

Next, calculate the full volume of trough which can be found by multiplying its length, width, and depth. The volume of the trough is \(12 * 3 * 3 = 108\) cubic feet.

Calculate the Depth of the Water in the Trough

Given that the volume of the water in the trough is 70 gallons, or approximately 9.3676 cubic feet after conversion, and that the area of the base of the trough is \(12 * 3 = 36\) square feet, the depth 'h' of the water in the trough can be determined by dividing the volume by the area. That is, \(h = \frac{9.3676}{36}.\)

Key concepts

Volume Conversion

Understanding volume conversion is essential when dealing with problems that involve different units of measurement for volume. In our example, we are given a volume in gallons and need to convert it to cubic feet to proceed with the calculation. Since 1 gallon is approximately equal to 0.13368 cubic feet, we use this as a conversion factor.

Here's the formula for converting gallons to cubic feet:
\[ \text{Volume in cubic feet} = \text{Volume in gallons} \times 0.13368 \].

Applying this formula to our exercise where the volume is 70 gallons, the calculation would be \( 70 \times 0.13368 \) cubic feet. After performing the multiplication, we'd get the volume of water in cubic feet, which is crucial for further calculations.

Trough Volume Computation

When computing the volume of a trough, or any three-dimensional object for that matter, it's important to apply geometric principles to find the right values. A trough in this context can be seen as a rectangular prism, where the volume is the product of its length, width, and depth.

The formula to calculate the volume of a rectangular prism is:
\[ \text{Volume} = \text{length} \times \text{width} \times \text{depth} \].

For our trough that is 12 feet long, 3 feet wide, and 3 feet deep, we multiply these dimensions together to find the total volume of the trough. Using the dimensions from the exercise, the volume calculation becomes \(12 \times 3 \times 3 = 108\) cubic feet, which represents the maximum volume of water the trough can hold.

Geometry in Algebra

The combination of geometry and algebra is often used to solve real-world problems. This is seen in the final step of our exercise, where we apply algebra to find the depth of the water in the trough. Knowing the volume of the water and the area of the base of the trough, we can find the depth by using the formula for the volume of a rectangular prism.

To find the depth 'h', we derive it from the formula for volume:
\[ h = \frac{\text{Volume of water}}{\text{Area of base}} \].

The area of the base is the product of the trough's length and width. In our case, the area is \(12 \times 3 = 36\) square feet. Thus, the depth is the volume of water (9.3676 cubic feet) divided by the base area (36 square feet). Following this formula, we find the water depth in the trough by dividing the converted volume of 9.3676 cubic feet by the area of the base, leading us to the final depth of the water.