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Answer :
Answer:
A. The gallons of water in the tank are:
- 793.61 gallons approximately.
The gallons missing to fill the tank are:
- 34457.52 gallons approximately.
Step-by-step explanation:
To solve the exercise we first find the volume of the tank with the formula:
- Volume of a cylinder = PI * r ^ 2 * h
Where:
r = Radius (remember that the radius is half the diameter)
h = Height (which in this case will be the length)
Replacing you get:
- Volume of a cylinder = PI * (10 feet) ^ 2 * 15 feet
- Volume of a cylinder = 4712.39 cubic feet approximately.
Having the volume of the cylinder, we find the volume of the water, using the volume formula, but this time you are going to take the measured 3 feet as the diameter, with which its radius would be 1.5 feet (since it is horizontal) :
- Water volume = PI * (1.5 feet) ^ 2 * 15 feet
- Water volume = 106.09 cubic feet approximately.
To find the volume of water in gallons it is not necessary to take into account the conversion factor to cubic inches, but you can directly use the conversion factor between cubic feet and gallons that I present below:
1 cubic foot = 7.48052 gallons.
With this conversion factor we operate:
- Water volume = 106.09 cubic feet * 7.48052 = 793.61 gallons
Now, to find the remaining volume to fill the tank, you must first subtract the volume of water from the volume of the tank:
- Remaining volume = 4712.39 cubic feet - 106.09 cubic feet
- Remaining volume = 4606.3 cubic feet.
Converting them to gallons gives:
- Remaining volume = 4606.3 cubic feet * 7.48052 = 34457.52 gallons.
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Rewritten by : Barada
The tank currently holds approximately 766.56 gallons of water. To fill the tank, it would require approximately 1490.33 more gallons.
To solve this problem, we need to calculate the volume of water contained in a horizontally lying cylindrical tank and then convert it into gallons. The tank diameter is 8 ft, giving a radius of 4 ft, and its length is 6 ft. The height of the water is 3 ft.
- First, calculate the volume of the water segment in the cylinder. Use the formula for the segment of a circle:
⇒ Volume of segment = Area of segment × Length.
- For a circle segment, the area A can be found using the formula:
⇒ A = [tex]\frac{1}{2}[/tex] × r² × (θ - sinθ),
where θ is the angle in radians at the circle center subtended by the segment. Here,
⇒ θ = 2 × [tex]\frac{r-h}{r}[/tex]
- In our case, r (radius) is 4 ft and h (height) is 3 ft, so:
θ = 2 × [tex]\frac{4-3}{4}[/tex]
= 2 × (0.25).
Using a calculator, θ ≈ 2.625 radians.
- Now, find the area of the segment:
⇒ Area = 0.5 × 4² × (2.625 - sin(2.625))
⇒ 0.5 × 16 × (2.625 - 0.49) ≈ 0.5 × 16 × 2.135
≈ 17.08 ft².
- Now, multiply by the length of the tank:
⇒ Volume of water = 17.08 ft² × 6 ft ≈ 102.48 ft³.
- Converting to Gallons:
⇒ 1 cubic foot = 7.48 gallons.
⇒ So, 102.48 ft³ ≈ 102.48 × 7.48 ≈ 766.56 gallons.
- Total volume of the tank is:
⇒ Volume = πr²h
⇒ π × 4² × 6 ≈ 301.59 ft³.
- Convert the total volume to gallons:
⇒ Total volume in gallons ≈ 301.59 × 7.48
≈ 2256.89 gallons.
- Finally, the additional gallons needed to fill the tank:
⇒ 2256.89 - 766.56 ≈ 1490.33 gallons.
Summary
The tank currently holds approximately 766.56 gallons of water. To fill the tank, it would need approximately 1490.33 more gallons.
Complete question:
A cylindrical tank is lying horizontally on the ground. Its diameter is 8ft. It's length is 6 ft. The height of the water in the tank is 3 ft.
a)how many gallons of water are in the tank?
b)how many more gallons of water will it take to fill the tank?
