a cylindrical tank 23 feet long and 9 feet in diameter is resting on its side in a horizontal position.
find the number of gallons foil in the tank if the depth of the oils 3.5?
find the total area of the cylindrical tank
make a sketch of the cylinder lying on its side, showing an end view.
Draw a line showing the level of oil, complete the isosceles triangle from the
centre of your circle to the end of the chord representing the level of oil.
You know the two equal sides are 4.5 ft each, they are radii, and the height
of that triangle is 1 ft
The volume of oil would the the (Area of the segment)*(23) ft^3
Let's concentrate on that segment.
Here are the steps to follow:
- Using the cosine law, you can find the central angle of the isosceles triangle, call it θ degrees.
- The sector's area is (θ/360)(π)(4/5^2)
- The area of the triangle is (1/2)(4.5)(4.5)sinθ
- subtract to get the area of the segment
volume of oil = area of segment * 23 ft^3 = ....
area of the cylinder = 2π (4.5)^2 + 2π(4.5)(23) ft^2 = ....
I have a typo, the line of
- The sector's area is (θ/360)(π)(4/5^2)
should say:
- The sector's area is (θ/360)(π)(4.5^2)
but I'm sure you probably recognized that anyway.
To find the number of gallons of oil in the tank, we need to calculate the volume of the oil. Since the tank is lying on its side, the height of the oil represents the depth.
First, we need to find the volume of the cylindrical tank.
The formula for the volume of a cylinder is:
Volume = π × r^2 × h,
where π is a constant approximately equal to 3.14159, r is the radius of the cylinder, and h is the height (or depth) of the cylinder.
Given that the diameter of the tank is 9 feet, we can find the radius by dividing the diameter by 2:
Radius = Diameter / 2 = 9 / 2 = 4.5 feet.
The height (or depth) of the oil is given as 3.5 feet.
Plugging these values into the formula for the volume, we get:
Volume = 3.14159 × (4.5)^2 × 3.5 = 237.7785 cubic feet.
Now, to convert the volume from cubic feet to gallons, we need to know the conversion factor. 1 cubic foot is equal to about 7.48052 gallons.
Multiplying the volume by the conversion factor, we get:
Number of gallons = 237.7785 × 7.48052 = 1775.7 gallons (approximately).
Therefore, the number of gallons of oil in the tank is approximately 1775.7 gallons.
To find the total area of the cylindrical tank, we need to consider both the lateral surface area and the top and bottom circular surfaces.
The formula for the lateral surface area of a cylinder is:
Lateral Surface Area = 2 × π × r × h,
where π is the constant, r is the radius of the cylinder, and h is the height (or depth) of the cylinder.
Plugging in the values we have:
Lateral Surface Area = 2 × 3.14159 × 4.5 × 23 = 642.5133 square feet.
The formula for the surface area of a circle is:
Surface Area = π × r^2,
where π is the constant and r is the radius of the circle.
To find the area of the top and bottom circular surfaces, we calculate:
Surface Area = 3.14159 × (4.5)^2 = 63.617 square feet.
Finally, to find the total area of the cylindrical tank, we sum up the lateral surface area and the top and bottom circular surfaces:
Total Area = Lateral Surface Area + 2 × Surface Area = 642.5133 + 2 × 63.617 = 769.7473 square feet.
Therefore, the total area of the cylindrical tank is approximately 769.7473 square feet.