water is stored in a tank in the shape of an interval cone of height 10ft and diameter 6ft.express the volume of water in the tank as a function of the height h of the water level.
To find the volume of water in the tank as a function of the height, you'll need to use the formula for the volume of a cone, which is given by:
V = 1/3 * π * r^2 * h,
where V is the volume, π is the mathematical constant pi (approximately 3.14), r is the radius of the base of the cone, and h is the height of the cone.
In this case, the tank is in the shape of an inverted cone, so we need to determine the radius at a given height in order to calculate the volume.
Let's consider the dimensions of the tank: the height is given as 10 ft and the diameter (which is twice the radius) is given as 6 ft. From the diameter, we can calculate the radius as follows:
Radius = Diameter/2 = 6 ft / 2 = 3 ft.
Now, we need to account for the fact that the water level (h) may not always be equal to the height of the tank (10 ft). We need to subtract the height of the water (h) from the total height of the tank to find the remaining height (h'):
h' = 10 ft - h.
Next, we can find the radius at the current water level using similar triangles. The ratio of corresponding lengths in similar triangles is equal, so:
r' / h' = r / h.
By substituting the known values, we have:
r' / (10 ft - h) = 3 ft / h,
Cross-multiplying, we get:
r' * h = 3 ft * (10 ft - h),
r' * h = 30 ft - 3h,
r' = (30 ft - 3h) / h.
Finally, we can substitute this expression for r' into the volume formula to find the volume (V) as a function of the water level (h):
V = 1/3 * π * [(30 ft - 3h) / h]^2 * h.
Simplifying further, we have:
V = 1/3 * π * (30 ft - 3h)^2 * h.
Therefore, the volume of water in the tank, expressed as a function of the height h of the water level, is given by:
V(h) = 1/3 * π * (30 ft - 3h)^2 * h.
To express the volume of water in the tank as a function of the water level height h, we need to find the formula for the volume of a cone and then adjust it according to the given dimensions.
The volume of a cone is given by the formula:
V = (1/3)πr^2h,
where V is the volume, π is a mathematical constant approximately equal to 3.14159, r is the base radius of the cone, and h is the height of the cone.
In the given tank, the height of the cone is 10ft, and the diameter (which is twice the radius) is 6ft. Therefore, the radius can be calculated as half of the diameter:
r = 6ft / 2 = 3ft.
Now we can express the volume of water in the tank as a function of the water level height h:
V(h) = (1/3)πr^2h,
where r is the base radius of the cone (3ft).