A tank in the form of an inverted right circular cone is 8m across the top and 10m deep. If the tank is filled to a height of 9m with water, find the work done in pumping the water to the top of the tank:

the radius is 4, the depth is 10

so

the coordinates of the line passes through <---This should be "the line passes through the coordinates"

(4,0) and (0,-10)

Equation of the line is: (y-(-10)) = 5/2(x - 0)

x = 2/5y + 4

work is :

d(W) = 9180pi[(2/5)y + 4)^2ydy

and multiplied by y is the change in length to the top of the cone right?

I assume your 9180 is the density of water in metric,

9180 newton/cu.m.

Should that not be 9800 newton/cu.m. ?

Or, using your numbers, 9810 N/cu.m.?

"the coordinates of the line passes ..."

That should be "the line passes through the coordinates"

"(4,0) and (0,-10)"

That means your (0,0) is the center of the circular top of the cone.

"and multiplied by y is the change in length to the top of the cone right?"

Should be:

"and it is multiplied by (-y) because (-y) is the change in height to the top....."

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is the math model correct?

The y in the "y dy" should be negative. Because all y's below the (0,0) are negative.

The boundaries for integration should be from y = -10 to y = -1.