A water tank had the shape of a cone...?

Question

A water tank has the shape of a cone. The tank is 10m high and has a radius of 3m at the top. If the water is 5 m deep (in the middle), what is the SURFACE AREA of the water?

If anyone could help, that would be great..

Answer 1

to find the radius of the surface area, use proportions 5/10 =r/3

r=1.5

pi r^2 =3.14 x 2.25 =7.07 m^2

Answer 2

Start by calculating the area of the container.

SA of a cone = πr² + πrs

where r = radius and s = "slant height"

The slant height is calculated using the pythagorean theorem. The slant height (s) is the hypotenuse of the right triangle. Your radius (3m) is the base of the triangle and you're given the height (10m).

s² = (3²) + (10²), s² = 109, s = √109 = 10.4m

   Inserting s into the formula SA =  ÃÂ€r² + πrs

SA (container) = π (3²)  + π(3)(10.4) =

              9π + 31.32π =

           28.3 + 98.4 = 126.67m²

   If the container is half full (5m of water in a 10m deep tank) than the SA of the water should equal 1/2 the SA of the container ... 1/2 (126.67m²) = 63.3m².

  To try to calculate the SA of the water using the formula we used to calculate the SA of the container, we would need to know the radius (width) of the cone at the 5m point. If we knew that, we could calculate the slant height using     h = 5, instead of h=10 ... however, since the radius narrows from top to bottom I don't know how to determine r at h = 5.

   I believe that it's safe to presume that the SA of the water at the 5m (halfway point) is equal to 1/2 the SA of the container or 63.3m².

Good luck!

UPDATE -

CORRECTION:

The radius of the cone (water) at h = 5 can be calculated using a ratio.

5/10 = x/3

15 = 10x, x = 1.5.

Calculating the "new" slant height using r = 1.5 and h = 3. s² =(1.5)² + (5)² = 2.25 + 25 = 27.25

s = 5.22 (half of slant height #1)

SA of the water at h = 5 using r = 1.5, s = 5.22

πr² + πrs = π(1.5)² + π(1.5)(5.22) =

                     π(2.25) + π(7.83)=

                         7.07 + 24.6 = 31.67 m²

I'm really surprised that the 1/2 approximation and this calculation are so far off. Mathematically the "new" radius should be accurate as it was calculated using a ratio. It also makes sense that the 2nd slant height was equal to 1/2 the first. I would use 31.67m² ... I believe my previous answer is incorrect ... SORRY :(

Best of luck!