Triangular pool needs to be filled with water. The corners are the coordinates. Units represent feet

Question

A triangular pool needs to be filled the coordinates are A(-6,-4) B(-2,12)C(4,-2). If the pool needs to be filled with 100 feet cube determine the depth of the pool? (assume the depth is in uniform)

Answer 1

First in order to find the depth we need to find the area of the triangle given. If we assume that the points are given to us in feet we can form a triangle, call it ABC.

 

The equation below describes the area of a triangle.

 

A=(1/2)bh

 

We will also have to use the distance formula later on shown below:

 

D= sqrt((x₁-x₂)^2+(y₁-y₂)^2)

 

 

 

now we can find the area, however, we must remember that our area formula only works for right triangles so we must form one.

 

 

Now we must find a line that goes through AC to do this we will use these formulas

 

y-y₁=m(x-x₁)

m=y₁-y₂/x₁-x₂

 

AC_m= 2/10 =-1/5

 

y-(-4)=-1/5*(x-6)

y+4=1/5x+6/5

y=-1/5x+6/5-20/5

y=1/5x-2.8

 

Now we must find a line perpendicular to this using the opposite reciprocal and point B

 

y-12=-5(x-(-2))

y=-5x-10+12

y=-5x+2

 

So with this perpendicular line, we split the triangle into two. Now we must find the point that the two lines intercept

 

-5x+2=1/5x-2.8

4.8=26/5x

x=4.8/5.2 or ~0.92

-5(4.8/5.2)+2=y

y= -2.62

 

Now we find the distance between this point (0.92,-2.62), call it D, and A, B, and C. 

 

DA=sqrt((0.92-(-6))^2+(-2.62-(-4)^2)= 7.13540... ~7.14

DB=sqrt((0.92-(-2))^2+(-2.62-12)^2)= 14.908749... ~14.91

DC=sqrt((0.92-(4))^2+(-2.62-(-2)^2)= 3.09095... ~3.09

 

Now we can find the area of our two triangles knowing that

 

DA = b₁

DB = h

DC = b₂ 

 

So we solve

 

Triangle 1 ABD

A₁ = b₁h/2

A₁ = (7.14*14.91)/2 = 53.2287 ft^2

 

Triangle 2 BCD

A₂= b₂h/2

A₂= (3.09*14.91)/2= 23.03595 ft^2

 

Entire Triangle ABC

 

A₁+A₂=53.2287+23.03595=76.26465 ft^2

 

Now we must find the depth. The formula shown below is for a triangular prism which is what our pool is. So to find our depth we must solve for d. 

 

V=1/2*b*h*d

 

 100=76.26465*d

100/76.26465=d

d=1.311 ft

 

So the depth of our pool is 1.311 ft

 

Hope this helps

 

 

 

Answer 2

You need Heron's Formula.

First calculate the length of each side using ((x1-x2)^2 + (y1-y2)^2)^.5 for each side.

Use these sides in Heron's formula area=(s*(s-a)*(s-b)*(s-c))^.5 to get the area of the pool, where s=(a+b+c)*.5.

100/area of the pool will give you the depth.

I got the area as 75.966 square feet and the depth as 1.32 feet.

However, this is a bunch of nasty calculator arithmetic so you need to do it all yourself and be sure.

There may be a way to do it directly from the co-ordinates, but I am not sure.

I just checked and there is a way directly from the co-ordinates using a 3x3 determinant and you can check on this too.