A question was posed a while back regarding a tough related rates problem regarding a cone. This got me to wondering.
"A conical tank with a 10 feet radius and 24 feet high has water coming in at 20 ft^3/min. How fast is the water height changing when it is 6 feet high?".
The spin on this problem is that the cone is lying on its side instead of the usual apex down.
What we have is a horizontal plane slicing the cone. This creates a parabola. The equation of a cone is \(\displaystyle \L\\z^{2}=a^{2}(x^{2}+y^{2})\). The equation of a plane is \(\displaystyle \L\\z=ay+c\)
I set these equal and got \(\displaystyle \L\\y=\frac{a}{2c}x^{2}-\frac{c}{2a}\)
Now, does anyone know how to derive a parabola equation that could solve this?.
Would we need the angle of rotation?. How do we find a and c?.
The side of the cone not lying on the x-axis could be a line that we could implement in our formula?.
"A conical tank with a 10 feet radius and 24 feet high has water coming in at 20 ft^3/min. How fast is the water height changing when it is 6 feet high?".
The spin on this problem is that the cone is lying on its side instead of the usual apex down.
What we have is a horizontal plane slicing the cone. This creates a parabola. The equation of a cone is \(\displaystyle \L\\z^{2}=a^{2}(x^{2}+y^{2})\). The equation of a plane is \(\displaystyle \L\\z=ay+c\)
I set these equal and got \(\displaystyle \L\\y=\frac{a}{2c}x^{2}-\frac{c}{2a}\)
Now, does anyone know how to derive a parabola equation that could solve this?.
Would we need the angle of rotation?. How do we find a and c?.
The side of the cone not lying on the x-axis could be a line that we could implement in our formula?.