How Do You Calculate the Surface Area of a Cone with Equal Diameter and Height?

[DGK]

Homework Statement

Determine a simplified, factorised expression, in terms of the radius (r), for the surface area of a cone where diameter (D) = perpendicular height (h)

Homework Equations

A = πr (r + √(h^2 + r^2))

The Attempt at a Solution

h=D=2r

A = πr (r + √(2r^2 + r^2))

A/π = r (r + √(2r^2 + r^2))

 

[andrewkirk]

You can factorise the inside of your square root, which will set the scene for further simplification. Ideally you want to get r outside of the square root.

 

[DGK]

A/pi = r (r + sqrt( r (2r + r))Is this right? I really don't know what I'm doing!

 

[DGK]

You can factorise the inside of your square root, which will set the scene for further simplification. Ideally you want to get r outside of the square root.

Thanks for your response!

See above for my reply...sorry I'm new to this!

 

[Nathanael]

A/pi = r (r + sqrt( r (2r + r))

You could've factored out another r from the red part. Then use $\sqrt{a^2b}=\sqrt{a^2}\sqrt{b}=a\sqrt{b}$.

Also no need to divide both sides by pi, you want to find A not A/pi.

And what is h²? You seem to say its 2r².

 

[andrewkirk]

DGK there's a further problem that the formula you give for A in the OP under '2. Homework Equations ' is wrong. Show the working by which you got to that formula, and somebody will show you where you went wrong.

 

[ehild]

Homework Statement

Determine a simplified, factorised expression, in terms of the radius (r), for the surface area of a cone where diameter (D) = perpendicular height (h)

Homework Equations

A = πr (r + √(h^2 + r^2))

The Attempt at a Solution

h=D=2r

A = πr (r + √((2r)^2 + r^2))

You miss the parentheses around 2r.

 

[ehild]

DGK there's a further problem that the formula you give for A in the OP under '2. Homework Equations ' is wrong. Show the working by which you got to that formula, and somebody will show you where you went wrong.

The relevant equation A = πr (r + √(h^2 + r^2)) is correct.

 

[andrewkirk]

The relevant equation A = πr (r + √(h^2 + r^2)) is correct.

It is if you include the base. I assumed he wasn't.

 

[ehild]

It is if you include the base. I assumed he wasn't.

Why? It would be the lateral surface area.

 

[HallsofIvy]

Here's another way of looking at it. A cone with base radius r and height h has "slant height" $\sqrt{r^2+ h^2}$ by the Pythagorean theorem. Imagine cutting a slit from the base to the tip of the cone, then flattening it. (A cone is a "developable surface" and can be flattened without warping.)

It will flatten to part of a disk with radius $\sqrt{r^2+ h^2}$. To see what part, look at the two circumferences. Since the cone had base radius r, the circumference of that base is $2\pi r$. A circle with radius $\sqrt{r^2+ h^2}$ has circumference $2\pi\sqrt{r^2+ h^2}$.

[ mod edit ]

 

[John Verghese]

Isn't the relevant equation
πr^2 + πrL
L = √(r^2+h^2)
= √(r^2+(2r)^2)
L^2 = r^2+(2r)^2
= (3r)^2
L = √3r

 

[ehild]

Isn't the relevant equation
πr^2 + πrL
L = √(r^2+h^2)
= √(r^2+(2r)^2)
L^2 = r^2+(2r)^2
= (3r)^2
L = √3r

It is wrong. Expand (2r)^2. It is not 2r^2!

 

[John Verghese]

Isn't the relevant equation
πr^2 + πrL
L = √(r^2+h^2)
= √(r^2+(2r)^2)
L^2 = r^2+(2r)^2
= (3r)^2
L = √3r

Then:
TSA = πr^2 + πr√3r
= πr^2 + √3πr^2
= πr^2(1+√3)

 

[John Verghese]

It is wrong. Expand (2r)^2. It is not 2r^2!

Ahhh ok thanks heaps (this exact question is in my assignment)
would it instead be:
L^2 = r^2 + (2r)^2
= r^2 + 4r^2
= 5r^2
L = √5r
or
L = 5√r
?

 

[ehild]

Which one do you think? How do you apply the square root to a product? What is √(ab)?

 

[John Verghese]

Which one do you think? How do you apply the square root to a product? What is √(ab)?

so is it:
L = √5r
or
L = 5√r

 

[ehild]

You have to know. Answer my question: √(ab)=?

 

[John Verghese]

The next part of the question is:
A rectangular prism has:
- Height equivalent to the slant height (L) of the cone in part a
- Length twice the diameter of the cone in part a
- width 5 times the radius of the cone in part a
Determine a simplified factorised expression, in terms of the radius (r), for the volume of the rectangular prism.

All i have so far is:
V = LxWxH
H = √5r or 5√r
L = 2D = 2(2r) = 4r
W = 5r
V = 4r x 5r x ...
and I don't know where to go from there.