# Math Help - Late night Geometry help-Urgent!

1. ## Late night Geometry help-Urgent!

Yes, I know this is the urgent forum. The title means this is EXTRA urgent. As in, before 8 am on Thursday morning (today). I've been working on this math project all night, and there's a few things I'm really having trouble with. If anyone can help me with this, they will earn my eternal gratitude.

It is: A cone is formed from a circle of radius 10 cm with a sector of $90^o$ cut out.

A) What is the slant height of the cone

B) What is the radius of the cone?

C) What is the volume of the cone, to the nearest cubic centimetre?

Ok, I know that "A" is 10 cm, because the slant height of a cone is equal to the radius of the circle it's formed from.

But what about "B"? I know I can't use the Pythagorean Theorem, because I only have slant height...

Once I know the radius, for C I should just be able to use the formula for volume.

There's my first question. here's the next.

There's a rectangular prism which has dimensions 4 by 5 by 10. There's also a rectangular prism which has dimensions three times those of the small box.

a) Compares the volume of the two boxes and explain your answer

b) compare the surface area of the boxes and compare your answer.

the volume of 4 by 5 by 10 is 200
The volume of 12 by 15 by 30 is 5400

I'm not really seeing a correlation... What am I missing?

And finally,

What happens to the volume of a cone under each set of conditions?

a) the radius is unchange but the height is doubled

b) the radius is doubled but the height is unchanged

c) The radius and the hieght are both doubled.

I'm really not sure how to do this one at all- all I know is "It gets bigger"

Your help is really, REALLY appreciated.

2. Originally Posted by cazcaz
...

It is: A cone is formed from a circle of radius 10 cm with a sector of $90^o$ cut out.

A) What is the slant height of the cone

B) What is the radius of the cone?

C) What is the volume of the cone, to the nearest cubic centimetre?

...
Hello,

to B)

the arc of the sector (with one quarter missing) is as large as the circumference of the base circle:

$\frac{270^\circ}{360^\circ}\cdot 2 \pi \cdot 10 = 15 \pi$

the circumference of the base circle is:

$p = 2\pi \cdot r = 15\pi~\implies~r = \frac{15}2$

to C)
Use Pythagorean theorem to calculate the height of the cone:

$h = \sqrt{10^2-7.5^2}=\sqrt{\frac{175}4}\approx 6.614$

The volume of the cone is calcutated by: $V = \frac13 \cdot \pi r^2 \cdot h$

Therefore the volume is:

$V = \frac13 \cdot \pi \cdot 7.5^2 \cdot \sqrt{\frac{175}2} \approx 255\ cm^3$

3. Originally Posted by cazcaz
...
There's a rectangular prism which has dimensions 4 by 5 by 10. There's also a rectangular prism which has dimensions three times those of the small box.

a) Compares the volume of the two boxes and explain your answer

b) compare the surface area of the boxes and compare your answer.

the volume of 4 by 5 by 10 is 200
The volume of 12 by 15 by 30 is 5400

...
Hello,

since every distance is enlarged by the factor 3 the volume is enlarged by the factor $3 \cdot 3 \cdot 3 =3^3 = 27$

In general: If the dimensions of a rectangular prism are enlarged by the faktor k and the original volume is $V_1 = a \cdot b \cdot c$ then

$V_2 = (k\cdot a) \cdot (k\cdot b) \cdot (k\cdot c)= k^3 \cdot a \cdot b \cdot c = k^3 \cdot V_1$

4. Originally Posted by cazcaz
...
What happens to the volume of a cone under each set of conditions?

a) the radius is unchange but the height is doubled

b) the radius is doubled but the height is unchanged

c) The radius and the hieght are both doubled.

...
Hello,

let the original volume be $V_1=\frac13 \cdot r^2 \cdot h$

to a) Plug in 2h instead of h:

$V_2=\frac13 \cdot r^2 \cdot (2h) = 2 \cdot \frac13 \cdot r^2 \cdot h = 2 \cdot V_1$

to b) Plug in 2r instead of r, expand and compare with the original volume.

to c) Plug in 2r instead of r, 2h instead of h, expand the term and compare with the original volume.

5. Thank you SO MUCH for your help!