# A cone has a surface area of #600π cm^2# and a base area of #225π cm^2#. Calculate: the apothem and volume of the cone?

Apothem is

In a cone whose radius at base is

While base area in such case is

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The apothem of the cone can be calculated using the formula: [ \text{Apothem} = \sqrt{\text{Surface Area} - \text{Base Area}} ]

Substitute the given values: [ \text{Apothem} = \sqrt{600\pi - 225\pi} ]

[ \text{Apothem} = \sqrt{375\pi} ]

The volume of the cone can be calculated using the formula: [ \text{Volume} = \frac{1}{3} \times \text{Base Area} \times \text{Height} ]

Given that the base area is ( 225\pi ), we can find the radius (( r )) of the base using the formula for the area of a circle (( A = \pi r^2 )): [ r = \sqrt{\frac{225\pi}{\pi}} = 15 \text{ cm} ]

The slant height (( l )) of the cone can be calculated using the Pythagorean theorem: [ l = \sqrt{r^2 + \text{Apothem}^2} ]

[ l = \sqrt{15^2 + (15\sqrt{375})^2} ]

[ l = \sqrt{225 + 15^2 \times 375} ]

[ l = \sqrt{225 + 84375} ]

[ l = \sqrt{84600} ]

[ l = 290 \text{ cm} ]

Now, the height (( h )) of the cone can be calculated using the Pythagorean theorem: [ h = \sqrt{l^2 - r^2} ]

[ h = \sqrt{290^2 - 15^2} ]

[ h = \sqrt{84100} ]

[ h = 290 \text{ cm} ]

Now, substitute the values into the formula for volume: [ \text{Volume} = \frac{1}{3} \times 225\pi \times 290 ]

[ \text{Volume} = 21750\pi ]

So, the apothem of the cone is ( 15\sqrt{375} ) cm and the volume is ( 21750\pi ) cubic centimeters.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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