Math / Geometry

Question9. The volume and surface area of a square pyramid are given, respectively, by the formulas $V=\frac{1}{3} s^{2} h$ and $S A=\frac{1}{2}(4 s) L+s^{2}$ . Which expression represents the surface area-to-volume ratio for a square pyramid in simplest form?

Studdy Solution

STEP 1

What is this asking? We need to find the simplest expression for the ratio of surface area to volume of a square pyramid. Watch out! Don't forget to simplify the expression as much as possible!
Also, remember the order: surface area to volume, not the other way around.

STEP 2

1. Set up the ratio.
2. Simplify the expression.

STEP 3

We're asked to find the surface area-to-volume ratio, which we can write as $\frac{SA}{V}$ .
This means surface area divided by volume.
Remember, order matters in ratios!

STEP 4

We know the formulas for both surface area and volume.
Let's plug them in: $\frac{SA}{V} = \frac{\frac{1}{2}(4s)L + s^2}{\frac{1}{3}s^2 h}$ Look at that beautiful ratio!

STEP 5

We can simplify the numerator by multiplying $\frac{1}{2}$ by $4s$ : $\frac{1}{2} \cdot 4s = \frac{4s}{2} = 2s$ So, the numerator becomes $2sL + s^2$ .

STEP 6

Remember, dividing by a fraction is the same as multiplying by its reciprocal.
The reciprocal of $\frac{1}{3}$ is $3$ , so we have: $\frac{2sL + s^2}{\frac{1}{3}s^2 h} = \frac{3(2sL + s^2)}{s^2 h} = \frac{6sL + 3s^2}{s^2 h}$ We're getting closer to a simpler form!

STEP 7

Notice that both terms in the numerator have a common factor of $s$ .
Let's factor it out: $\frac{6sL + 3s^2}{s^2 h} = \frac{s(6L + 3s)}{s \cdot s \cdot h}$ Now we can divide to one by canceling out one $s$ from the top and bottom: $\frac{s(6L + 3s)}{s \cdot s \cdot h} = \frac{6L + 3s}{sh}$ Look how much cleaner that looks!

STEP 8

We can also write the fraction as the sum of two simpler fractions: $\frac{6L + 3s}{sh} = \frac{6L}{sh} + \frac{3s}{sh} = \frac{6L}{sh} + \frac{3}{h}$ This might be useful in some cases, but both forms are considered simplified.

STEP 9

The simplified surface area-to-volume ratio for a square pyramid is $\frac{6L + 3s}{sh}$ or $\frac{6L}{sh} + \frac{3}{h}$ .

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Question9. The volume and surface area of a square pyramid are given, respectively, by the formulas $V=\frac{1}{3} s^{2} h$ and $S A=\frac{1}{2}(4 s) L+s^{2}$ . Which expression represents the surface area-to-volume ratio for a square pyramid in simplest form?

Studdy Solution

STEP 1

What is this asking? We need to find the simplest expression for the ratio of surface area to volume of a square pyramid. Watch out! Don't forget to simplify the expression as much as possible!
Also, remember the order: surface area to volume, not the other way around.

STEP 2

1. Set up the ratio.
2. Simplify the expression.

STEP 3

We're asked to find the surface area-to-volume ratio, which we can write as $\frac{SA}{V}$ .
This means surface area divided by volume.
Remember, order matters in ratios!

STEP 4

We know the formulas for both surface area and volume.
Let's plug them in: $\frac{SA}{V} = \frac{\frac{1}{2}(4s)L + s^2}{\frac{1}{3}s^2 h}$ Look at that beautiful ratio!

STEP 5

We can simplify the numerator by multiplying $\frac{1}{2}$ by $4s$ : $\frac{1}{2} \cdot 4s = \frac{4s}{2} = 2s$ So, the numerator becomes $2sL + s^2$ .

STEP 6

Remember, dividing by a fraction is the same as multiplying by its reciprocal.
The reciprocal of $\frac{1}{3}$ is $3$ , so we have: $\frac{2sL + s^2}{\frac{1}{3}s^2 h} = \frac{3(2sL + s^2)}{s^2 h} = \frac{6sL + 3s^2}{s^2 h}$ We're getting closer to a simpler form!

STEP 7

STEP 8

We can also write the fraction as the sum of two simpler fractions: $\frac{6L + 3s}{sh} = \frac{6L}{sh} + \frac{3s}{sh} = \frac{6L}{sh} + \frac{3}{h}$ This might be useful in some cases, but both forms are considered simplified.

STEP 9

The simplified surface area-to-volume ratio for a square pyramid is $\frac{6L + 3s}{sh}$ or $\frac{6L}{sh} + \frac{3}{h}$ .

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Studdy Solution

STEP 1

What is this asking? We need to find the simplest expression for the ratio of surface area to volume of a square pyramid. Watch out! Don't forget to simplify the expression as much as possible!Also, remember the order: surface area to volume, not the other way around.

STEP 2

1. Set up the ratio.2. Simplify the expression.

STEP 3

We're asked to find the surface area-to-volume ratio, which we can write as SAV\frac{SA}{V}VSA​.This means surface area divided by volume.Remember, order matters in ratios!

STEP 4

We know the formulas for both surface area and volume.Let's **plug them in**: SAV=12(4s)L+s213s2h \frac{SA}{V} = \frac{\frac{1}{2}(4s)L + s^2}{\frac{1}{3}s^2 h} VSA​=31​s2h21​(4s)L+s2​ Look at that beautiful ratio!

STEP 5

We can simplify the numerator by multiplying 12\frac{1}{2}21​ by 4s4s4s: 12⋅4s=4s2=2s \frac{1}{2} \cdot 4s = \frac{4s}{2} = 2s 21​⋅4s=24s​=2s So, the numerator becomes 2sL+s22sL + s^22sL+s2.

STEP 6

STEP 7

STEP 8

We can also write the fraction as the sum of two simpler fractions: 6L+3ssh=6Lsh+3ssh=6Lsh+3h \frac{6L + 3s}{sh} = \frac{6L}{sh} + \frac{3s}{sh} = \frac{6L}{sh} + \frac{3}{h} sh6L+3s​=sh6L​+sh3s​=sh6L​+h3​ This might be useful in some cases, but both forms are considered simplified.

STEP 9

The simplified surface area-to-volume ratio for a square pyramid is 6L+3ssh\frac{6L + 3s}{sh}sh6L+3s​ or 6Lsh+3h\frac{6L}{sh} + \frac{3}{h}sh6L​+h3​.

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What is this asking? We need to find the simplest expression for the ratio of surface area to volume of a square pyramid. Watch out! Don't forget to simplify the expression as much as possible!
Also, remember the order: surface area to volume, not the other way around.

1. Set up the ratio.
2. Simplify the expression.

We're asked to find the surface area-to-volume ratio, which we can write as $\frac{SA}{V}$ .
This means surface area divided by volume.
Remember, order matters in ratios!

We know the formulas for both surface area and volume.
Let's plug them in: $\frac{SA}{V} = \frac{\frac{1}{2}(4s)L + s^2}{\frac{1}{3}s^2 h}$ Look at that beautiful ratio!

We can simplify the numerator by multiplying $\frac{1}{2}$ by $4s$ : $\frac{1}{2} \cdot 4s = \frac{4s}{2} = 2s$ So, the numerator becomes $2sL + s^2$ .

We can also write the fraction as the sum of two simpler fractions: $\frac{6L + 3s}{sh} = \frac{6L}{sh} + \frac{3s}{sh} = \frac{6L}{sh} + \frac{3}{h}$ This might be useful in some cases, but both forms are considered simplified.

The simplified surface area-to-volume ratio for a square pyramid is $\frac{6L + 3s}{sh}$ or $\frac{6L}{sh} + \frac{3}{h}$ .