Dimensional Analysis Practice II - Comcast.net - Free Printable

Let's solve each problem step by step.

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1. What are the correct units for the answer to a problem if the following series of conversion factor units are used?

Given:

```
zog² / zog × lump / twillip × dunks / zog × dunks / twillip
```

We need to simplify this expression by canceling out units.

Let’s write it as:

$$
\frac{zog^2}{zog} \times \frac{lump}{twillip} \times \frac{dunks}{zog} \times \frac{dunks}{twillip}
$$

Now simplify each term:

- $ \frac{zog^2}{zog} = zog $
- So now: $ zog \times \frac{lump}{twillip} \times \frac{dunks}{zog} \times \frac{dunks}{twillip} $

Now cancel $ zog $ in numerator and denominator:

- $ zog \times \frac{1}{zog} = 1 $ → cancels out

So we’re left with:

$$
\frac{lump}{twillip} \times \frac{dunks}{1} \times \frac{dunks}{twillip} = \frac{lump \cdot dunks \cdot dunks}{twillip \cdot twillip}
$$

Which is:

$$
\frac{lump \cdot dunks^2}{twillip^2}
$$

✔ Answer:
lump·dunks² / twillip²

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2. Evaluate the following:

$$
\frac{(6.02 \times 10^{-23})(7.11 \times 10^{-31})(3.98 \times 10^{24})(3.82 \times 10^8)}{(3.92 \times 10^{-16})(4 \times 10^8)(6.99 \times 10^{16})(2.99 \times 10^{30})}
$$

#### Step 1: Multiply numerators and denominators separately

Numerator:
$$
(6.02)(7.11)(3.98)(3.82) \times 10^{-23 -31 +24 +8}
$$

First, calculate exponents:
$$
-23 -31 +24 +8 = (-54) + (32) = -22
$$

Now multiply coefficients:
$$
6.02 × 7.11 = 42.7822 \\
42.7822 × 3.98 ≈ 169.962956 \\
169.962956 × 3.82 ≈ 650.25
$$

So numerator ≈ $ 6.5025 \times 10^2 \times 10^{-22} = 6.5025 \times 10^{-20} $

Wait — better to keep full precision and do all at once.

Alternatively, let's compute more accurately:

Use calculator-style approximation:

- 6.02 × 7.11 = 42.7822
- 42.7822 × 3.98 ≈ 169.962956
- 169.962956 × 3.82 ≈ 650.25 → ~6.5025 × 10²

So numerator: $ 6.5025 \times 10^{-20} $? Wait — exponent was -22, so:

Actually: $ 6.5025 \times 10^2 \times 10^{-22} = 6.5025 \times 10^{-20} $

But wait — 6.5025 × 10² is 650.25, and 650.25 × 10^{-22} = 6.5025 × 10^{-20}

Yes.

Now Denominator:

Coefficients: $ 3.92 × 4 × 6.99 × 2.99 $

Exponent: $ -16 + 8 + 16 + 30 = (-16 + 8) + (16 + 30) = (-8) + 46 = 38 $

So exponent: $ 10^{38} $

Now coefficients:

- 3.92 × 4 = 15.68
- 15.68 × 6.99 ≈ ?

15.68 × 7 = 109.76
But 6.99 is 0.01 less → subtract 15.68 × 0.01 = 0.1568
So 109.76 - 0.1568 = 109.6032

Then × 2.99:

109.6032 × 3 = 328.8096
Subtract 109.6032 × 0.01 = 1.096032 → 328.8096 - 1.096032 = 327.713568

So denominator ≈ $ 3.2771 × 10^2 \times 10^{38} = 3.2771 × 10^{40} $

Wait — no: 327.713568 = 3.2771 × 10²

So total denominator: $ 3.2771 × 10^{40} $

Now divide:

$$
\frac{6.5025 \times 10^{-20}}{3.2771 \times 10^{40}} = \frac{6.5025}{3.2771} \times 10^{-20 - 40} = \frac{6.5025}{3.2771} \times 10^{-60}
$$

Now divide:

6.5025 ÷ 3.2771 ≈ 1.984

So result ≈ $ 1.984 \times 10^{-60} $

Let’s double-check:

Better to use exact values.

Alternatively, let's do it step-by-step with logs or scientific notation.

But since this is practice, we can accept approx.

Let me recalculate more carefully:

Numerator:

6.02 × 7.11 = 42.7822
42.7822 × 3.98 = ?

42.7822 × 4 = 171.1288
Minus 42.7822 × 0.02 = 0.855644
→ 171.1288 - 0.855644 = 170.273156

Then × 3.82:

170.273156 × 3.82

Break into:

170.273156 × 3 = 510.819468
170.273156 × 0.8 = 136.2185248
170.273156 × 0.02 = 3.40546312

Add: 510.819468 + 136.2185248 = 647.0379928
+ 3.40546312 = 650.44345592

So numerator coefficient: ~650.44
Exponent: -23 -31 +24 +8 = (-54) + 32 = -22 → $ 6.5044 \times 10^2 \times 10^{-22} = 6.5044 \times 10^{-20} $

Denominator:

3.92 × 4 = 15.68
15.68 × 6.99 = ?

15.68 × 7 = 109.76
Minus 15.68 × 0.01 = 0.1568 → 109.76 - 0.1568 = 109.6032

Then × 2.99 = 109.6032 × (3 - 0.01) = 109.6032×3 = 328.8096; minus 1.096032 = 327.713568

So 3.2771 × 10²

Exponent: -16 + 8 + 16 + 30 = 38 → $ 3.2771 \times 10^{40} $

Now:

$$
\frac{6.5044 \times 10^{-20}}{3.2771 \times 10^{40}} = \frac{6.5044}{3.2771} \times 10^{-60} \approx 1.984 \times 10^{-60}
$$

✔ Answer:
≈ $ 1.98 \times 10^{-60} $ (rounded to three significant figures)

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3. Given the following equivalents, convert 1 fizzle to frizzles.

Given:

- 3 swizzles = 7 twizzles
- 1 fizzle = 2 drizzles
- 2 twizzles = 14 sizzles
- 1 swizzle = 20 frizzles
- 8 drizzles = 6 sizzles

We want: 1 fizzle → ? frizzles

Start from fizzle → drizzles → sizzles → twizzles → swizzles → frizzles

Step-by-step:

1. 1 fizzle = 2 drizzles (given)

2. drizzles → sizzles:
8 drizzles = 6 sizzles → so 1 drizzle = 6/8 = 0.75 sizzles
So 2 drizzles = 2 × 0.75 = 1.5 sizzles

3. sizzles → twizzles:
2 twizzles = 14 sizzles → so 1 twizzle = 7 sizzles
Therefore, sizzles → twizzles: divide by 7
1.5 sizzles = 1.5 / 7 = 0.2142857... twizzles

4. twizzles → swizzles:
We have: 3 swizzles = 7 twizzles → so 1 twizzle = 3/7 swizzles
So 0.2142857 twizzles = (3/7) × 0.2142857 = ?

Compute:
0.2142857 × 3 / 7 = (0.2142857 × 3) / 7 = 0.6428571 / 7 ≈ 0.0918367 swizzles

5. swizzles → frizzles:
1 swizzle = 20 frizzles → so 0.0918367 swizzles = 0.0918367 × 20 = 1.836734 frizzles

So 1 fizzle ≈ 1.84 frizzles

But let’s do it exactly using fractions.

---

Exact calculation:

1. 1 fizzle = 2 drizzles

2. 8 drizzles = 6 sizzles → so 2 drizzles = (6/8) × 2 = 12/8 = 3/2 sizzles

3. 2 twizzles = 14 sizzles → so 1 twizzle = 7 sizzles → 1 sizzle = 1/7 twizzle
So 3/2 sizzles = (3/2) × (1/7) = 3/14 twizzles

4. 3 swizzles = 7 twizzles → so 1 twizzle = 3/7 swizzles
So 3/14 twizzles = (3/14) × (3/7) = 9/98 swizzles

5. 1 swizzle = 20 frizzles → so 9/98 swizzles = (9/98) × 20 = 180/98 = 90/49 frizzles

Simplify: 90 ÷ 49 ≈ 1.8367

So 1 fizzle = 90/49 frizzles ≈ 1.84 frizzles

✔ Answer:
$ \frac{90}{49} $ frizzles or ≈ 1.84 frizzles

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4. Convert 20,000 leagues to yards

Given:

- 12 in = 1 ft
- 1 ft = 1 yd
- 1 fathom = 2 yards
- 1 statute mile = 5280 ft
- 1 nautical mile = 6080 ft
- 1 league = 3 nautical miles

We want: 20,000 leagues → ? yards

Step-by-step:

1. 1 league = 3 nautical miles
So 20,000 leagues = 20,000 × 3 = 60,000 nautical miles

2. 1 nautical mile = 6080 ft
So 60,000 nautical miles = 60,000 × 6080 ft

Calculate:
60,000 × 6080 = 6 × 10⁴ × 6.08 × 10³ = 3.648 × 10⁸ ft

3. Now convert feet to yards:
1 yard = 3 feet → so divide by 3

$$
\frac{3.648 \times 10^8}{3} = 1.216 \times 10^8 \text{ yards}
$$

So:

20,000 leagues = 121,600,000 yards

✔ Answer:
1.216 × 10⁸ yards

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✔ Final Answers:

1. lump·dunks² / twillip²
2. ≈ $ 1.98 \times 10^{-60} $
3. $ \frac{90}{49} $ frizzles or ≈ 1.84 frizzles
4. 1.216 × 10⁸ yards

Let me know if you'd like these formatted neatly!