Problem Analysis #27

To solve the problem, we are given that a maximum area has value $100$$\sqrt{8}$$$, and we are asked to express this in the form:

$$\frac{p\sqrt{q}}{r}$$

where:

• $p$, $q$, and $r$ are positive integers,
• $q$ is squarefree (i.e., it has no repeated prime factors),
• $\gcd(p, r) = 1$.

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Step 1: Simplify the Given Expression

We start with:

$$100\sqrt{8}$$

Recall that:

$$\sqrt{8} = \sqrt{4 \cdot 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2}$$

So:

$$100\sqrt{8} = 100 \cdot 2\sqrt{2} = 200\sqrt{2}$$

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Step 2: Express in the Required Form

We now write:

$$200\sqrt{2} = \frac{200\sqrt{2}}{1}$$

This matches the form:

$$\frac{p\sqrt{q}}{r}$$

with:

• $p = 200$,
• $q = 2$,
• $r = 1$.

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Step 3: Verify Conditions

Let’s check each condition:

• Positive integers: All values $p = 200$, $q = 2$, $r = 1$ are positive integers.
• Squarefree $q$: $2$ is squarefree since its only prime factor is $2$, and it does not repeat.
• $\gcd(p, r) = 1$: $\gcd(200, 1) = 1$, which satisfies the requirement.

Thus, the expression is correctly written in required form.

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Step 4: Compute $p + q + r$

$$p + q + r = 200 + 2 + 1 = \boxed{203}$$

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Final Answer

$$\boxed{203}$$