Suppose a, b are positive real numbers such that `asqrt(a) + b sqrt(b) = 183, asqrt(b) + bsqrt(a) = 182`. Find `9/5(a-b)`.

Answer

To solve the problem, we start with the given equations:

1. \( a \sqrt{a} + b \sqrt{b} = 183 \)
2. \( a \sqrt{b} + b \sqrt{a} = 182 \)

Let's denote:
- \( \sqrt{a} = p \)
- \( \sqrt{b} = q \)

Then we can rewrite the equations in terms of \( p \) and \( q \):

1. \( p^3 + q^3 = 183 \)
2. \( p^2 q + q^2 p = 182 \)

The second equation can be factored as:
\[ pq(p + q) = 182 \]

Now, let’s multiply the second equation by 3:
\[ 3pq(p + q) = 3 \times 182 = 546 \]

Next, we add the first equation and the modified second equation:
\[ p^3 + q^3 + 3pq(p + q) = 183 + 546 = 729 \]

Using the identity \( p^3 + q^3 + 3pq(p + q) = (p + q)^3 \), we can write:
\[ (p + q)^3 = 729 \]

Taking the cube root of both sides, we find:
\[ p + q = 9 \]

Now, we know \( p + q = 9 \).

Next, we will use the identity for \( p^3 + q^3 \):
\[ p^3 + q^3 = (p + q)(p^2 - pq + q^2) \]

We can express \( p^2 + q^2 \) in terms of \( p + q \) and \( pq \):
\[ p^2 + q^2 = (p + q)^2 - 2pq = 9^2 - 2pq = 81 - 2pq \]

Substituting this into the equation for \( p^3 + q^3 \):
\[ 183 = 9(81 - 3pq) \]
\[ 183 = 729 - 27pq \]
\[ 27pq = 729 - 183 = 546 \]
\[ pq = \frac{546}{27} = 20.2222 \approx 20.22 \]

Now we have \( p + q = 9 \) and \( pq \approx 20.22 \).

Using these values, we can find \( p \) and \( q \) by solving the quadratic equation:
\[ x^2 - (p + q)x + pq = 0 \]
\[ x^2 - 9x + 20.22 = 0 \]

Using the quadratic formula:
\[ x = \frac{9 \pm \sqrt{9^2 - 4 \cdot 20.22}}{2} \]
\[ x = \frac{9 \pm \sqrt{81 - 80.88}}{2} \]
\[ x = \frac{9 \pm \sqrt{0.12}}{2} \]
\[ x = \frac{9 \pm 0.3464}{2} \]

Calculating the two possible values for \( x \):
1. \( x_1 = \frac{9 + 0.3464}{2} \approx 4.1732 \)
2. \( x_2 = \frac{9 - 0.3464}{2} \approx 4.8268 \)

Thus, we have:
- \( \sqrt{a} \approx 4.1732 \) and \( \sqrt{b} \approx 4.8268 \) (or vice versa).

Now, we find \( a \) and \( b \):
\[ a \approx (4.1732)^2 \approx 17.42 \]
\[ b \approx (4.8268)^2 \approx 23.27 \]

Finally, we need to find \( \frac{9}{5}(a - b) \):
\[ a - b \approx 17.42 - 23.27 \approx -5.85 \]
\[ \frac{9}{5}(a - b) \approx \frac{9}{5} \times (-5.85) \approx -10.53 \]

Thus, the final answer is:
\[ \frac{9}{5}(a - b) \approx -10.53 \]