Final answer:
To find the equation for y in terms of x, two systems of equations are solved to find the constants a and b. The resulting equation is y = 5x² + 32/x. Using this equation, the value of y when x is 3 is approximately 55.67.
Step-by-step explanation:
Finding the Equation and Value of y in terms of x
We are given that y is the sum of two quantities: one that varies directly as x² and one that varies inversely as x. This can be expressed as:
y = ax² + b/x where a and b are constants.
First, we'll plug in the given values to find the constants:
We can solve these equations simultaneously to find the values of a and b.
Multiplying the first equation by 2 to eliminate the fraction, we get:
We'll multiply the second equation by 4 to align the b terms:
Subtracting the first from the second gives us:
Solving for a gives a = 5 and substituting a back into any of the initial equations gives us b = 32.
The equation for y is:
y = 5x² + 32/x
To find the value of y when x=3, we substitute 3 into our equation for x:
y = 5(3²) + 32/3
y = 45 + 32/3
y = 45 + 10.67
y ≈ 55.67
Therefore, when x is 3, y is approximately 55.67.