Question

The sum of two quantities is y, one of the quantities varies directly as x ² and the other inversely as x. If y =32 , when x=2 and y=86, when x=4, find

A. An equation for y in terms of x.
B. the value of y when x=3

Answer 1

To address the student's question, we first establish an equation for y in terms of x considering one quantity varies directly as x² and the other inversely as x.

Step-by-step explanation:

The student is asking to find an equation for y in terms of x where the sum of two quantities is y, one quantity varies directly as x2 and the other inversely as x. With given values of y for corresponding values of x, we can establish the equation and solve for y when x=3.

Step-by-step explanation:

Let's assume a is the constant of variation for the quantity that varies directly as x2, and b is the constant of variation for the quantity that varies inversely as x.

Together, their sum gives us y:

y = ax2 + b/x

We are given two sets of values to create a system of equations:

Solving the system of equations gives us the values of a and b, which we can then substitute back into the equation for y to find y when x=3.

Finding the values of a and b

1. 32 = 4a + b/2

2. 86 = 16a + b/4

After solving these, we get:

Our equation for y now is y = 6x2 + 64/x.

Value of y when x=3

Substituting x=3 into our equation gives us:

y = 6(3)2 + 64/3 = 54 + 64/3 = 54 + 21.33 ≈ 75.33