if x is directy proportinal to y cube and y=2 when x= 1, (a) find an equation connecting x and y. (b) find the value of x when y = 1 (c) find the value of y when x=108.
x = ky^3
1 = k*2^3
k = 1/8
now just crank out the answers
x ∝ y^3 ---> x = ky^3 , where k is a constant
given: y=2 when x=1
1 = 8k
k = 1/8
x = (1/8)y^3
when y = 1
x = (1/8)(1^3) = 1/8
when x = 108
108 = (1/8)y^3
864 = y^3
y = appr 9.5244
To solve this problem, we will use the concept of direct proportionality. Remember that when two variables, x and y, are directly proportional, their relationship can be represented by the equation x = ky^n, where k is the constant of proportionality.
(a) To find an equation connecting x and y, we need to determine the value of k and the exponent, n. We are given that when y = 2, x = 1. Substituting these values into the equation, we get:
1 = k * 2^n
Next, we need to eliminate k by dividing both sides by k:
1/k = 2^n
Now, we can use logarithms to solve for n. Taking the logarithm of both sides, we have:
log(1/k) = log(2^n)
Using the property of logarithms, we can bring down the exponent:
log(1/k) = n * log(2)
Solving for n, we divide both sides by log(2):
n = log(1/k) / log(2)
Now, we can substitute the given values of x and y (1 and 2, respectively) into the equation:
n = log(1/k) / log(2) = log(1/k) / 0.301
To get the value of n, we can use a scientific calculator or an online logarithm calculator to find the logarithm value. Let's say n is approximately 3.
Therefore, our equation connecting x and y is x = ky^3.
(b) To find the value of x when y = 1, we can substitute the values into our equation:
1 = k * 1^3
1 = k * 1
k = 1
Substituting the value of k into the equation, we find:
x = 1 * y^3
x = y^3
When y = 1, we get:
x = 1^3
x = 1
Therefore, when y = 1, x = 1.
(c) Finally, to find the value of y when x = 108, we can rearrange the equation:
x = ky^3
Substituting the given value of x, we have:
108 = k * y^3
To find the value of y, let's rearrange the equation again by solving for y:
y^3 = 108 / k
y^3 = 108 / 1
y^3 = 108
y ≈ ∛108 ≈ 4.326
Therefore, when x = 108, the approximate value of y is 4.326.