Questions: Suppose that y is proportional to the 4th power of x, and that y=17 when x=6. What is y when x=4? Round your answer to two decimal places if necessary.
Transcript text: Suppose that $y$ is proportional to the $4^{\text {th }}$ power of $x$, and that $y=17$ when $x=6$. What is $y$ when $x=4$ ? Round your answer to two decimal places if necessary.
Solution
Solution Steps
To solve this problem, we need to use the concept of direct proportionality. Since y is proportional to the 4th power of x, we can write the relationship as y=k⋅x4, where k is the proportionality constant. First, we will find the value of k using the given values of y and x. Then, we will use this value of k to find y when x=4.
Solution Approach
Use the given values y=17 and x=6 to find the proportionality constant k.
Substitute x=4 into the equation y=k⋅x4 to find the new value of y.
Step 1: Establish the Proportional Relationship
Given that y is proportional to the 4th power of x, we can express this relationship mathematically as:
y=k⋅x4
where k is the proportionality constant.
Step 2: Calculate the Proportionality Constant
Using the provided values y=17 when x=6, we can substitute these values into the equation to find k:
17=k⋅64
Calculating 64:
64=1296
Thus, we have:
k=129617≈0.013117283950617283
Step 3: Find y when x=4
Now, we substitute x=4 into the equation to find the new value of y:
y=k⋅44
Calculating 44:
44=256
Substituting k into the equation:
y=0.013117283950617283⋅256≈3.3580246913580245
Rounding this value to four significant digits gives:
y≈3.36