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.

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.
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Solution

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Solution Steps

To solve this problem, we need to use the concept of direct proportionality. Since y y is proportional to the 4th 4^{\text{th}} power of x x , we can write the relationship as y=kx4 y = k \cdot x^4 , where k k is the proportionality constant. First, we will find the value of k k using the given values of y y and x x . Then, we will use this value of k k to find y y when x=4 x = 4 .

Solution Approach
  1. Use the given values y=17 y = 17 and x=6 x = 6 to find the proportionality constant k k .
  2. Substitute x=4 x = 4 into the equation y=kx4 y = k \cdot x^4 to find the new value of y y .
Step 1: Establish the Proportional Relationship

Given that y y is proportional to the 4th 4^{\text{th}} power of x x , we can express this relationship mathematically as: y=kx4 y = k \cdot x^4 where k k is the proportionality constant.

Step 2: Calculate the Proportionality Constant

Using the provided values y=17 y = 17 when x=6 x = 6 , we can substitute these values into the equation to find k k : 17=k64 17 = k \cdot 6^4 Calculating 64 6^4 : 64=1296 6^4 = 1296 Thus, we have: k=1712960.013117283950617283 k = \frac{17}{1296} \approx 0.013117283950617283

Step 3: Find y y when x=4 x = 4

Now, we substitute x=4 x = 4 into the equation to find the new value of y y : y=k44 y = k \cdot 4^4 Calculating 44 4^4 : 44=256 4^4 = 256 Substituting k k into the equation: y=0.0131172839506172832563.3580246913580245 y = 0.013117283950617283 \cdot 256 \approx 3.3580246913580245 Rounding this value to four significant digits gives: y3.36 y \approx 3.36

Final Answer

Thus, the value of y y when x=4 x = 4 is 3.36\boxed{3.36}.

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