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.

Solution Steps

To solve this problem, we need to use the concept of direct proportionality. Since $y$ is proportional to the $4^{\text{th}}$ power of $x$, we can write the relationship as $y = k \cdot x^4$, 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 $x = 4$.

1. Use the given values $y = 17$ and $x = 6$ to find the proportionality constant $k$.
2. Substitute $x = 4$ into the equation $y = k \cdot x^4$ to find the new value of $y$.

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

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

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

Final Answer