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