y is inversely proportional to the square of (x + 1). y = 0.875 when x = 1. Find y when x = 4.
Understand the Problem
The question involves a mathematical relationship where y is inversely proportional to the square of (x + 1). We are tasked with finding the value of y when x is 4, given that y is 0.875 when x is 1. This involves calculating the proportionality constant and then using it to find y for the new value of x.
Answer
The value of \( y \) when \( x = 4 \) is $0.14$.
Answer for screen readers
The value of ( y ) when ( x = 4 ) is $0.14$.
Steps to Solve
- Identify the relationship
Since $y$ is inversely proportional to the square of $(x + 1)$, we can express this mathematically as: $$ y = \frac{k}{(x + 1)^2} $$ where $k$ is a proportionality constant we need to determine.
- Determine the proportionality constant ( k )
Given that $y = 0.875$ when $x = 1$, substitute these values into the equation: $$ 0.875 = \frac{k}{(1 + 1)^2} $$ Calculate $(1 + 1)^2 = 2^2 = 4$: $$ 0.875 = \frac{k}{4} $$ Now, solve for ( k ): $$ k = 0.875 \times 4 = 3.5 $$
- Substitute ( k ) back into the equation
Now that we have ( k ), we rewrite the equation for $y$: $$ y = \frac{3.5}{(x + 1)^2} $$
- Calculate ( y ) when ( x = 4 )
Substitute ( x = 4 ) into the equation: $$ y = \frac{3.5}{(4 + 1)^2} $$ Calculate ( (4 + 1)^2 = 5^2 = 25 ): $$ y = \frac{3.5}{25} $$
- Perform the final calculation
Now, divide: $$ y = 0.14 $$
The value of ( y ) when ( x = 4 ) is $0.14$.
More Information
This problem demonstrates the concept of inverse proportionality. Finding the proportionality constant is crucial for using the relationship correctly.
Tips
One common mistake is miscalculating the square when substituting ( x ). Make sure always to add first, then square the result.
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