$f=12$, what is the value of $p$ in this equation?
$p=\frac{x^2}{6}-4$
28
24
22
20
Answer (1)
Substitute each given value of x into the equation p = 6 x 2 − 4 .
Calculate the value of p for each x .
Check which value of x results in an integer value for p .
The value of p when x = 24 is p = 6 2 4 2 − 4 = 96 − 4 = 92 .
Therefore, the value of p is 92 .
Explanation
Understanding the Problem We are given the equation p = 6 x 2 − 4 and four possible values for x : 28, 24, 22, and 20. We need to find the value of p when x is one of these given values.
Testing the values of x Let's test each value of x to see which one results in an integer value for p . This will help us determine which of the given options is the correct value for x .
Calculating p for x=28 If x = 28 , then p = 6 2 8 2 − 4 = 6 784 − 4 = 3 392 − 4 = 3 392 − 3 12 = 3 380 ≈ 126.67. This is not an integer.
Calculating p for x=24 If x = 24 , then p = 6 2 4 2 − 4 = 6 576 − 4 = 96 − 4 = 92. This is an integer.
Calculating p for x=22 If x = 22 , then p = 6 2 2 2 − 4 = 6 484 − 4 = 3 242 − 4 = 3 242 − 3 12 = 3 230 ≈ 76.67. This is not an integer.
Calculating p for x=20 If x = 20 , then p = 6 2 0 2 − 4 = 6 400 − 4 = 3 200 − 4 = 3 200 − 3 12 = 3 188 ≈ 62.67. This is not an integer.
Final Answer Since x = 24 gives an integer value of p = 92 , we can conclude that the value of p is 92 when x = 24 .
Conclusion Therefore, the value of p is 92.
Examples
Understanding how to substitute values into an equation and solve for an unknown variable is a fundamental skill in algebra. This skill is useful in many real-world scenarios, such as calculating the trajectory of a projectile, determining the optimal dimensions for a container, or modeling population growth. For example, if you are designing a bridge, you need to calculate the forces acting on the bridge and ensure that the bridge can withstand those forces. This involves substituting values into equations and solving for unknown variables.
