If $p$ is inversely proportional to the square of $q$ , and $p$ is $14$ when $q$ is $8$ , determine $p$ when $q$ is equal to $4$ . $\newline$ Answer:

View step-by-step help

Home

Math Problems

Algebra 1

Write and solve inverse variation equations

Full solution

Q. If

p

is inversely proportional to the square of

q

, and

p

14

when

q

8

, determine

p

when

q

is equal to

4

\newline

Answer:

Write Inverse Proportionality Equation: Given that $p$ is inversely proportional to the square of $q$ , we can write this relationship as $p = \frac{k}{{q^2}}$ , where $k$ is the constant of proportionality.
Find Constant of Proportionality: We know that $p = 14$ when $q = 8$ . Let's substitute these values into the equation to find $k$ . $\newline$ $14 = \frac{k}{8^2}$
Substitute $q=4$ to Find $p$ : Calculating the square of $8$ and solving for $k$ gives us: $\newline$ $14 = \frac{k}{64}$ $\newline$ To find $k$ , multiply both sides by $64$ : $\newline$ $14 \times 64 = k$
Substitute $q=4$ to Find $p$ : Calculating the square of $8$ and solving for $k$ gives us: $\newline$ $14 = \frac{k}{64}$ $\newline$ To find $k$ , multiply both sides by $64$ : $\newline$ $14 \times 64 = k$ Performing the multiplication, we get: $\newline$ $k = 896$ $\newline$ Now we have the constant of proportionality.
Substitute $q=4$ to Find $p$ : Calculating the square of $8$ and solving for $k$ gives us: $\newline$ $14 = \frac{k}{64}$ $\newline$ To find $k$ , multiply both sides by $64$ : $\newline$ $14 \times 64 = k$ Performing the multiplication, we get: $\newline$ $k = 896$ $\newline$ Now we have the constant of proportionality.With $k$ found, we can now write the complete inverse proportionality equation as: $\newline$ $p$ $0$
Substitute $q=4$ to Find $p$ : Calculating the square of $8$ and solving for $k$ gives us: $\newline$ $14 = \frac{k}{64}$ $\newline$ To find $k$ , multiply both sides by $64$ : $\newline$ $14 \times 64 = k$ Performing the multiplication, we get: $\newline$ $k = 896$ $\newline$ Now we have the constant of proportionality.With $k$ found, we can now write the complete inverse proportionality equation as: $\newline$ $p$ $0$ To find $p$ when $p$ $2$ , substitute $p$ $3$ into the equation for $p$ $4$ : $\newline$ $p$ $5$
Substitute $q=4$ to Find $p$ : Calculating the square of $8$ and solving for $k$ gives us: $\newline$ $14 = \frac{k}{64}$ $\newline$ To find $k$ , multiply both sides by $64$ : $\newline$ $14 \times 64 = k$ Performing the multiplication, we get: $\newline$ $k = 896$ $\newline$ Now we have the constant of proportionality.With $k$ found, we can now write the complete inverse proportionality equation as: $\newline$ $p$ $0$ To find $p$ when $p$ $2$ , substitute $p$ $3$ into the equation for $p$ $4$ : $\newline$ $p$ $5$ Calculating the square of $p$ $3$ and solving for $p$ gives us: $\newline$ $p$ $8$
Substitute $q=4$ to Find $p$ : Calculating the square of $8$ and solving for $k$ gives us: $\newline$ $14 = \frac{k}{64}$ $\newline$ To find $k$ , multiply both sides by $64$ : $\newline$ $14 \times 64 = k$ Performing the multiplication, we get: $\newline$ $k = 896$ $\newline$ Now we have the constant of proportionality.With $k$ found, we can now write the complete inverse proportionality equation as: $\newline$ $p$ $0$ To find $p$ when $p$ $2$ , substitute $p$ $3$ into the equation for $p$ $4$ : $\newline$ $p$ $5$ Calculating the square of $p$ $3$ and solving for $p$ gives us: $\newline$ $p$ $8$ Performing the division, we get: $\newline$ $p$ $9$ $\newline$ This is the value of $p$ when $p$ $4$ is $p$ $3$ .