Question

Use the given informationTo answer the question The distance s that an object falls varies directly with the square of the time t of the fall if an object falls 16 feet in one second how long does it take for the object to fall 144 feet?

Answer 1

Given the question, the following are the solution steps to answer the question.

STEP 1: Define direct variation

Direct variation describes a simple relationship between two variables . We say y varies directly with x (or as x , in some textbooks) if: y=kx. for some constant k , called the constant of variation or constant of proportionality.

STEP 2: Represent the statements in mathematical terms

`\begin{gathered} s\text{ varies directly as the square of t} \\ s\propto t^2 \end{gathered}`

STEP 3: Express the variation in form of an equation

`\begin{gathered} s\propto t^2 \\ Introducing\text{ the constant of variation, we have;} \\ s=kt^2----equation\text{ 1} \\ \\ Making\text{ k the subject of the equation, we have;} \\ Divide\text{ both sides by t}^2 \\ (s)/(t^2)=(kt^2)/(t^2) \\ k=(s)/(t^2)-----equation\text{ 2} \end{gathered}`

STEP 4: Calculate the value of k using the given values and the equation in step 3

`\begin{gathered} k=(s)/(t^2) \\ s=16,t=1 \\ By\text{ substitution,} \\ k=(16)/(1^2)=(16)/(1)=16 \\ \\ \therefore k=16 \end{gathered}`

STEP 5: Make t the subject of the equation 1 in step 3

`\begin{gathered} From\text{ equation 1 in step 3,} \\ \begin{equation*} s=kt^2 \end{equation*} \\ Make\text{ t the subject of the formula, divide both sides by k} \\ (s)/(k)=(kt^2)/(k) \\ (s)/(k)=t^2 \\ \\ Find\text{ the square root of both sides} \\ \sqrt{(s)/(k)}=√(t^2) \\ t=\sqrt{(s)/(k)} \end{gathered}`

STEP 6: Use the derived equation in step 5 to find the value of t when s = 144

`\begin{gathered} t=\sqrt{(s)/(k)} \\ t=\sqrt{(144)/(16)} \\ t=√(9) \\ t=3secs \end{gathered}`

Hence, the time it takes the object to fall from 144 feet is 3 seconds