$r$ is inversely proportional to $\sqrt{3}$. $s$ is directly proportional to $\sqrt{t}$. Given that $r=10$ and $t=16$ when $s=49$, find a formula for $r$ in terms of $t$.
Answer (1)
Express r in terms of s as r = s k 1 .
Express s in terms of t as s = k 2 t .
Substitute s into the equation for r to get r = 4 t k .
Use the given values to find k = 20 , so the formula is r = 4 t 20 .
Explanation
Understanding the Problem We are given that r is inversely proportional to s and s is directly proportional to t . We are also given that r = 10 and t = 16 when s = 49 . Our goal is to find a formula for r in terms of t .
Expressing r in terms of s Since r is inversely proportional to s , we can write this relationship as: r = s k 1 ,where k 1 is a constant of proportionality.
Expressing s in terms of t Since s is directly proportional to t , we can write this relationship as: s = k 2 t ,where k 2 is a constant of proportionality.
Substituting s into r Now, we substitute the expression for s in terms of t into the equation for r : r = k 2 t k 1 = k 2 t 1/4 k 1 = 4 t k ,where k = k 2 k 1 is a constant.
Finding the Constants We are given that r = 10 when t = 16 and s = 49 . Let's use this information to find the constant k . First, let's find k 2 using s = 49 and t = 16 : 49 = k 2 16 = k 2 ( 4 ) k 2 = 4 49 = 12.25 Next, let's find k 1 using r = 10 and s = 49 : 10 = 49 k 1 = 7 k 1 k 1 = 10 × 7 = 70 Now we can find k : k = k 2 k 1 = 4 49 70 = 2 7 70 = 70 × 7 2 = 20
Final Formula Therefore, the formula for r in terms of t is: r = 4 t 20
Verification To verify, let's plug in t = 16 : r = 4 16 20 = 2 20 = 10 This matches the given information, so our formula is correct.
Examples
Imagine you are designing a water filtration system. The flow rate r of water through a filter is inversely proportional to the square root of the filter's surface area s . The surface area s is directly proportional to the square root of the filter's production time t . Knowing the flow rate at a specific production time helps you determine the relationship between flow rate and production time, allowing you to optimize the filter's design for desired water output.
