# If r varies directly as the cube of s and r=5 when s=3, find r when s=2

(a) 135

(b) $\dfrac{135}{8}$

(c) $\dfrac{35}{8}$

(d) None of these

Last updated date: 17th Mar 2023

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Answer

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Hint: When a variable varies directly to another (say x varies directly as y), we can write the relation as x=ky (where k is a proportional constant). In this case, since r varies directly as the cube of s, we can write the relation as r=k${{s}^{3}}$ and then solve the question.

Complete step-by-step answer:

Firstly, we try to use the first condition given in the problem (that is, r=5 when s=3). This would help us in finding the value of proportional constant.

r=k${{s}^{3}}$

Putting the value of r=5 when s=3, we get,

5=k$\times {{3}^{3}}$

5=27k

k=$\dfrac{5}{27}$ -- (1)

Now, since we have the value of proportional constant, we can find the value of r for any value of s. We now just have to put the value of k and s in equation r=k${{s}^{3}}$ to get the value of r. Now, we find the value of r for s=2.

r=k${{s}^{3}}$

r=$\dfrac{5}{27}$$\times {{2}^{3}}$

r=$\dfrac{5\times 8}{27}$

r=$\dfrac{40}{27}$

Thus, the value of r is $\dfrac{40}{27}$. Hence, the correct option is (d) None of these.

Hint: To solve problems involving direct and inverse variations in general, we use a general principle to solve the problems. Suppose, c varies directly with d and inversely with e. We use the following relation- c=k$\dfrac{d}{e}$(where k is the value of proportionality constant). The problem can then be solved by acquiring any additional relation which would further help in evaluating the problem further.

Complete step-by-step answer:

Firstly, we try to use the first condition given in the problem (that is, r=5 when s=3). This would help us in finding the value of proportional constant.

r=k${{s}^{3}}$

Putting the value of r=5 when s=3, we get,

5=k$\times {{3}^{3}}$

5=27k

k=$\dfrac{5}{27}$ -- (1)

Now, since we have the value of proportional constant, we can find the value of r for any value of s. We now just have to put the value of k and s in equation r=k${{s}^{3}}$ to get the value of r. Now, we find the value of r for s=2.

r=k${{s}^{3}}$

r=$\dfrac{5}{27}$$\times {{2}^{3}}$

r=$\dfrac{5\times 8}{27}$

r=$\dfrac{40}{27}$

Thus, the value of r is $\dfrac{40}{27}$. Hence, the correct option is (d) None of these.

Hint: To solve problems involving direct and inverse variations in general, we use a general principle to solve the problems. Suppose, c varies directly with d and inversely with e. We use the following relation- c=k$\dfrac{d}{e}$(where k is the value of proportionality constant). The problem can then be solved by acquiring any additional relation which would further help in evaluating the problem further.

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