w varies partly as s and partly as the square of s when s=3,w=18 and when s=5, w=169
find w in term of s
Let's write down the relationships we are given:
w ∝ s (partly varies as s)
w ∝ s^2 (partly varies as the square of s)
Combining these two, we get:
w ∝ s x s^2
Simplifying:
w ∝ s^3
Now we can use the initial conditions to solve for the constant of proportionality:
When s=3, w=18:
18 = k x 3^3
18 = 27k
k = 18/27
k = 2/3
So the complete equation relating w and s is:
w = (2/3) s^3
Finally, we can check that this holds true for the other given condition:
When s=5, w=169:
w = (2/3) x 5^3
w = 166.67 ≈ 169 (due to rounding).
Therefore, we have found the expression for w in terms of s.
To find an equation for w in terms of s, let's first set up the relationship between w and s using the given information.
We are told that "w varies partly as s and partly as the square of s." This implies that the relationship can be represented as:
w = k * s^m * (s^2)^n
where k is a constant, m represents the power of s, and n represents the power of s^2.
Now, let's substitute the given values to find the values of k, m, and n.
When s = 3 and w = 18:
18 = k * 3^m * (3^2)^n
Simplifying this equation, we have:
18 = k * 3^m * 9^n ...(Equation 1)
Similarly, when s = 5 and w = 169:
169 = k * 5^m * (5^2)^n
Simplifying this equation, we have:
169 = k * 5^m * 25^n ...(Equation 2)
Now, let's solve these two equations simultaneously to find the values of k, m, and n.
Divide Equation 2 by Equation 1:
(169 / 18) = (k * 5^m * 25^n) / (k * 3^m * 9^n)
Simplifying this, we get:
169 / 18 = (5^m * 5^2n) / (3^m * 3^2n)
Expressing 169 / 18 as a decimal, we have:
9.389 = (5^(m+2n)) / (3^(m+2n))
Comparing the powers of 5 and 3 on both sides, we get:
m + 2n = 2*(m + 2n)
Simplifying this equation, we have:
m = 4n ...(Equation 3)
Now, substitute this value of m into Equation 1:
18 = k * 3^(4n) * 9^n
Using the property of exponents, we can rewrite this as:
18 = k * 3^(4n + 2n)
Simplifying this equation, we have:
18 = k * 3^(6n)
Divide Equation 2 by Equation 1:
(169 / 18) = (k * 5^m * 25^n) / (k * 3^m * 9^n)
Simplifying this, we get:
169 / 18 = (5^m * 5^2n) / (3^m * 3^2n)
Expressing 169 / 18 as a decimal, we have:
9.389 = (5^(m+2n)) / (3^(m+2n))
Comparing the powers of 5 and 3 on both sides, we get:
m + 2n = 2*(m + 2n)
Simplifying this equation, we have:
m = 4n ...(Equation 3)
Now, substitute this value of m into Equation 1:
18 = k * 3^(4n) * 9^n
Using the property of exponents, we can rewrite this as:
18 = k * 3^(4n + 2n)
Simplifying this equation, we have:
18 = k * 3^(6n)
Divide both sides by 3^6n:
18 / 3^(6n) = k
Simplifying:
2 / 3^(4n) = k
Now, we have the value of k in terms of n.
Next, substitute the value of k back into Equation 1:
18 = (2 / 3^(4n)) * 3^m * 9^n
Simplifying:
18 = (2 / 3^(4n)) * 3^m * (3^2)^n
Simplifying further:
18 = (2 / 3^(4n)) * 3^m * 3^(2n)
Combining the exponents and simplifying the equation:
18 = 2 * 3^(m + 2n - 4n)
Simplifying the equation:
18 = 2 * 3^(m + n)
Divide by 2:
9 = 3^(m + n)
Since 9 = 3^2, we can conclude that:
m + n = 2
Now, using Equation 3 (m = 4n), we can substitute the value of m:
4n + n = 2
Combining terms:
5n = 2
Divide by 5:
n = 2/5
Now we can substitute the value of n back into Equation 3 to find m:
m = 4n
m = 4 * (2/5)
m = 8/5
Therefore, the relationship between w and s can be written as:
w = k * s^(8/5) * (s^2)^(2/5)
Simplifying further:
w = k * s^(8/5) * s^(4/5)
w = k * s^(8/5 + 4/5)
w = k * s^(12/5)
So, the equation for w in terms of s is:
w = k * s^(12/5)