Two variables P and L are such that P varies partly as L and partly as the square root of L.Determine the relationship between P and L given that when L=16,P=500 and when L=25,P=800.
P = k L + m L^.5
500 = 16 k + 4 m
800 = 25 k + 5 m
2500 = 80 k + 20 m
3200 = 100 k + 20 m
---------------------subtract
-700 = -20 k
k = 35 etc
To determine the relationship between the variables P and L, we need to express P as a function of L.
Let's express the relationship between P and L partly as L and partly as the square root of L. We can set up two equations based on the given information:
When L = 16, P = 500
P ∝ L^a * √L^b
Substituting the values, we get:
500 ∝ 16^a * √16^b
Simplifying further:
500 ∝ 16^a * 4^b
Dividing both sides by 500:
1 ∝ (16^a * 4^b) / 500
Similarly, using the second set of information:
When L = 25, P = 800
800 ∝ 25^a * √25^b
Simplifying:
800 ∝ 25^a * 5^b
Dividing both sides by 800:
1 ∝ (25^a * 5^b) / 800
We have two equations now:
1 ∝ (16^a * 4^b) / 500
1 ∝ (25^a * 5^b) / 800
To solve for the values of 'a' and 'b', we can equate the two equations:
(16^a * 4^b) / 500 = (25^a * 5^b) / 800
Now, we can write 16 and 4 as powers of 2, and 25 and 5 as powers of 5:
(2^4a * 2^2b) / 500 = (5^2a * 5^b) / 800
Simplifying further, we get:
(2^(4a+2b)) / 500 = (5^(2a+b)) / 800
Cross multiplying:
800 * 2^(4a+2b) = 500 * 5^(2a+b)
To simplify the equation further, let's rewrite 800 and 500 as powers of 2 and 5 respectively:
(2^2 * 2^9 * 2^2a * 2^2b) = (2^2 * 5^2 * 5^(2a+b))
Cancelling out the common bases:
2^(2+9+2a+2b) = 5^(2+2a+b)
Now we have:
2^(11+2a+2b) = 5^(2+2a+b)
Since the bases on both sides are different, the exponents must be equal:
11 + 2a + 2b = 2 + 2a + b
Simplifying:
9 + a = b
Therefore, the relationship between P and L can be expressed as:
P ∝ L^a * √L^a/9
or
P = k * L^a * √L^a/9
where 'k' is a constant.