S is partly constant and partly varies with T. When S =530,T=1600,when S=730,T=3600.find
(A)the formula connecting S and T
(B)S when T=1300
s = a + b t
530 = a + 1600 b
730 = a + 3600 b
---------------------------- subtract
-200 = -2000 b
b = 0.10
530 =a + 160
a = 370
so
s = 370 + 0.10 t
etc
To find the formula connecting S and T, we can use the method of variation. Let's start by finding the constant part of S.
Given:
When S = 530 and T = 1600
When S = 730 and T = 3600
Since S is partly constant and partly varies with T, we can assume that the formula connecting S and T can be written as:
S = C + kT
Where C is the constant part and k is the coefficient of variation. To find the constant part C, we can substitute one set of values into the equation. Let's use the first set of values:
530 = C + (k * 1600)
Now, let's use the second set of values to find the coefficient of variation k:
730 = C + (k * 3600)
Equating the constant parts C, we have:
C + (k * 1600) = C + (k * 3600)
Simplifying and rearranging, we get:
(k * 3600) - (k * 1600) = 730 - 530
Simplifying further:
k * (3600 - 1600) = 200
k * 2000 = 200
k = 200 / 2000
k = 0.1
Now that we have the value of k, we can substitute it back into either equation to solve for C. Let's use the first equation:
530 = C + (0.1 * 1600)
Simplifying:
530 = C + 160
C = 530 - 160
C = 370
So, the formula connecting S and T is:
S = 370 + 0.1T
Now, to find S when T = 1300, we can substitute the value of T into the formula:
S = 370 + 0.1 * 1300
S = 370 + 130
S = 500
Therefore, when T = 1300, S = 500.
(A) Let's assume that the constant part of S is represented by "k" and the varying part is represented by "mT". Therefore, the formula connecting S and T can be written as S = k + mT.
To find the values of "k" and "m", we can substitute the given values of S and T into the formula and solve the resulting system of equations.
When S = 530 and T = 1600:
530 = k + m(1600) ----(1)
When S = 730 and T = 3600:
730 = k + m(3600) ----(2)
We can solve this system of equations using various methods such as substitution or elimination. Once we find the values of "k" and "m", we can write the formula connecting S and T.
(B) To find the value of S when T = 1300, we can now substitute T = 1300 into the formula we obtained in part (A) and solve for S.
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To find the formula connecting S and T, we can first determine the constant part and the varying part of S.
Step 1: Determine the constant part of S
When S = 530 and T = 1600, we have one data point. Let's call the constant part of S as "a".
So, we can say that when T = 1600, S is composed of the constant part (a) and the varying part.
Hence, S = a + (varying part).
Step 2: Determine the varying part of S
We can subtract the constant part of S from the actual value of S to find the varying part.
(varying part) = S - a
Step 3: Find the varying part using the second data point
When S = 730 and T = 3600, we can substitute these values into the equation from Step 1:
730 = a + (varying part)
Since the constant part remains the same, the varying part will also be the same.
Subtracting the equation in Step 1: 730 = a + (varying part) from the equation in Step 1: 530 = a + (varying part) gives us:
(530 - 730) = a + (varying part) - (a + (varying part))
-200 = 0
This means that the varying part cancels out, leaving us with the conclusion that the varying part is zero.
Step 4: Determine the formula connecting S and T
Since the varying part is zero, we can say that S is solely dependent on the constant part (a) and doesn't depend on T.
Therefore, the formula connecting S and T is:
S = a
Step 5: Find the constant part (a)
Using the first data point, where S = 530 and T = 1600, we substitute these values into the formula:
530 = a
Hence, a = 530.
So, the formula connecting S and T is S = 530.
To find S when T = 1300, we can substitute the value of T into the formula we obtained:
S = 530, T = 1300
S = 530.
Therefore, when T = 1300, S = 530.