A quantity T is partly constant and partly varies as the square root of s.

a) Using constant a and b, write down an equation connecting T and S.
b) If S = 16, and when T = 24 and S = 36 when T = 32, find the value of the constants a and b.

T = a + b√s

Now write two equations using the given numbers, and solve for a and b.

Detail

a=31.662

b=3.032

I asked the question.

a) The equation connecting T and S can be written as T = a + b√s.

b) Let's use the given values to find the value of constants a and b.

When S = 16 and T = 24, we can substitute these values into the equation:
24 = a + b√16

Simplifying, we get:
24 = a + 4b

When S = 36 and T = 32, we can substitute these values into the equation:
32 = a + b√36

Simplifying, we get:
32 = a + 6b

Now we have a system of two equations:
24 = a + 4b
32 = a + 6b

We can solve this system of equations to find the values of a and b.

Multiplying the first equation by 3 and the second equation by 2, we get:
72 = 3a + 12b
64 = 2a + 12b

Subtracting the second equation from the first, we get:
8 = a

Substituting this value back into the second equation, we get:
64 = 2(8) + 12b
64 = 16 + 12b
48 = 12b
b = 4

So, the values of the constants are a = 8 and b = 4.

To write down an equation connecting T and S, we can express T as a sum of a constant term and a term that varies with the square root of S. Let's call the constant term a and the coefficient of the square root of S b.

a) The equation connecting T and S would be:

T = a + b√(S)

b) To find the values of the constants a and b, we can use the given information.

When S = 16 and T = 24:
24 = a + b√(16)
24 = a + 4b (since √(16) = 4)

When S = 36 and T = 32:
32 = a + b√(36)
32 = a + 6b (since √(36) = 6)

Now we have a system of two equations with two unknowns (a and b). We can solve this system to find the values of a and b.

From the first equation: a = 24 - 4b
Substituting this value of a into the second equation:

32 = (24 - 4b) + 6b
32 = 24 - 4b + 6b
32 = 24 + 2b
2b = 32 - 24
2b = 8
b = 4

Now substituting the value of b back into the first equation to find a:

a = 24 - 4(4)
a = 24 - 16
a = 8

So, the values of the constants are a = 8 and b = 4.

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