The length of time f(x), in seconds, that it takes a pendulum of length x meters to swing from one side to the other and back is approximated by a square root function.
A pendulum of length 4 meters has a period of about 4 seconds, and a pendulum of length 9 meters has a period of about 6 seconds. Sketch a graph the function.
To sketch the graph of the function, we can use the information given to find the equation of the square root function.
Let f(x) be the period of the pendulum in seconds and x be the length of the pendulum in meters.
From the given information:
f(4) = 4 seconds
f(9) = 6 seconds
We can use these points to find the equation of the function. Let's assume the function is of the form f(x) = a*sqrt(x) + b.
Using the point (4, 4):
4 = a*sqrt(4) + b
4 = 2a + b (1)
Using the point (9, 6):
6 = a*sqrt(9) + b
6 = 3a + b (2)
Now, we can solve these equations simultaneously to find the values of a and b:
Subtracting (1) from (2):
6 - 4 = 3a - 2a
2 = a
Substitute a = 2 back into (1):
4 = 2*2 + b
4 = 4 + b
b = 0
Therefore, the function is f(x) = 2*sqrt(x).
Now, let's sketch the graph of this function:
The graph will start at the origin (0, 0) and increase as x increases. It will be a curve that gets steeper as x increases since it is a square root function. The graph will pass through the points (4, 4) and (9, 6) as calculated above.
The graph will be concave up and continue to increase as x increases.
This graph represents the relationship between the length of the pendulum and the time it takes to swing from one side to the other and back.