Question

A pendulum hangs from a clock and swings over its base.

The distance between the pendulum and the base of the

clock, f(x), over the horizontal distance, x, is modeled by

the function shown.

f(x) = 0.25x² – 2x + 10

What is the minimum distance between the pendulum and

the base of the clock?

XI XII

Answer 1

To find the minimum distance, we need to find the minimum value of the function f(x) = 0.25x² - 2x + 10. The minimum value occurs at the vertex of the parabola, which can be found using the formula x = -b/(2a). Substituting the values, we get x = 2. Plugging this value of x into the function, we get f(2) = 6.5. Therefore, the minimum distance between the pendulum and the base of the clock is 6.5 units.

Step-by-step explanation:

To find the minimum distance between the pendulum and the base of the clock, we need to find the minimum value of the function f(x) = 0.25x² - 2x + 10.

This function represents a quadratic equation in standard form. Since the coefficient of the x² term is positive, the graph of the function opens upwards. Therefore, the minimum value of the function occurs at the vertex of the parabola.

The x-coordinate of the vertex can be found using the formula x = -b/(2a) where a, b, and c are the coefficients of the quadratic equation. Substituting the values, we get x = -(-2)/(2*0.25) = 2.

Plugging this value of x into the function, we get f(2) = 0.25(2)² - 2(2) + 10 = 6.5.

Therefore, the minimum distance between the pendulum and the base of the clock is 6.5 units.