QuestionQuestion 19 (1 point) The height, $h$ , in centimetres, of a piston moving up and down in an engine cylinder can be modelled by the function $h(t)=18 \sin (50 \pi t)+18$ , where $t$ is the time, in seconds. What is the piston's minimum height? a) -18 cm b) 18 cm c) 0 cm d) 9 cm

Studdy Solution

STEP 1

1. The function $h(t) = 18 \sin(50 \pi t) + 18$ models the height of a piston.
2. The sine function, $\sin(x)$ , oscillates between -1 and 1.
3. We need to determine the minimum value of $h(t)$ .

STEP 2

1. Analyze the sine function's range.
2. Determine the minimum value of the function $h(t)$ .

STEP 3

The sine function, $\sin(x)$ , has a range of $[-1, 1]$ . This means that the sine function can take any value between -1 and 1, inclusive.

STEP 4

Substitute the minimum value of $\sin(x)$ into the function $h(t) = 18 \sin(50 \pi t) + 18$ .
The minimum value of $\sin(x)$ is -1. Therefore, substitute $-1$ into the function:
$h(t) = 18(-1) + 18$ $h(t) = -18 + 18$ $h(t) = 0$
Thus, the piston's minimum height is $\boxed{0}$ cm.
The correct answer is $\boxed{c) \, 0 \, \text{cm}}$ .

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