Question

Michelle is balancing the wheel on her bicycle. She has marked a point on the tire that when rotated can be modelled by the function \(h(t)=59+24 \sin 125 t\) where \(h\) is the height, in centimetres, and \(t\) is the time, in seconds. Determine the height of the mark, to the nearest tenth of a centimetre, when \(t=17.5 \mathrm{s}.\)

Short answer

The height is approximately 59.0 cm.

Step-by-step solution

Substitute the Value of t

Substitute the given value of time, \( t = 17.5 \) seconds into the function \( h(t) = 59 + 24 \sin(125t) \).

Calculate the Angle

Calculate the angle for the sine function by multiplying 125 by 17.5 to get \(125 \times 17.5 = 2187.5 \) radians.

Evaluate the Sine Function

Evaluate \( \sin (2187.5) \) using a calculator to get the approximate value.

Compute the Height

Substitute the value of the sine function back into the equation \( h(t) = 59 + 24 \sin(2187.5) \) and compute the overall height.

Round to the Nearest Tenth

Round the calculated height to the nearest tenth of a centimeter.

Key concepts

Sinusoidal Functions

Sinusoidal functions are mathematical functions that describe waves. They take the form of sine or cosine functions, usually written as \(y = A \sin(Bt + C) + D\) or \(y = A \cos(Bt + C) + D\). These functions are used in various real-world applications, including sound waves, light waves, and even the modeling of a bike tire's height over time.

In the problem, we have the sinusoidal function \(h(t) = 59 + 24 \sin(125t)\), where:

• \textbf{h(t)} represents the height over time (in centimeters).
• \textbf{59} is the vertical shift, indicating the mean height.
• \textbf{24} is the amplitude, showing the maximum variation from the mean height.
• \textbf{125t} is the argument inside the sine function, affecting the frequency of the oscillations.

By substituting the value of time \(t = 17.5\) into this function, we can determine the height of the mark at a given time.

Angle Calculation

Angle calculation is crucial when dealing with trigonometric functions, especially sine and cosine. The angle in radians can be found by multiplying the coefficient of t inside the sine function by the time value, i.e., \(125 \times 17.5 \). Here's how:

• To calculate the angle for the sine function, we multiply the frequency coefficient by the time: \(125 \times 17.5 = 2187.5 \) radians.

This results in radians, the standard unit of angular measurement used in many areas of mathematics. Use this angle to compute the sine value, which helps to determine further calculations involving height.

Height Calculation

Calculating the height involves a few straightforward steps following the calculation of the angle. After finding that \(125 \times 17.5 = 2187.5\) radians:

• Evaluate \(\text{sine} (2187.5)\) using a calculator to obtain an approximate value. Typically, a scientific calculator, or a calculator app, can be used.
• Inserting the sine value back into the height equation: \(h(t) = 59 + 24 \sin(2187.5)\).

For the provided problem, if \( \sin(2187.5)\ \) approximates to a certain value, replace this back into the equation to find the specific height. This gives the height at that particular moment in time.

Rounding

Rounding is essential to simplify results for better understanding or practical application. After calculating the height using sine and substituting back into the equation, you may end up with a decimal value. Here's how to proceed:

• After computing the value, say the height comes out to be something like 70.256 cm.
• Rounding off to the nearest tenth implies checking the digit in the hundredths place.
• If this digit (hundredths place) is 5 or more, you increase the tenths place by 1. Otherwise, it remains as-is.

For example, the value 70.256 cm rounds to 70.3 cm when considering the tenths place due to the 5 in the hundredths place. This rounding helps present a clean, easy-to-read answer, crucial for homework or real-world use.