Question

a weather balloon is released from the top of a 25 m tower. the height of the balloon increases 50 m every minute after it is released.

the height of the balloon, \\(h\\), can be modelled by a linear relation where \\(t\\) is the time in minutes.

which option does not represent this relationship?

\\(h = 50t + 25\\)
\\(h = 25(2t + 1)\\)

Explanation:

Identify the initial value and rate

The balloon starts at a height of \(25\text{ m}\) when \(t = 0\).
The height increases by \(50\text{ m}\) every minute.
Using Constant Rate of Change, the rate is \(50\text{ m/min}\).

Formulate the linear equation

Using Linear Relationships, we write the equation:

$$H = 50t + 25$$

This represents the height \(H\) at any time \(t\).

Analyze the algebraic options

We factor the equation to check the second option:

$$H = 25(2t + 1)$$

Both algebraic options correctly represent the relationship.

Analyze the graphical options

For the graph, when \(t = 0\), the vertical intercept \(H\) must be \(25\).
The left graph shows a line starting below \(200\) on the vertical axis.
The right graph shows a line starting below \(200\) on the vertical axis.
Let's calculate the \(t\)-intercept where \(H = 0\):

$$0 = 50t + 25 \implies t = -0.5$$

The left graph has a horizontal intercept at \(t = -0.5\) (half a grid unit left of the vertical axis).
The right graph has a horizontal intercept at \(t = -4\) (four grid units left of the vertical axis).
Therefore, the right graph does not represent this relationship.

Answer:

• (A) \(H = 50t + 25\)
• (B) \(H = 25(2t + 1)\)
• (C) The graph on the bottom-left
• (D) The graph on the bottom-right (Correct answer)