Question

use the drawing tool(s) to form the correct answer on the provided grid.
lisa and penny are guests at a beachside hotel. lisa is laying on a towel at the beach which is 50 feet from the base of the hotel. when she looks up at an angle of 54.5 degrees, she can see her sister penny waving to her from the hotel window.
draw a scale representation of the triangle that models the situation and can be solved to find the straight-line distance between penny and lisa. round calculations to the nearest foot. each unit on the grid represents five feet.
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Explanation:

Step1: Identify the triangle type

This is a right - triangle problem where we know the adjacent side (distance from Lisa to the base of the hotel, 50 feet) and the angle of elevation (54.5 degrees). We can use the cosine function to find the hypotenuse (distance between Lisa and Penny), but first, we need to recall the cosine formula: $\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}$.

Step2: Rearrange the formula

From $\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}$, we can rearrange it to $\text{hypotenuse}=\frac{\text{adjacent}}{\cos(\theta)}$. Here, $\theta = 54.5^{\circ}$ and the adjacent side $a = 50$ feet.

Step3: Calculate the hypotenuse

First, find $\cos(54.5^{\circ})$. Using a calculator, $\cos(54.5^{\circ})\approx0.5807$. Then, $\text{hypotenuse}=\frac{50}{0.5807}\approx86$ feet (rounded to the nearest foot).

Step4: Scale and grid consideration

Each unit on the grid represents 5 feet. So the length in grid units would be $\frac{86}{5} = 17.2$ units. But since the problem is about drawing the triangle, we can also think in terms of the right - triangle with adjacent side 10 units (since 50 feet / 5 feet per unit = 10 units) and angle $54.5^{\circ}$. Then the hypotenuse in units is $\frac{10}{\cos(54.5^{\circ})}\approx\frac{10}{0.5807}\approx17.2$ units.

To draw the triangle:

• Plot a point for the base of the hotel (let's say at the origin (0,0)).
• Plot a point for Lisa at (10,0) (since 50 feet is 10 units at 5 feet per unit).
• Then, using the angle of $54.5^{\circ}$ from the point representing Lisa, draw a line segment to represent the line of sight to Penny. The length of this line segment should be approximately 17.2 units. We can use the line - segment tool to draw from (10,0) at an angle of $54.5^{\circ}$ with a length of approximately 17.2 units. The vertical side (opposite side) can be found using $\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}$, $\text{opposite}=\text{hypotenuse}\times\sin(\theta)=86\times\sin(54.5^{\circ})\approx86\times0.8142\approx70$ feet, which is 14 units on the grid ($70\div5 = 14$ units). So the point for Penny would be at (10,14) approximately? Wait, no. Wait, if Lisa is at (10,0) (50 feet from the base (0,0)), and the angle of elevation is $54.5^{\circ}$, then the vertical change (opposite side) is $50\times\tan(54.5^{\circ})\approx50\times1.393\approx69.65$ feet, which is about 14 units (69.65 / 5≈13.93≈14 units). So the coordinates of Penny would be (10,14) (if we consider the base of the hotel at (0,0) and Lisa at (10,0)). Then we can draw the line segment from (10,0) to (10,14) (the vertical side) and from (0,0) to (10,14) (the hypotenuse), but actually, the angle is from Lisa's position, so the angle between the horizontal line (from Lisa to the base of the hotel) and the line of sight to Penny is $54.5^{\circ}$. So we can draw a horizontal line from Lisa's point (10,0) to the left to (0,0) (the base of the hotel), then from Lisa's point (10,0), draw a line at $54.5^{\circ}$ above the horizontal. The intersection of this line with the vertical line from the base of the hotel (the y - axis) will be Penny's position.

Answer:

To draw the triangle:

1. Plot the base of the hotel at a point (e.g., (0,0)).
2. Plot Lisa's position at (10,0) (since 50 feet is 10 units with 5 feet per unit).
3. Using the line - segment tool, draw a horizontal line from (10,0) to (0,0) (adjacent side).
4. Using the angle tool (if available) or by calculating the slope, draw a line from (10,0) at an angle of $54.5^{\circ}$ above the horizontal line. The length of this line (hypotenuse) should be approximately 17.2 units (or the vertical side should be approximately 14 units). The intersection of this line with the vertical line through (0,0) (the y - axis) is Penny's position. The distance between Lisa and Penny is approximately 86 feet.