Question

Pam and Erin start at the same point and begin rollerblading in different directions. Pam is rollerblading west at a speed of 2

miles per hour. Erin is rollerblading north at a speed of 4 miles per hour.

Answer 1

The question relates to the calculation of the distance between two moving individuals, Pam and Erin, who are traveling in perpendicular directions at different speeds. By using the Pythagorean theorem, we can calculate Pam and Erin would be √20 miles apart.

Step-by-step explanation:

The question involves determining how far apart Pam and Erin will be after a certain amount of time if Pam is rollerblading west at 2 miles per hour and Erin is rollerblading north at 4 miles per hour. This is a problem related to vectors and can be visualized using a coordinate plane where their paths represent perpendicular vectors. The resulting distance between them after a certain time is the hypotenuse of the right-angled triangle formed by their separate paths.

To find this distance, we would use the Pythagorean theorem: c² = a² + b², where 'a' and 'b' are the distances traveled by Pam and Erin in a given time, and 'c' is the distance between them. For example, after one hour, Pam would have traveled 2 miles, and Erin would have traveled 4 miles. Plugging these into the formula, we get c² = 2² + 4², which gives us c = √(4 + 16), resulting in c = √20 miles. Thus, after one hour, Pam and Erin would be √20 miles apart.

Answer 2

Pam and Erin, starting at the same point, rollerblade in different directions. Pam moves west at a speed of 2 miles per hour, while Erin heads north at a speed of 4 miles per hour.

Step-by-step explanation:

Pam's motion can be described as a horizontal displacement, represented by her speed of 2 miles per hour to the west. Erin's motion is vertical, with a speed of 4 miles per hour to the north. To determine the overall displacement and direction, we can use vector addition.

The horizontal and vertical components of their displacements form a right-angled triangle, and we can use the Pythagorean theorem to find the magnitude of the resultant displacement. Letting D represent the overall displacement, we have:

`\[ D = \sqrt{(2 \, \text{miles/hour})^2 + (4 \, \text{miles/hour})^2} \]\[ D = √(4 + 16) = √(20) \, \text{miles/hour} \]`

The direction of the displacement can be found using trigonometric functions. The angle \( \theta \) formed by Pam's displacement can be expressed as:

`\[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = (4)/(2) \]\[ \theta = \tan^(-1)\left((4)/(2)\right) = \tan^(-1)(2) \]`

So, Pam and Erin's overall displacement is
`\( √(20) \)`miles per hour at an angle of
`\( \tan^(-1)(2) \)` with respect to the horizontal.