Question

1. an aeroplane takes off and reaches a height of 1500 m after travelling an overland distance of 12 km. what is the distance covered by the plane, in km?

1. a. if the area of a square is 25 cm\\(^2\\), find the length of the diagonal. leave your answer in exact form.

hint: let \\(x\\) = length of the diagonal.

Explanation:

Convert units to kilometers

We need to find the distance covered by the plane in kilometers.
The vertical height is given as \(1500\text{ m}\).
Convert the height to kilometers:

$$1500\text{ m} = 1.5\text{ km}$$

Apply the Pythagorean theorem for Question 1

The path of the plane forms a right-angled triangle.
The horizontal distance is \(12\text{ km}\) and the vertical height is \(1.5\text{ km}\).
Let \(x\) be the hypotenuse representing the distance covered:

$$x^2 = 12^2 + 1.5^2$$

$$x^2 = 144 + 2.25$$

$$x^2 = 146.25$$

$$x = \sqrt{146.25} \approx 12.09\text{ km}$$

Find the side length of the square for Question 2

The area of a square is given by:

$$\text{Area} = s^2$$

Given the area is \(25\text{ cm}^2\):

$$s^2 = 25 \implies s = 5\text{ cm}$$

Calculate the diagonal of the square

The diagonal \(x\) of a square with side length \(s\) forms a right-angled triangle:

$$x^2 = s^2 + s^2$$

$$x^2 = 5^2 + 5^2$$

$$x^2 = 25 + 25 = 50$$

$$x = \sqrt{50} = 5\sqrt{2}\text{ cm}$$

Answer:

Question 1

The distance covered by the plane is approximately \(12.09\text{ km}\) (or exactly \(\sqrt{146.25}\text{ km}\)).

Question 2

The length of the diagonal is \(5\sqrt{2}\text{ cm}\).