Question

a 25 ft. ladder needs to be placed against a wall 15 ft. high. how far do you place the ladder away from the base of the wall?
problem 7. given the rectangle below find its area

Explanation:

Step1: Apply Pythagorean theorem

For the ladder - wall problem, by Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c = 25\) (ladder length), \(a = 15\) (wall height), and \(b\) is the distance from the base of the wall.

$$b^{2}=c^{2}-a^{2}$$

$$b^{2}=25^{2}-15^{2}$$

$$b^{2}=625 - 225$$

$$b^{2}=400$$

$$b = 20$$

Step2: Find the length of the rectangle

For the rectangle problem, let the length of the rectangle be \(l\) and width \(w = 2\). Using Pythagorean theorem (diagonal \(d=\sqrt{20}\)), \(l^{2}+w^{2}=d^{2}\)

$$l^{2}+2^{2}=(\sqrt{20})^{2}$$

$$l^{2}+4 = 20$$

$$l^{2}=16$$

$$l = 4$$

Step3: Calculate the area of the rectangle

Area of rectangle \(A=l\times w\)

$$A=4\times2$$

$$A = 8$$

Answer:

The ladder should be placed \(20\) ft away from the base of the wall. The area of the rectangle is \(8\) square units.