Question

a 20 - ft ladder is standing up against a wall. the distance between the base of the ladder and the wall is 4 ft less than the distance between the top of the and the base of the wall. find the distance between the base of the ladder and the wall. the distance between the base of the ladder and the wall is ft.

Explanation:

Step1: Define variables

Let the distance between the top of the ladder and the base of the wall be $x$ ft. Then the distance between the base of the ladder and the wall is $(x - 4)$ ft.

Step2: Use Pythagorean theorem

Since the ladder, wall and ground form a right - triangle with the ladder as the hypotenuse, by the Pythagorean theorem $a^{2}+b^{2}=c^{2}$, where $a=x - 4$, $b = x$ and $c = 20$. So we have $(x - 4)^{2}+x^{2}=20^{2}$.

Step3: Expand the equation

Expand $(x - 4)^{2}$ using the formula $(a - b)^{2}=a^{2}-2ab + b^{2}$. We get $x^{2}-8x + 16+x^{2}=400$.

Step4: Combine like terms

Combine the $x^{2}$ terms: $2x^{2}-8x+16 = 400$. Then subtract 400 from both sides to get a quadratic equation in standard form: $2x^{2}-8x-384 = 0$. Divide through by 2: $x^{2}-4x - 192=0$.

Step5: Solve the quadratic equation

Factor the quadratic equation: $x^{2}-4x - 192=(x - 16)(x + 12)=0$. Set each factor equal to zero: $x-16 = 0$ gives $x = 16$ and $x + 12=0$ gives $x=-12$. Since distance cannot be negative, we take $x = 16$.

Step6: Find the required distance

The distance between the base of the ladder and the wall is $x - 4$. Substitute $x = 16$ into it, we get $16-4=12$ ft.

Answer:

12