Question

1. a wheelchair ramp is 39 feet long. the ramp covers a horizontal distance of 36 feet. what is the vertical height of the ramp? 8. solve for x. 9. solve for x.

Explanation:

7.

Step1: Apply Pythagorean theorem

The ramp forms a right - triangle, where the length of the ramp is the hypotenuse $c = 39$ feet and the horizontal distance is one leg $a = 36$ feet. Let the vertical height be $b$. The Pythagorean theorem is $a^{2}+b^{2}=c^{2}$, so $b=\sqrt{c^{2}-a^{2}}$.

Step2: Substitute values

Substitute $c = 39$ and $a = 36$ into the formula: $b=\sqrt{39^{2}-36^{2}}=\sqrt{(39 + 36)(39 - 36)}$ (using the difference - of - squares formula $x^{2}-y^{2}=(x + y)(x - y)$). Then $b=\sqrt{75\times3}=\sqrt{225}=15$ feet.

Step1: Use the geometric mean theorem in a right - triangle

In a right - triangle, if the altitude drawn to the hypotenuse divides the hypotenuse into two segments of lengths $a$ and $b$ and the length of the altitude is $h$, then $h^{2}=a\times b$. Here, $12^{2}=x\times16$.

Step2: Solve for $x$

We have the equation $144 = 16x$. Divide both sides by 16: $x=\frac{144}{16}=9$.

Step1: Use the geometric mean theorem in a right - triangle

In a right - triangle, if the altitude drawn to the hypotenuse divides the hypotenuse into two segments of lengths $a$ and $b$ and the length of the hypotenuse is $c$, then $c^{2}=x\times(x + 36)$. Also, using the similarity of triangles, we can use the proportion $\frac{x}{60}=\frac{60}{x + 36}$. Cross - multiply to get $x(x + 36)=60^{2}$, which is $x^{2}+36x-3600 = 0$.

Step2: Solve the quadratic equation

For the quadratic equation $ax^{2}+bx + c = 0$ (here $a = 1$, $b = 36$, $c=-3600$), we use the quadratic formula $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. First, calculate the discriminant $\Delta=b^{2}-4ac=36^{2}-4\times1\times(-3600)=1296 + 14400=15696$. Then $x=\frac{-36\pm\sqrt{15696}}{2}=\frac{-36\pm125.29}{2}$. We take the positive root since length cannot be negative. $x=\frac{-36 + 125.29}{2}\approx44.64$. Another way: from $x(x + 36)=60^{2}$, we can also factor the quadratic as $(x + 60)(x - 24)=0$. So $x = 24$ or $x=-60$. We discard $x=-60$ as length is non - negative.

Answer:

15 feet

8.