Question

1. three rods will be used to make a right - angled triangle. the yellow rod is 17 cm longer than the purple rod. the green rod (which happens to be the longest) is 53 cm long. what is the length of the yellow rod? 5. the path of an aerial skier is given by the equation h=-0.2d² + 2.5d + 8.

Explanation:

Step1: Identify the longest rod

The green rod is the longest and is 53 cm long. Let the length of the purple rod be $x$ cm, and the length of the yellow rod be $y$ cm.

Step2: Set up the relationship between yellow and purple rods

We know that $y=x + 17$.

Step3: Use the Pythagorean theorem for right - angled triangle

Since it's a right - angled triangle, $x^{2}+y^{2}=53^{2}$. Substitute $y=x + 17$ into the Pythagorean equation: $x^{2}+(x + 17)^{2}=2809$.
Expand: $x^{2}+x^{2}+34x+289 = 2809$.
Combine like terms: $2x^{2}+34x+289-2809=0$, so $2x^{2}+34x - 2520 = 0$.
Divide by 2: $x^{2}+17x - 1260 = 0$.
Factor the quadratic equation: $(x + 45)(x - 28)=0$.
So $x=-45$ or $x = 28$. Since length cannot be negative, $x = 28$.

Step4: Find the length of the yellow rod

Substitute $x = 28$ into $y=x + 17$. Then $y=28+17=45$.

Answer:

45 cm