use the following information to answer the next question.\nthe piston in a water pump engine is attached to…

use the following information to answer the next question.\nthe piston in a water pump engine is attached to a connecting rod that is, in turn, attached to a crankshaft.\na technician studied the crankshaft as it moved counter - clockwise from its resting position to a new\nposition, as shown below.\nthe distance, x, to the nearest tenth of a centimetre, is ________.

use the following information to answer the next question.\nthe piston in a water pump engine is attached to a connecting rod that is, in turn, attached to a crankshaft.\na technician studied the crankshaft as it moved counter - clockwise from its resting position to a new\nposition, as shown below.\nthe distance, x, to the nearest tenth of a centimetre, is ________.

Answer

Explanation:

Step1: Use cosine - law in the triangle formed

We can consider a triangle with sides related to the positions of the connecting rod and crankshaft. Let the length of the connecting rod be $l = 25$ cm, the length of the crankshaft be $r=5$ cm. In the new - position, we want to find the side $x$. By the cosine - law, for a triangle with sides $a$, $b$, $c$ and the angle $\theta$ opposite to side $c$, $c^{2}=a^{2}+b^{2}-2ab\cos\theta$. Here, $a = 25$, $b = 5$, and $\theta = 20^{\circ}$. [x^{2}=25^{2}+5^{2}-2\times25\times5\times\cos(20^{\circ})]

Step2: Calculate the values

First, calculate the squares and the product terms: [25^{2}=625], [5^{2}=25], and [2\times25\times5\times\cos(20^{\circ})=250\times\cos(20^{\circ})\approx250\times0.9397 = 234.925] Then, [x^{2}=625 + 25-234.925=415.075]

Step3: Find the value of $x$

Take the square - root of $x^{2}$: [x=\sqrt{415.075}\approx20.4]

Answer:

$20.4$