1
$\begingroup$

A wheel 5 feet in diameter rolls up with an incline of 18 degrees 20 minutes. What is the height of the center of the wheel above the base of the incline when the wheel has rolled 5 ft up the incline?

  • 3
    Draw a picture. Do you see a right triangle that might give you some clues.2012-07-03
  • 0
    I drew a picture with 5 being hypotnouse 18 degrees 20 being an angle and I got 1.57. When i did sin 18 degrees 20= x/5.2012-07-03
  • 0
    Hmm I am not sure how to proceed.2012-07-03
  • 0
    @Bishop did you figure it out?2012-07-03
  • 0
    No I cant seem to figure it out.2012-07-03
  • 0
    By the way welcome to Math Stackexchange :)2012-07-03
  • 0
    Bishop, it does seem that you are able to write what you've already tried for solving a problem. Please write those out as part of your question the next time you ask one...2012-07-04

4 Answers 4

3

Does this diagram help? It is not drawn to scale but it can help in visualizing the problem.

enter image description here

So note that $|BD|=5ft$, $|CD|=2.5ft$, $\angle BDC=90^\circ$. Having settled that $$|CM|=\dfrac{2.5}{\sin 72}=2.628655561\ldots$$ we now need to find $$|MD|=\dfrac{2.5}{\tan 72}=.812229924\ldots$$ Since $|BD|=|BM|+|MD|$, then $$|BM|=5-|MD|=4.187700759\ldots$$ Calculating now $$|ME|=|BM|\sin 18=1.294070702\ldots$$ Finally, $$|CE|=|CM|+|ME|.$$

  • 0
    Similar trangles galore!2012-07-03
  • 0
    Yes I did something similar and I got BE is 4.74 when I did 5 cos 18 degrees 20 minutes the thing is my final answer is 3.75 ft and I am not sure how to get there. I mean I know from E to right below the circle is 1.57.2012-07-03
  • 0
    Maybe I add 1.57 plus the radius which is 2.5 feet.2012-07-03
  • 0
    @azetina Good Drawing!2012-07-03
  • 0
    @ncmathsadist thanks. I did it with GeoGebra. Very nice program indeed.2012-07-03
  • 0
    @Bishop If my answer responds to the question, do hit the check mark to accept as an answer. :)2012-07-03
  • 0
    I think it's not quite clear from the problem description which position the wheel _starts from_ before it moves the 5 feet along the incline. If the wheel starts being tangent to the incline at your point $B$, then it would initially protrudes _below the floor_. It might be a more realistic assumption that the wheel is initially tangent to _both_ the floor and the incline, in which case the answer would be simply $2\frac 12 + 5\sin 18^\circ 20'$.2012-07-04
  • 0
    @HenningMakholm I understand that clearly but what I suggested would give an idea on how to solve the matter at hand.2012-07-04
0

Place an additional label where the segment joining $C$ to $E$ and the segment joining $B$ to $D$ intersect. Let's agree to call that point $M$. The length of $\overline{EM}$ is $|\overline{BM}|\sin(18^\circ)$. The angle $\angle CMD$ is $72^\circ$. Can you figure the rest?

Now use the fact that $$\sin(72^\circ) = {2.5\over |\overline{CM}|}.$$ so $$|\overline{CM}| = {2.5\over \sin(72^\circ)} = 2.629. $$

Also we have $|\overline{BM}| = 5 - | \overline{MD}|$. Can you find the length of $\overline { MD}$?

  • 0
    I know the lenght of EM is about 1.57 but no I am not sure about figuring out the rest. Am I to use 72 degrees to find a certain length?2012-07-03
  • 0
    @ncmathsadist I disagree with you since EM is in $\triangle BEM$ and therefore its hyp0tenuse is not 5 ft.2012-07-03
  • 0
    I will say the final answer is 3.95 I know BE is 4.74 the question is how to reach the final answer?2012-07-03
  • 0
    would MD be .81240 I did 2.629 cos 722012-07-03
0

Rotation of axes. Plug in

$$ x = 5^{'}, y =2.5 ^{'}, \alpha = 18 \frac 13^{0}$$

into $$ y_2 = x \sin \alpha + y \cos \alpha. $$

-1

You'll just have to imagine that the radius (Half of the diameter which is 2.5 feet) is aligned with the height of the triangle. So You just have to get the height of the triangle. To get it's height. Simply...

Sin 18.33 = height / 5 feet Sin 18.33 x 5 feet = height (0.3145)(5 feet) = height 1.5725 = Height

So, to get the height of the center of the circle to the ground. Simply...

Height of the center of the circle w/respect to the base = 1.5725 + 2.5 The answer would be...

4.0725 feet