Problem 4Suppose a mark is made at the top of a wheel with an 18 inch radius and then the wheel is rolledto a point 6t inches to the right.a. Calculate the new height of the mark above the ground (in inches). Provide a diagramthat justifies your reasoning, and explain how you used it to set up an equation. Yourexplanation should touch on the concepts and connection between arc length, anglemeasurement and right triangle trigonometry explicitly.b. Explain why the height after we roll the wheel any given distance x can be modeled byh(x) = 18 + 18 cos18P/le/content/612319/Homee to searchhulu

Name: Problem 4Suppose a mark is made at the top of a wheel with an 18 inch
Uploaded: 2024-03-20T06:46:14.000Z
Channel: Bartleby
Description: Problem 4Suppose a mark is made at the top of a wheel with an 18 inch radius and then the wheel is rolledto a point 6t inches to the right.a. Calculate the new height of the mark above the ground (in inches). Provide a diagramthat justifies your rea

Consider the provided question, (a) According to the given question we draw a diagram, The figure represent the problem when point A is moved to point B. when we have to find the length of BF. We know, Arc length = rθ where, r = radius θ = angle of arcArc length = distance moved form A to B = 6π So, 6π=rθ⇒θ= 6πr⇒θ= 6π18 Given, radius=18 inch⇒θ= π3 In the figure, triangle BCD is a right angle triangle in which ∠ACB=π3. So, ∠BCD=π2-π3=π6Since, ∠ACB=π3So, ∠CBD=π3Now, In triangle CBD, cosπ3=BDBCWe can see, BC in the figure is radius.So, BC=18⇒cosπ3=BD18⇒12=BD18⇒BD=9Since, DF=CECE=radius=18, So, DF=18thus, BF=BD+DF=9+18=27So, height of the point =27 inches(b) From the figure, we can understood that when a point A at the top of the circle is moved to a point x to the angle of rotation will be angle ACB in the figure. So, ∠ACB=xr⇒∠ACB=x18 From the figure, we can see that ∠ACB=∠CBD, because AC and BD are parallel lines. .In triangle CBD, cosx18=BDBC⇒cosx18=BD18⇒BD=18cosx18Since, DF will equal to radius which is 18. So, height=DF+BD⇒height=18+18cosx18So, the function of the height can be written as,h(x)=18+18cos(x18)We know that cosine function is decreasing from 0 to π, At π2⇒cosπ2=0, So, height = radiusfrom π2 to π, cosine function is negative.So, the equation will be correct for all values of x.