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Chapter 6: Q. 92 (page 412)

Bridge Clearance A one-lane highway runs through a tunnel in the shape of one-half a sine curve cycle. The opening is 28 feet wide at road level and is 15 feet tall at its highest point.

(a) Find an equation for the sine curve that fits the opening. Place the origin at the left end of the sine curve.
(b) If the road is 14 feet wide with 7 -foot shoulders on each side, what is the height of the tunnel at the edge of the road?

Short Answer

Expert verified

(a) The required sin functiony=15sin蟺28t

(b) The height of the tunnel at the edge of road is 152units

Step by step solution

01

Step 1.(a) Find an equation for the sine curve that fits the opening. Place the origin at the left end of the sine curve. 

The equation for the sine curve that fits the opening can be found out if we consider the curve with left end as the origin and half cycle as the half curve of sine function above the x-axis as follows-:


Now, the half-curve begins at x=0and ends at x=28, so the period of the function will be T=56as the full cycle will end at x=56. Therefore,

蝇=2蟺T=2蟺56=蟺28

Also, the amplitude of the function will beA=15because the maximum value of this sine function atx=0is 15 units.
Since the sine function is of the form y=Asin(蝇t), we get the required sine function as-:

localid="1646480248236" y=15sin蟺28t

02

Step 2.(b) If the road is 14 feet wide with 7 -foot shoulders on each side, what is the height of the tunnel at the edge of the road? 

If the road is 14 feet wide, the road will be 7 feet away from the origin point from where the sine curve starts.
So, we need to find the value of function atx=7to find the height of tunnel at the edge of road which is-:

y=15sin蟺28(7)=15sin蟺4=1512=152

Hence,the height of the tunnel at the edge of road is152units

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Most popular questions from this chapter

From a parking lot, you

want to walk to a house on the beach. The house is located

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