Brooke is standing at the location \"A\" on the shore of Circle Lake. The lake h
ID: 3372719 • Letter: B
Question
Brooke is standing at the location "A" on the shore of Circle Lake. The lake has a circular shoreline and a radius of 2 miles. In this problem, we will study the minimum and maximum time it takes Brooke to go from the location "A" to the location "C" diametrically opposite, subject to several scenarios regarding how fast she can walk and row. In all cases, assume Brooke rows in a straight line and walks along a portion of the circumference, as indicated by the arrows. We have labeled an angle q in the picture. Note:
Brooke is standing at the location "A" on the shore of Circle Lake. The lake has a circular shoreline and a radius of 2 miles. In this problem, we will study the minimum and maximum time it takes Brooke to go from the location "A" to the location "C" diametrically opposite, subject to several scenarios regarding how fast she can walk and row. In all cases, assume Brooke rows in a straight line and walks along a portion of the circumference, as indicated by the arrows. We have labeled an angle q in the picture. When q=0, Brooke rows straight from "A" to "C" and never walks. When q=pi/2, Brooke walks from "A" to "C" along the upper half circumference and never rows. When q is strictly between 0 and pi/2, Brooke rows from "A" to "B" and then walks from "B" to "C". In what follows, be careful to convert "ft/sec" to "miles/hr" units using the fact that there are 5280 feet/mile and 3600 seconds/hour. Assume that Brooke can walk c ft/sec and row d miles/hr. Find a formula for the time (in hours) it takes Brooke to go from "A" to "C" via "B"; your formula should involve c,d and q: Assume that Brooke can walk 10 ft/sec and row 2 miles/hr. If Brooke wants to reach "C" as soon as possible, then q = and it takes her hours to reach "C". If Brooke wants to maximize the time required to reach "C", then q= and it takes her hours to reach "C". Assume that Brooke can walk 4 ft/sec and row 2 miles/hr. If Brooke wants to reach "C" as soon as possible, then q = and it takes her hours to reach "C". If Brooke wants to maximize the time required to reach "C", then q= and it takes her hours to reach "C". Assume that Brooke is injured and can only walk 2 ft/sec and row 2 miles/hr. If Brooke wants to reach "C" as soon as possible, then q = and it takes her hours to reach "C". If Brooke wants to maximize the time required to reach "C", then q= and it takes her hours to reach "C".Explanation / Answer
Given
radius 2 mi
walk c ft/sec = c*(mi/5280ft)(3600sec/hr)
= (15/22)c
= 0.6818c mi/hr
row d mi/hr
(a)
Let O be the center of the Circle Lake
Then OA = OB = 2 (radius).
So ?OAB is an isosceles triangle.
m?OAB = m?OBA = q
The distance AB = 2*cos(q)
m?BOC = m?OAB + m?OBA = 2q
The distance BC = 2(2q) = 4q
time (A to B) = 2cos(q)/d
time (B to C) = 4q/(0.6818c)
= 5.8667(q/c)
Total time = 2cos(q)/d + 5.8667(q/c)
(b)
c = 10*0.6818 = 6.818 mi/hr
(b1)
time (walk ABC) = 2*?/6.818 = 0.9215 hr = 55 min
time (row AC) = 4/2 = 2 hr
Answer: q=?/2 it takes 55 min.
(b2)
Answer: q=0 it takes 2 hr.
(c)
c = 4*0.6818 = 2.7272 mi/hr
(c1)
time (walk ABC) = 2*?/2.7272 = 2.3039 hr = 2 hr 18.23 min
Answer: q=0 it takes 2 hr.
(c2)
Answer: q=?/2 it takes 2 hr 18 min.
(d)
c = 2*0.6818 = 1.3636 mi/hr
(d1)
Answer: q=0 it takes 2 hr.
(d2)
time (walk ABC) = 2*?/1.3636 = 4.6078 hr = 4 hr 36 min
Answer: q=?/2 it takes 4 hr 36 min.
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