On a day when the sun passes directly overhead at noon, a 1.80-m-tall man casts a shadow of length S(t)=1.8|cot(πt/12)|, where S is the length of shadow in meters and t is the number of hours since 6am, at what time from the following list the length of the shadow would equal to the man’s height?
(A) 3pm (B) 4pm (C) 5pm (D) 6pm (E) never happen
clearly, when cot(πt/12) = 1
πt/12 = π/4
t = 3
Huh. I get 9am
Note that at noon, t=6, and the shadow has zero length.
For the length of the shadow to be equal to the man's height, then S(t)=1.8, meaning the equation becomes:
1.8=1.8|cot(πt/12)|
1=|cot(πt/12)|
1=1/tan(πt/12)
tan(πt/12)=1
πt/12=π/4
πt=3π
t=3
This means 3 hours after 6am would be 9am, which means none of the times on the list are correct, making option E correct.
To find the time at which the length of the shadow is equal to the man's height, we need to set the shadow length equation equal to the man's height, and solve for t.
The shadow length equation is given as S(t) = 1.8|cot(πt/12)|.
And the man's height is 1.80 meters.
So we have the equation: 1.8|cot(πt/12)| = 1.80.
To solve for t, we first isolate the absolute value expression:
|cot(πt/12)| = 1.
Now, we consider two cases, where the cotangent function is negative and positive.
Case 1: cot(πt/12) < 0.
In this case, the absolute value expression becomes -cot(πt/12).
So we have: -cot(πt/12) = 1.
Simplifying: cot(πt/12) = -1.
Case 2: cot(πt/12) > 0.
In this case, the absolute value expression remains as cot(πt/12).
So we have: cot(πt/12) = 1.
Now let's solve each case separately.
Case 1: cot(πt/12) = -1.
To find the values of t that satisfy this equation, we can look at the range of the cotangent function.
The cotangent function is negative in the second and fourth quadrants of the unit circle.
Let's find the values of t in the second quadrant: π < πt/12 < 3π/2.
π < πt/12 < 3π/2.
Multiply by 12/π: 12 < 12t/2 < 18.
Simplify: 6 < 6t < 18.
Divide by 6: 1 < t < 3.
Therefore, in the second quadrant, the values of t that satisfy cot(πt/12) = -1 are 1 < t < 3.
Now let's find the values of t in the fourth quadrant: 3π/2 < πt/12 < 2π.
3π/2 < πt/12 < 2π.
Multiply by 12/π: 18 < 12t/2 < 24.
Simplify: 9 < 6t < 24.
Divide by 6: 3/2 < t < 4.
Therefore, in the fourth quadrant, the values of t that satisfy cot(πt/12) = -1 are 3/2 < t < 4.
Now let's move on to Case 2.
Case 2: cot(πt/12) = 1.
To find the values of t that satisfy this equation, we can again look at the range of the cotangent function.
The cotangent function is positive in the first and third quadrants of the unit circle.
Let's find the values of t in the first quadrant: 0 < πt/12 < π/2.
0 < πt/12 < π/2.
Multiply by 12/π: 0 < 12t/2 < 6.
Simplify: 0 < 6t < 6.
Divide by 6: 0 < t < 1.
Therefore, in the first quadrant, the values of t that satisfy cot(πt/12) = 1 are 0 < t < 1.
Now let's find the values of t in the third quadrant: π < πt/12 < 3π/2.
π < πt/12 < 3π/2.
Multiply by 12/π: 12 < 12t/2 < 18.
Simplify: 6 < 6t < 18.
Divide by 6: 1 < t < 3.
Therefore, in the third quadrant, the values of t that satisfy cot(πt/12) = 1 are 1 < t < 3.
Now, let's combine all the solutions for each case.
For Case 1, the values of t that satisfy cot(πt/12) = -1 are 1 < t < 3 and 3/2 < t < 4.
For Case 2, the values of t that satisfy cot(πt/12) = 1 are 0 < t < 1 and 1 < t < 3.
From these ranges of t-values, we can see that there is an overlap between Case 1 and Case 2. The overlapping range is 1 < t < 3.
So, the length of the shadow would equal to the man's height at 1pm and 2pm.
None of the options provided exactly match the time. Therefore, the answer is (E) never happen.
To find the time when the length of the shadow is equal to the man's height, we need to set the equation for the length of the shadow, S(t), equal to the man's height, which is 1.80 meters.
S(t) = 1.8 * |cot(πt/12)| = 1.80
Now, let's solve this equation to find the value of t.
1.8 * |cot(πt/12)| = 1.80
Since we have an absolute value function, we need to consider both the positive and negative cases.
Case 1: cot(πt/12) > 0
We can remove the absolute value by dropping the bars.
1.8 * cot(πt/12) = 1.80
Divide both sides by 1.8:
cot(πt/12) = 1
Taking the inverse cotangent (cot^(-1)) of both sides:
πt/12 = cot^(-1)(1)
πt/12 = π/4
Divide both sides by π/12:
t = 3
So, when cot(πt/12) > 0, the length of the shadow would be equal to the man's height at 3 pm.
Case 2: cot(πt/12) < 0
Again, remove the bars by dropping them and negating the expression.
1.8 * -cot(πt/12) = 1.80
Divide both sides by 1.8:
-cot(πt/12) = 1
Multiply both sides by -1 to isolate the cotangent:
cot(πt/12) = -1
Take the inverse cotangent of both sides:
πt/12 = cot^(-1)(-1)
πt/12 = 3π/4
Divide both sides by π/12:
t = 9
So, when cot(πt/12) < 0, the length of the shadow would be equal to the man's height at 9 pm.
Therefore, the length of the shadow would be equal to the man's height at both 3 pm and 9 pm. Since 9 pm is not an option in the given list, the correct answer would be (A) 3 pm.