The slope of the minute hand on an analog clock is given by the expression cot(π/30 t), where t is time in minutes. Which of the following best represents the first time when the slope is 0.5?
-9.7 min.
-10.0 min.
-10.3 min.
-10.6 min.
its 10.6 but i have no clue how to solve
1. C (28.3)
2. E (sec x)
3. B (tan x and sec x)
4. E (sec x)
5. A (2.5 seconds)
6. D (a=1; b=-1; c=pi)
7. C (f(x)=2sin(-1.5x+0.5)-2)
8. B (theta(t)=6cos(2pi/3 t)
9. C (x=[20,30])
10. A (a decreasing function defined in Quadrants I and II)
11. B (51°)
12. D (sqrt 1-x^2)
13. E (arccos (cosx)=x)
14. D (10.6 mins)
To find the first time when the slope is 0.5, we need to solve the equation cot(π/30 t) = 0.5.
Using an inverse trigonometric function, we can rewrite the equation as arccot(0.5) = π/30 t.
Using a calculator, we find that arccot(0.5) is approximately 63.43 degrees or 1.1106 radians.
Now we can solve for t by multiplying both sides of the equation by (30/π):
(π/30)t = arccot(0.5)
t = (30/π) * arccot(0.5)
t ≈ 10.639
Therefore, the closest answer choice is -10.6 min.
To find the first time when the slope of the minute hand is 0.5, we need to solve the equation cot(π/30 t) = 0.5. Let's break down the steps to find the solution.
Step 1: Rewrite the equation
We know that cot(x) = 1/tan(x), so we can rewrite the equation as 1/tan(π/30 t) = 0.5.
Step 2: Isolate the tangent function
Multiplying both sides of the equation by tan(π/30 t), we get 1 = 0.5 * tan(π/30 t).
Step 3: Solve for t
Taking the arctangent (tan^(-1)) of both sides, we have arctan(1) = arctan(0.5 * tan(π/30 t)).
Simplifying further, we get π/4 = arctan(0.5 * tan(π/30 t)).
Step 4: Solve for π/30 t
To isolate the variable t, multiply both sides of the equation by 30/π:
(π/4) * (30/π) = (30/π) * arctan(0.5 * tan(π/30 t)).
This simplifies to 30/4 = 30 * arctan(0.5 * tan(π/30 t)).
Step 5: Find the inverse tangent
Dividing both sides by 30, we have 30/4 / 30 = arctan(0.5 * tan(π/30 t)).
This simplifies to 1/4 = arctan(0.5 * tan(π/30 t)).
Step 6: Evaluate the inverse tangent
Using a calculator, find the inverse tangent of both sides to get:
0.244979 = 0.5 * tan(π/30 t).
Step 7: Solve for π/30 t
Dividing both sides by 0.5, we have 0.244979 / 0.5 = tan(π/30 t).
This simplifies to 0.489958 = tan(π/30 t).
Step 8: Find the angle with a tangent of 0.489958
Using a calculator or a table of tangent values, find the angle whose tangent is approximately 0.489958. We find that the angle is approximately 26.6 degrees.
Step 9: Solve for t
We know that the angle π/30 t corresponds to 26.6 degrees. Setting up the proportion π/30 = 26.6/180, we can solve for t:
π/30 = 26.6/180
π * 180 = 30 * 26.6
180π = 798
π = 798/180
π ≈ 4.4333
Therefore, t = (798/180) * 30 ≈ 133.2 minutes.
Step 10: Determine the closest given time
Out of the given options, the closest time to 133.2 minutes is -10.0 minutes.
Therefore, the best representation of the first time when the slope is 0.5 is -10.0 minutes.