Find the absolute maximum and absolute minimum values of f on the given interval. f(t) = 9t + 9 cot(t/2), [π/4, 7π/4]
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Final answer:
The absolute maximum and minimum of a function on a given interval can be found by calculating the function's critical points and evaluating the function at these points and the interval endpoints, then comparing these values.
Explanation:
In order to find the absolute maximum and absolute minimum values of a function on a given interval, you must first find the critical points of the function within the interval. Critical points occur where the derivative of the function is equal to zero or is undefined. In this case, the derivative of f(t) = 9t + 9 cot(t/2) is f'(t) = 9 - (9/2) csc2(t/2). Set this to zero and solve for t to find the critical points. Additionally, the endpoints of the interval, π/4 and 7π/4, could be the absolute maximum or minimum, so these should be evaluated as well. Once you have found the values of the function at these points and the endpoints, compare them to determine the absolute maximum and minimum values.
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Final answer:
To find the absolute maximum and minimum values of a function, we find the critical points and endpoints. Evaluating the function at these points gives the maximum and minimum values.
Explanation:
To find the absolute maximum and absolute minimum values of a function on a given interval, we need to find the critical points and endpoints of the interval.
To find the critical points of f, we need to find where the derivative of f is equal to zero or undefined. The derivative of f(t) = 9t + 9cot(t/2) is f'(t) = 9 - 9csc^2(t/2).
Setting f'(t) = 0, we have 9 - 9csc^2(t/2) = 0. Solving this equation, we get csc^2(t/2) = 1, which means sin^2(t/2) = 1. This gives us sin(t/2) = ±1. The critical points occur when t/2 = π/2 or t/2 = 3π/2. Solving for t, we get t = π or t = 3π as the critical points.
The endpoints of the interval are π/4 and 7π/4.
Now we evaluate the function f at the critical points and endpoints:
- f(π/4) = 9(π/4) + 9cot(π/8) ≈ 6.566
- f(π) = 9π + 9cot(π/2) = 9π
- f(3π) = 9(3π) + 9cot(3π/2) = 27π
- f(7π/4) = 9(7π/4) + 9cot(7π/8) ≈ 46.607
From these evaluations, we can see that the absolute maximum value occurs at t = 7π/4 and is approximately 46.607, while the absolute minimum value occurs at t = π/4 and is approximately 6.566.
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The area of a playground is 216 yd^2. the width of the playground is 6 yd longer than its length find the length and width.A. length=18, width=12 B. length=12, width=18 C. length=18, width=24 D.length=24, width=18
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A rectangular lot that measures 125 feet by 185 feet is completely fenced. What is the length, in feet, of the fence?Can you please explain with some details?
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Answer:
CD = √11 and CE = √11
Step-by-step explanation:
We know that m∠D is 45° (by using the sum of interior angles in a triangle) so therefore, ΔDCE is a 45 - 45 - 90 triangle (the 45, 45, and 90 refer to the angle measures). The ratio of sides in a 45 - 45 - 90 triangle is 1 : 1 : √2 where the 1s are the sides and the √2 is the hypotenuse. We need to solve for x in x : x : √22. If you notice that √22 = √2 * √11, we can use this to find x, therefore, x = 1 * √11 = √11 so CD = √11 and CE = √11.
Answer:
Step-by-step explanation:
The triangle is a right triangle.
We can apply trigonometric functions to solve for the missing sides.
sin θ = opp/hyp
sin 45 = CD /√22
Multiply both sides by √22.
√22 sin 45 = CD
√11 = CD
cos θ = adj/hyp
cos 45 = CE /√22
Multiply both sides by √22.
√22 cos 45 = CE
√11 = CE
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Answer:
A or B
Step-by-step explanation:
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12 + 12 = 24
12 x 12 = 144
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OR
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1. You find, that the number 9.2 is 100% - because it's the output value of the task.
2. You need to find out what x is.
3. If 100% equals 9.2, you can write it down as 100%=9.2.
4. Now you know, that x% equals 43.7 of the output value, so we can write it down as x%=43.7.
5. Now you have two simple equations:
1) 100%=9.2
2) x%=43.7
where left sides of both of them have the same units, and both right sides have the same units, so we can do something like that:
100%/x%=9.2/43.7
7. Solution for 43.7 is what percent of 9.2
100%/x%=9.2/43.7
(100/x)*x=(9.2/43.7)*x - you multiply both sides of the equation by x
100=0.210526315789*x - you divide both sides of the equation by (0.210526315789) to get x
100/0.210526315789=x
475=x
x=475
So the answer is 475%! Hope it works!