Santino bought a book for $23.54. The price of the book was $22. What was the sales-tax rate?

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 equation of the perpendicular line is y = 6.

Final answer:

To create a sample data set with n=7, a mean of 9, and a standard deviation of 0, all seven data points must be identical to the mean, resulting in the data set {9, 9, 9, 9, 9, 9, 9}.

Explanation:

To construct a sample data set with n=7, where \(\overline{x}=9\) (x bar is the sample mean), and s=0 (s is the sample standard deviation), follow these steps:

1. Understand that if the standard deviation s is zero, all data points in the sample must be identical.
2. Since the sample mean \(\overline{x}\) is given as 9, all seven data points in the sample must also be 9 to achieve a standard deviation of 0.
3. Thus, the sample data set will simply be {9, 9, 9, 9, 9, 9, 9}.

Remember that the sample mean is calculated as the sum of all data values divided by the number of values (n), and the sample standard deviation measures the amount of variation or dispersion of a set of values. A standard deviation of zero indicates that all values are identical to the mean.