Question

1. for the mean of 7 numbers to increase by 4, by how much would the sum of the 7 numbers have to increase?

f. 4
g. 7
h. 11
j. 28

1. which of the following is an equation of the circle in the standard \\((x,y)\\) coordinate plane whose center is at \\((4,-3)\\) and whose radius is 5 coordinate units long?

a. \\((x-4)^2 + (y+3)^2 = 25\\)
b. \\((x-3)^2 + (y+4)^2 = 25\\)
c. \\((x+3)^2 + (y-4)^2 = 5\\)
d. \\((x+4)^2 + (y-3)^2 = 25\\)

Explanation:

Relate mean and sum for Question 22

Using the Algebraic Modeling knowledge point, let the sum of the 7 numbers be \(S\) and their mean be \(M\). The relationship between the sum and the mean is:

$$ M = \frac{S}{7} \implies S = 7M $$

If the mean increases by 4, the new mean is \(M' = M + 4\). The new sum \(S'\) is:

$$ S' = 7(M + 4) = 7M + 28 = S + 28 $$

Thus, the sum must increase by 28.

Identify the standard equation of a circle for Question 23

The standard equation of a circle with center \((h, k)\) and radius \(r\) is:

$$ (x - h)^2 + (y - k)^2 = r^2 $$

Substitute the given values for Question 23

We are given the center \((h, k) = (4, -3)\) and the radius \(r = 5\). Substituting these values into the standard equation:

$$ (x - 4)^2 + (y - (-3))^2 = 5^2 $$

Simplifying the signs and calculating the square of the radius:

$$ (x - 4)^2 + (y + 3)^2 = 25 $$

This matches option A.

Answer:

Question 22

• F. 4
• G. 7
• H. 11
• J. 28 (Correct answer)

Question 23

• A. \((x-4)^2 + (y+3)^2 = 25\) (Correct answer)
• B. \((x-3)^2 + (y+4)^2 = 25\)
• C. \((x+3)^2 + (y-4)^2 = 5\)
• D. \((x+4)^2 + (y-3)^2 = 25\)