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### Question 58:
We are given the following probabilities:
- Probability of selecting a long-sleeved shirt: \( P(\text{Long-sleeved}) = \frac{13}{23} \)
- Probability of selecting a white shirt: \( P(\text{White}) = \frac{8}{23} \)
- Probability of selecting a long-sleeved and white shirt: \( P(\text{Long-sleeved and White}) = \frac{3}{23} \)

We are asked to find the probability that the selected shirt is either long-sleeved or white or both. This is the union of the two events (long-sleeved or white), which is given by:

\[
P(\text{Long-sleeved or White}) = P(\text{Long-sleeved}) + P(\text{White}) - P(\text{Long-sleeved and White})
\]

Substituting the values:

\[
P(\text{Long-sleeved or White}) = \frac{13}{23} + \frac{8}{23} - \frac{3}{23} = \frac{18}{23}
\]

Thus, the answer is **C. \( \frac{18}{23} \)**.

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### Question 59:
The equation of an ellipse is given by:

\[
\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1
\]

where:
- \( (h, k) \) is the center of the ellipse,
- \( a \) is the semi-major axis (horizontal axis),
- \( b \) is the semi-minor axis (vertical axis).

From the diagram, the center of the ellipse is at the origin \( (0, 0) \), so \( h = 0 \) and \( k = 0 \). The lengths of the major and minor axes are determined by the points \( (8, 0) \) and \( (0, 5) \), meaning:
- \( a = 8 \) (half the major axis),
- \( b = 5 \) (half the minor axis).

Thus, the equation becomes:

\[
\frac{x^2}{8^2} + \frac{y^2}{5^2} = 1
\]

or:

\[
\frac{x^2}{64} + \frac{y^2}{25} = 1
\]

Thus, the correct answer is **E**.

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Would you like to review any further steps or explanations? Here are some related practice questions:

1. What is the probability of two independent events both occurring?
2. How is the equation of an ellipse related to its foci?
3. What is the formula for the union of two events in probability?
4. How do you determine the major and minor axes of an ellipse from its equation?
5. What is the difference between mutually exclusive and independent events in probability?

**Tip:** When solving problems involving geometric shapes like ellipses, carefully analyze the graph to extract the correct parameters such as axis lengths and center coordinates.

58. A shirt will be randomly selected from a display of 23 shirts. The probability that the selected shirt will be long-sleeved is 13/23. The probability that the selected shirt will be white is 8/23. The probability that the selected shirt will be long-sleeved and white is 3/23. What is the probability that the selected shirt will be long-sleeved or white or both?
59. The ellipse in the standard (x, y) coordinate plane below is centered at the origin. The endpoints of the major and minor axes of the ellipse are labeled. Which of the following equations determines this ellipse?

Solve probability problems involving the union of two events and determine the equation of an ellipse based on its center and axis lengths. This problem involves core concepts in probability and coordinate geometry, ideal for students in Grades 11-12.

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