Final answer:
The question is about finding the dimensions of an ellipse given the foci and the major axis length. The major axis length is 10 units, the semi-major axis is 5 units, and the semi-minor axis can be calculated using the relationship between the major axis, minor axis, and foci, resulting in approximately 4.58 units.
Step-by-step explanation:
The question relates to the mathematical concept of an ellipse, which is a shape characterized by two focal points and a major and minor axis. The given foci of the ellipse are (7, 4) and (3, 4), indicating that the ellipse is oriented horizontally. The major axis has a total length of 10 units, denoted as 2a in many geometry texts. Therefore, the semi-major axis (a) is half of the major axis, which is 5 units in this case. Since the distance between the two foci (2c) is the length of the segment joining the foci, we can calculate this as the horizontal distance between the two points, resulting in 7 - 3 = 4 units.
Using the relationship c² = a² - b² (where c is the distance from the center of the ellipse to each focus, a is the semi-major axis, and b is the semi-minor axis), we can find b, the semi-minor axis. Since we have a = 5 and c = 2 (half the distance between the foci), we can calculate b as follows:
b² = a² - c² = 25 - 4 = 21
Thus, b = √21, which is approximately 4.58 units. This finds the necessary parts of the ellipse.