Problem Statement:

The problem is asking us to find the length of the longest straight line that lies entirely within the volume defined by the given closed surface.

Given:

= 3,
= 5,
= 1000,
= 1300,
z = 3 and z = 4.5

Approach:

To find the longest straight line that lies entirely within the given closed surface, we need to find the largest distance between any two points on the surface. We can do this by finding the distance between each pair of points and then selecting the largest distance.

We can use the distance formula to find the distance between any two points in three-dimensional space:

d = sqrt((x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2)

We will need to find the distance between each pair of points on the surface and then select the largest distance.

Solution:

Let's find the distance between all pairs of points on the surface:

Distance between and :
d = sqrt((5-3)^2 + (1000-3)^2 + (0-0)^2) = 997.008

Distance between and :
d = sqrt((3-5)^2 + (1300-1000)^2 + (0-0)^2) = 300

Distance between and :
d = sqrt((1000-5)^2 + (1300-1000)^2 + (0-0)^2) = 1301.947

Distance between and :
d = sqrt((1000-3)^2 + (1300-3)^2 + (0-0)^2) = 1829.554

Distance between and :
d = sqrt((5-3)^2 + (1000-1300)^2 + (3-4.5)^2) = 300.167

Distance between and :
d = sqrt((3-5)^2 + (1300-1000)^2 + (3-4.5)^2) = 300.167

Distance between and :
d = sqrt((1000-5)^2 + (1300-1000)^2 + (3-4.5)^2) = 1301.947

Distance between and :
d = sqrt((1000-3)^2 + (1300-3)^2 + (3-4.5)^2) = 1829.554

Distance between and :
d = sqrt((5-1000)^2 + (1000-1300)^2 + (3-4.5)^2) = 1299.22

Distance between and :
d = sqrt((3-1000)^2 + (1300-1000)^2 + (3-4.5)^2) = 1299.22

Distance between and :
d = sqrt((5-1000)^2 + (1000-3)^2 + (3-4.5)^2) = 1829.554

Distance between and :
d = sqrt((3-1000)^2 + (1300-3)^2 + (3-4.5)^2) = 1829.554

The largest distance is between and , which is 3.21.

Therefore, the correct answer is option A.