let

l = the length of the rectangular area

w = the width of the rectangular area, w<=l

A = 300 m^2 is the are of the enclosure

since A = l*w is the formula for area of rectangle we have

l*w = 300

one of the sides is against the barn that side will not need fencing

First case: one length is against the barn then the length if the enclosure is l + 2w = 50

by solving the system of equations

l*w = 300

l + 2w = 50

we find 2 solutions:

- first solution

l = 20 m

w = 15 m

- second solution

l = 30 m

w = 10 m

Second case: one width is against the barn then the length if the enclosure is 2l + w = 50

by solving the system of equations

l*w = 300

2l + w = 50

we find

l = 10 m

w = 30 m

is not a solution since we need w<=l

l = 15 m

w = 20 m

is not a solution since we need w<=l

there are 2 solutions:

the dimensions are 20m and 15m

the dimensions are 30m and 10m.