Of the 42 students in a certain Spring 2010 Math 147 class, 19 passed both the mid-term and the final examinations. A total of 25 students passed the mid-term examination and a total of 30 students passed the final exam.

(a) Represent this information on a Venn diagram. (Take the set of the 52 students to be the universal set.)

(b) How many students failed both examinations?

you forgot an image

step-by-step explanation:

part a: triangles gcb and epb

part b. sides cg and rp are parallel by definition of parallelogram.

angles cgb and e are congruent by being alternate interior angles of parallel lines cut by a transversal. angles gbc and ebp are congruent because they are vertical angles. by aa similarity, the triangles are similar.

part c. since the triangles are similar, the ratios of the lengths of corresponding sides are equal.

calculation of be:

gb/be = cb/bp

(400 ft)/be = 300/200

(400 ft)/be = 3/2

3be = 2 * 400 ft

3be = 800 ft

be = 800/3 ft

calculation of pe:

gc/pe = cb/bp

(350 ft)/pe = 300/200

(350 ft)/pe = 3/2

3pe = 2 * 350 ft

3pe = 700 ft

pe = 700/3 ft