Question

The area of a trapezoidal field is $1400$ square yards. Its altitude is $50$ yards. Find the two bases, if the number of yards in each base is an integer divisible by $8.$ The number of solutions to this problem is:

None

One

Two

Three

More than three

Answer: D

Concepts:trapezoid divisibility basic counting

Difficulty rating: 1450

Small Hint:

Use the trapezoid area formula to find the sum of the two bases

Big Hint:

List the unordered pairs of positive multiples of $8$ with that sum

Solution:

If the bases are $b_1,b_2,$ then $1400=\frac12(b_1+b_2)(50),$ so $b_1+b_2=56.$ The unordered positive pairs of multiples of $8$ are $(8,48),\quad(16,40),\quad(24,32).$ There are three solutions.

Thus, the correct answer is D.

1952 AMC 12 Problem 38

Answer: D

Difficulty rating: 1450

Small Hint:

Big Hint:

Solution:

Problem 38 in Other Years