Question

Spheres of equal size are piled in the form of complete pyramid with regular base, find the total number of spheres in the pile if the number of spheres in the long side is 5 and that of the short side is 4 until the top layer consists of a single row of spheres.

A. 30 B. 50 C. 40 D. 60 E. None of the above
Given f(x) = (x-4)(x+3)+4, when f(x) is divided by (x-K), the remainder is K. Fine the value of K.
A. 2 B.4 C.-4 D.-2 E. None of the above
A messenger travels from points A to B. If he will leave A at 8:00 am and travel at 2kph, he wil arrive at B 3 mins. earlier than his expected time of arrival. However, if he will leave at 8:30 am and travels at 3kph, he will arrive 6 minutes later than the expected time of arrival. What is the expected time of arrival ?
A. 8:06 B. 9:06 C. 10:06 D. 7:06 E. none of the above

Answer 1

The response explains how to approach the geometric, algebraic, and kinematic aspects of the problem set, highlighting the need for knowledge of pyramidal numbers, the remainder theorem, and rate-time-distance relationships.

Step-by-step explanation:

The task involves a geometric arrangement of spheres to form a pyramidal structure, solving an algebraic equation involving the remainder theorem, and using rate, time, and distance calculations to determine an expected time of arrival. To calculate the number of spheres in a pyramidal pile with 5 spheres on a long side and 4 on a short side, we would use the formula for the sum of the first n integers, as each level's number of spheres can be represented as a triangle number. For the remainder theorem part, the remainder when f(x) is divided by (x - K) is f(K); therefore specifying that the remainder is K allows us to set up an equation to solve for K. Finally, the rate, time, and distance problem involves setting up equations representing two different scenarios and solving for the expected time of arrival.