Lesson 6 · from §102 of the 1910 text
Problems of Rate, Work, and Falling Bodies
More problems
These problems are mostly about rates, work, and falling bodies. Recall:
- Distance = rate × time, so time = \dfrac{\text{distance}}{\text{rate}}. If one train needs 1 hour less than another for the same distance, the difference of the two times is 1.
- Work: if a pump fills a tank in x hours, it fills \dfrac1x of the tank in one hour.
- Falling bodies: the distance S through which a body falls from rest in t seconds is S = \dfrac{gt^2}{2}, g being 32 feet; that is, S = 16t^2.
- Right triangles: the square of the hypotenuse equals the sum of the squares of the legs. Two people going north and east (or northwest and southwest) move at right angles to each other.
Fact from geometry: if one angle of a right triangle is 30 degrees, the hypotenuse is twice the shorter leg. Conversely, if the hypotenuse of a right triangle is double one leg, one angle of the triangle is 30 degrees.
Try one:
Quick check
A stone is dropped from a tower and strikes the ground 3 seconds later. How high is the tower?
Exercises
How to type math
Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.
Reject all answers which do not satisfy the conditions of the problems.
The dimensions of a certain rectangle and the longest straight line which can be drawn on its surface are represented in inches by three consecutive even numbers. Find its dimensions.
The dimensions of a rectangular box are in the ratio of 1 : 2 : 3. Find the edges, if the entire outer surface is 792 square inches.
The edges of two cubical bins differ by one yard. Their volumes differ by 61 cubic yards. Find the edge of each bin.
The rates of two trains differ by 5 miles per hour. The faster requires 1 hour less time to run 280 miles. Find the rate of each train.
An automobile made a round trip of 160 miles in 9 hours. Returning, the rate was increased 4 miles per hour. Find the rate each way.
A page of a certain book is 2 inches longer than it is wide. The printed portion covers half the area of the page and the margin is 1 inch wide. Find the length and width of the page.
A man paid $16,000 for a farm. Later he sold all but 40 acres of it for the same sum, thereby gaining $20 on each acre sold. Find the number of acres in the farm.
The price of oranges being raised 10 cents per dozen, one gets 5 fewer oranges for 50 cents. Find the original price.
Two pumps together can fill a standpipe in 45 minutes. One pump alone requires 2 hours less time than the other. Find the time each requires alone.
Each of two trains ran 200 miles. One ran 7 miles per hour faster than the other and required 1 hour and 45 minutes less time. Find the rate of each train.
A and B leave point P at the same time, A going north and B east. Five hours later A has traveled 17 miles more than B and the distance between them is 53 miles. Find the rate of each.
A and B leave point P at the same time, A going northwest and B southwest. Five hours later A has traveled 9 miles less and B has traveled 8 miles less than the distance between them. Find the rate of each.
A stone, dropped from a balloon which was passing over a river, struck the water 15 seconds later. How high was the balloon at the time the stone was dropped?
A man drops a stone over a cliff and hears it strike the ground below 6\tfrac12 seconds later. If sound travels 1152 feet per second, find the height of the cliff.