whence 9y=27y2,
and y = 3;

.*. x=27 y2=243;

.. the numbers are 243, and 3.

29. It is required to find the three sides of a right-angled triangle from the following data. The number of square feet in the area is equal to the number of feet in the hypothenuse the sum in the other two sides; and the square described upon the hypothenuse is less than the square described upon a line equal in length to the two sides, by half the product of the numbers representing the base and area.

2

Let the number of feet in the altitude,

=

and y the number in the base;

2

+ y2 = the number in the hypothenuse, (Eucl.

2

.. { xy = √√ x2 + y2+x+y;

also x2 + y2 = (x + y) −4xy2 =) x2 + 2xy + y2 — ‡ xy2; .. by transposition, xy=2xy,

and y = 8;

hence from the first equation, 4x = √x2+64+x+8, and by transposition, 3x-8=√√√x2+64; .. 9x3 −48x+64 = x2+64,

and 8x48x;

.. x=6;

whence the hypothenuse = √√64 + 36 = 10;
.. the sides are 6, 8, and 10 feet, respectively.

30. A Farmer has 2 cubical stacks of hay. The side of one is 3 yards longer than the side of the other; and the dif

ference of their contents is 117 solid yards. Required

cubing the latter equation, x3-3x2y + 3x y2 − y3 = 27;

and the sides of the stacks are 5, and 2 yards, respectively.

31. When a parish was enclosed, the allotment of one of the proprietors consisted of two pieces of ground; one of which was in the form of a right-angled triangle; the other was a rectangle, one of the sides of which was equal to the hypothenuse of the triangle, the other, to

FF

half the greater side; but wishing to have his land in one piece, he exchanged his allotments for a square piece of ground of equal area, one side of which equalled the greater of the sides of the triangle which contained the right-angle. By this exchange he found that he had saved ten poles of railing. What are the respective areas of the triangle and rectangle; and what is the length of each of their sides?

Let 2x the greater side of the triangle,

2

and y the less;
y= =

..√4x+y=the hypothenuse; and also the greater side of the rectangle,

and the less side of the rectangle ; .. xy=the area of the triangle, and x√√4x2+y=the area of the rectangle ;

••• 4x2=xy + x/4x2+y2,

or 4x-y=4x2+y2;

also 8x+10=2x+y+√4x2+y2+2x+2 √ √ 4x2 + y2. or 4x+10= y + 3/4x+y; in which equation substituting the value of √4x+y found above;

2

... 4x+10=y+3(4x-y)=12x-2y;
.. by transposition, 2y=8x-10,

and y=4x-5;

.. from the first equation, 54x2+(4x-5)2,
and 25=4x+16x2-40x +25;

by transposition, 40x=20x2;

.. 2=x,

and y=4x-5=3;

.. the sides of the triangle are 3, 4, and 5; the sides of the rectangle are 2, and 5; and the areas of the triangle and rectangle are 6, and 10, respectively.

227

SECT. IX.

Examples of the Solution of Problems producing Adfected Quadratic Equations.

1. A MERCHANT sold a quantity of brandy for £.39, and gained as much per cent. as the brandy cost him. What was the price of the brandy?

100

or =3900-100 x.;

by transposition, x2 + 100x=3900,

completing the square, x+100x+50=3900+2500

=6400;

extracting the root, x+50= ± 80;

.. x=30, or 130;

.. the price was £.30.

2. There are two numbers whose difference is 9, sum multiplied by the greater produces 266. those numbers?

Let x the greater;

..x-9= the less,

and x.(2x-9) = 266;

and their

What are

and both values answer the conditions of the problem.

3. It is required to find two numbers, the first of which may be to the second as the second is to 16; and the sum of the squares of the numbers may be equal to 225.

completing the square, x2+16x+64225 +64 = 289; extracting the root, x+8=17;

but as this latter value of x makes the second number an impossible quantity, 9 is the only value of r which answers the conditions, and therefore the numbers are 9 and 12.

4. Bought two sorts of linen for 6 crowns.

An ell of the

finer cost as many shillings as there were ells of the finer. Also 28 ells of the coarser (which was the whole quantity) were at such a price that 8 ells cost as many shillings as 1 ell of the finer. How many ells were there of the finer, and what was the value of each piece?

Let x the number of ells of the finer;

..the price of the finer (in shillings,)