Rajesh's Responses in WyzAnt Answers

Ratio of height to new height = height:new height = 147/30.5 = 4.82

This is a ratio problem and not a proportion problem. 

Ratio is the answer obtained by dividing two similar quantities (quantities with same units). We use symbol : to specify ratio.

Equality of two ratios is called proportion. We indicate that ratio of a/b is same as ratio of c/d as follows.

a/b=c/d or a:b = c:d

In a proportion problem, normally one of the 4 variables is unknown and we can solve for it as follows.

e.g.

3/4 = x/12, find x.

3/4 = x/12....multiply both sides by 12

3/4 * 12 = x/12 * 12 ....simplify

3 * 3 = x

x = 9

Let us first simplify the equation.

(1-cox5x)/(1-cos7x) .... multiply numerator and denominator by (1+cos5x)(1+cos7x)

=[(1-cos5x)(1+cos5x)(1+cos7x)]/[(1-cos7x)(1+cos7x)(1+cos5x)]......remember that (a+b)(a-b)=a²-b²

=[(1-cos²5x)(1+cos7x)]/[(1-cos²7x)(1+cos5x)]  ....1-cos²x = sin²x

=[sin²5x(1+cos7x)]/[sin²7x(1+cos5x)]....rearrange and multiply and divide by 25x².49x²

=[25x².sin²5x/25x²][49x²/(49x²sin²7x)][(1+cos7x)/(1+cos5x).....rearrange

=(sin5x/5x)².(7x/sin7x)2.(25x²/49x²).(1+cos7x)/(1+cos5x)

now simplify and take limit. Remember that limit _x->0 (sinx/x) = 1

=(25/49) . lim _x->0 (sin5x/5x)² . lim _x->0 (7x/sin7x)² . lim _x->0 (1+cos7x)/(1+cos5x)

=(25/49).1.1.(1+cos0)/(1+cos0) = (25/49).(2/2)

=25/49 

Let the moving point P be (x, y)

remember that slope m of line joining points (x1, y1) and (x2, y2) is m = (y1-y2)/(x1-x2)

slope of line joining point P(x, y) to A(1, 3) is (y-3)/(x-1)

slope of line joining point P(x, y) to B(3, 1) is (y-1)/(x-3)

We have been told that, slope of PA = 3 x slope of PB

(y-3)/(x-1) = 3 (y-1)/(x-3) ......multiply both sides by (x-1) (x-3)

(x-3)(y-3) = 3 (y-1)(x-1)

xy-3x-3y+9=3(xy-y-x+1)=3xy-3y-3x+3....subtract xy-3x-3y+9 from both sides

0 = 2xy -6 = 2(xy-3) = 0

xy -3 = 0 ...add 3 to both sides

xy = 3

y = 3/x is the equation of the locus

Let me restate this cryptic question.

2.4 million lb (pounds) of almonds are sold when the selling price is $5.50 per lb.

4.8 million lb (pounds) of almonds are sold when the selling price is $4.50 per lb.

What linear function expresses the amount of almonds sold as a function of the price per pound?

Answer:

Here selling price is the independent variable (x-axis) and almond sales in pound is a function of price (y-axis).

We can see that as price of almonds goes up, its sales decrease. So we expect a negative slope for this linear equation line.

Here we need to find the equation of line that passes through two points (x1, y1)=(5.5, 2.4) and (x2, y2)=(4.5, 4.8).

Solution using equation of line in 2 points form

(y-y1)/(y1-y2) = (x-x1)/(x1-x2) substitute co-ordinates of above two points in this equation

(y - 2.4)/(2.4 - 4.8) = (x - 5.5)/(5.5 - 4.5)

(y-2.4)/(-2.4) = (x - 5.5)/1

(y-2.4)/(-2.4) * -2.4 = (x - 5.5)/1 * -2.4 multiply both sides by -2.4

y - 2.4 = -2.4 (x - 5.5)

y - 2.4 = -2.4x + 13.2 ...remember when you multiply 2 negative numbers, you get a positive number

add 2.4 to both sides

y - 2.4 + 2.4 = -2.4x + 13.2 + 2.4

y = -2.4x + 15.6 (where x is price per pound of almonds and y is almonds sold in million pounds)

Alternative method by using slope-intercept form

Equation of line is given by

y = mx + c where m is the slope and c is the y-intercept

Slope of line passing through two points (x1, y1) and (x2, y2) is m = (y1-y2)/(x1-x2)

y = mx + c ....substitute above value of slope m.

y = (y1-y2)/(x1-x2)*x + c where c is the constant y-intercept

Substitute values of (x1, y1) and (x2, y2) in this equation

y= (2.4 - 4.8)/(5.5 - 4.5) *x +c

y=-2.4/1 *x +c y=-2.4x+c...................(1)

substitute (x1, y1) = (5.5, 2.4) in above equation to solve for c

2.4 = -2.4*5.5 +c

2.4 = -13.2 +c ....add 13.2 on both sides to get value of c

2.4 + 13.2 = -13.2 +c + 13.2

15.6 = c Substitute this value in equation (1) above to get

y = -2.4x + 15.6

Check:

We see that slope of the linear equation is negative. This is as expected.

substitute x = 5.5 in above equation and see if you get y = 2.4

substitute x = 4.5 in above equation and see if you get y = 4.8

1) use perfect square formula a²+2ab+b²=(a+b)²

16y² + 24y + 9 = (4y)² + 2.4y.3 + 3² = (4y+3)²

2) use difference of squares formula a² - b² = (a+b)(a-b)

49x² - 100y² = (7x)² - (10y)² = (7x-10y)(7x+10y)

3) use sum of cubes formula, a³ + b³ = (a+b)(a²-ab+b²)

27x³ + 64y³ = (3x)³ + (4y)³ = (3x+4y)(9x²-12xy+16y²)

4) use difference of cubes formula, a³ - b³ = (a-b)(a²+ab+b²)

x³ - 125 = x³ - 5³ = (x-5)(x²+5x+25)