Find the equations of the normals to the curve xy = 3 which are parallel to the line 3x - y - 2 = 0.

y = 3x + 8.

y = 3x - 8

Step-by-step explanation:

The slope of the tangent to the curve at a given point is obtained from the derivative.

xy = 3

y = 3/x

y = 3 x^-1

y' = -3x^-2

The slope of the normal (which is perpendicular to the tangent)

= -1 / -3x^-2 = 1/3 x^2.

Consider the given line:

3x - y - 2 = 0

y = 3x - 2 , so the slope of this line = 3.

Now this line is parallel to the normal  so 1/3 x^2 = 3

Therefore  x^2 = 9  and x = -3, 3.

Find the equations of the normals:

At x = -3, y = 3/-3 = -1, which gives the equation:

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

y = 3x + 8.

At x = 3 , y = 1

y - 1 = 3(x - 3)

y = 3x - 8.

fabric (square meters)                   cloaks

7                                                           21

4/3                                                         4

step-by-step explanation:

we are told that harry needs 2 square meters of fabric for every 6 wizard cloaks he makes.

1. first of all we will find how many cloaks can be made from 1 square meter of fabric by dividing 6 by 2.

we can see that 3 cloaks can be made from 1 square meter of fabric. now we will multiply 3 by 7 to find number of cloaks made from 7 square meters of fabric.

therefore, harry will make 21 cloaks from 7 square meters of fabric.

2. let us find how many square meters of fabric will be needed to make 4 cloaks.

first of all we will find fabric needed for 1 cloak by dividing 2 by 6.

we can see that fabric needed for 1 cloak is   square meter. now we will multiply   by 4 to find fabric needed for 4 cloaks.