hectictar

slope   =   \(\frac{y_2-y_1}{x_2-x_1}\ =\ \frac{7-6}{7--6}\ =\ \frac{1}{13}\)

Using the point  (-6, 6)  and the slope  \(\frac{1}{13}\)  we can make the equation of the line:

y - 6   =   \(\frac{1}{13}\)( x  -  -6)

y - 6   =   \(\frac{1}{13}\)( x  +  6)

I'm not totally sure what it means by "fully reduced" because if we simplified it any more then I don't think it would be in point-slope form anymore, so I would just leave it at that.

The total surface area to paint  =  12 m  *  80 m   =   960 square meters

1 ft   ≈   0.3048 m

1 ft²  ≈   0.09290304 m²

200 ft²   ≈   18.580608 m²

And so one gallon of paint covers about  18.581  square meters.

960 m²  /  ( 18.581 m² per gallon)  ≈   51.67 gallons

So  51  gallons would not be enough... it would take  52  gallons of paint to cover the wall.

(Note to previous answerer:  960 m²  ≈  10333.354 ft²   ≉  3150 ft²)

.

The sum of the interior angle measures in a hexagon is  180° * (6 - 2)  =  720°

135°  +  62°  +  151°  +  140°  +  n°  +  (n + 6)°   =   720°

135  +  62  +  151  +  140  +  n  +  n + 6   =   720

494  +  2n   =   720

2n   =   226

n   =   113

area of shaded region   =   area of larger rectangle  -  area of smaller rectangle

area of shaded region   =   ( 15 cm * 10 cm )  -  ( 9 cm * 6 cm )

area of shaded region   =   ( 150 sq cm )  -  ( 54 sq cm )

area of shaded region   =   96 sq cm

(f  o  g)(x)   =   f( g(x) )   =   3( g(x) ) - 1   =   3( x² + 2 ) - 1   =   3x² + 6 - 1   =   3x² + 5

Let  n  be the number of nickels in the jar  and let  d  be the number of dimes.

There are  22  coins in the jar, so we can make this equation:

n + d  =  22     Let's subtract  d  from both sides of this equation

n  =  22 - d

The total value of the coins is  $1.60,  so we can make this equation:

0.05n  +  0.10d  =  1.60

                                              We can multiply both sides of the equation through by  10

5n  +  10d   =   160

                                              Since  n = 22 - d  we can substitute  22 - d  in for  n

5(22 - d)  +  10d   =  160

                                              Distribute  5  to each term in parenthesees

110 - 5d  +  10d   =   160

                                              Combine  -5d  and  10d  to get  5d

110 + 5d   =   160

                                              Subtract 110  from both sides of the equation

5d   =   50

                                              Divide both sides of the equation by  5

d   =   10

Now that we know  d  =  10   we can find  n  by substituting  1  for  d  in the first equation.

n   =   22 - d   =   22 - 10   =   12

So there are  12  nickels and  10  dimes.

1.  Yes, △LMN is similar to △WXY because  21/30  =  7/10  =  28/40  and the included angles are equal because they are both 90°, so by the SAS similarity theorem, the two triangles are similar.

2. The one with the larger base, which is the one with △WXY as the base, will have the greater volume.

volume of smaller prism  =   (area of base) * (height)

volume of smaller prism   =   (1/2 * 30 * 21) * (height)

volume of smaller prism   =   (315) * (height)

volume of smaller prism   =   315h        ,  where  h  is the height of the prism

volume of larger prism   =   (area of base) * (height)

volume of larger prism   =   (1/2 * 40 * 28) * (height)

volume of larger prism   =   (560) * (height)

volume of larger prism   =   560h

The larger prism is how many times greater than the smaller prism? That is,  560h  is how many times greater than  315h  ?

560h/315h   =   16/9         so...         560h  =  (16/9) * 315h

In other words,  560h  is  16/9  times  315h ,  which means the volume of the larger larger prism is  16/9  times the volume of the smaller prism.

The surface area of a prism is the perimeter of the base times the height, plus two times the area of the base. In this case the base is a triangle so we have a triangular prism.

surface area   =   ( perimeter of base ) * ( height of prism )  +  2 * ( area of base )

surface area   =   ( 2 ft  +  1.5 ft  +  2.5 ft ) * ( 10 ft )  +  2 * ( 1/2  *  2 ft  *  1.5 ft )

surface area   =   ( 6 ft ) * ( 10 ft )  +  2 * ( 1.5 ft² )

surface area   =   60 ft²  +  3 ft²

surface area   =   63 ft²

Values of  x  that make the denominator 0 must be excluded from the domain.

Values of  x  that make the expression inside the square root negative must be excluded from the domain.

So.... any values of  x  that make  x² - 5x  less than or equal to zero must be excluded from the domain.

x² - 5x  ≤  0

To find which values of  x  make this negative, first let's find which values make it equal zero.

x² - 5x  =  0

x(x - 5)  =  0

x  =  0     or     x  =  5

The graph of   y = x² - 5x  is a parabola that intersects the  x-axis when  x = 0  and  x = 5.

So to find which values of  x  make the expression negative, we can test a point in each of the following intervals:

(-∞, 0)  or  (0, 5)  or  (5, ∞)

(-1)² - 5(-1)   =   1 + 5   =   6   >   0    So the values of  x  in (-∞, 0)  make the expression positive

(1)² - 5(1)   =   1 - 5   =   -4   <   0   So the values of  x  in  (0, 5)  make the expression negative

(6)² - 5(6)   =   36 - 30   =   6   >   0  So the values of  x  in  (5, ∞)  make the expression positive

And so the values of  x  that fall in the range [0, 5]  must be excluded from the domain.

So the domain is:

(-∞, 0)  U  (5, ∞)

Let  m  be the mystery number.

9x² + 6x + m  =  (3x)² + 2(3x) + m

Now we can substitute  a  for  3x  like this:

a² + 2a + m

Now we don't have to worry about the coefficient of  9  on the first term....and so  m  is half the coefficient of the middle term squared.

m  =  ((1/2) * 2)^2   =   1^2   =   1

Check:

9x² + 6x + 1   =   (3x + 1)(3x + 1)   =   (3x + 1)²