geno3141

Start by drawing this triangle on a sheet of graph paper.

Place the base on the x-axis, centered at the origin [point O].

Call the left-hand corner point A(-4,0) and the right-hand corner point B(4,0).

Place the vertex of the triangle [point C] on the positive y-axis.

The first problem is to find the coordinates of C.

The distance from A to O is 4; the distance from A to C is sqrt(80).

Use the Pythagorean Theorem to find the distance from ) to C; it iis 8; so the coordinates of C are (0,8).

Find the equation of line CB.

Its slope is (8 - 0) / (0 - 4) = -2.

The equation is:  y = -2x + 8.

Draw a rectangle inscribed in the triangle with its base on the x-axis.

Let the right-hand corner point of this base be (x,0) and the left-hand corner point of this base be (-x,0).

To find the height of the rectangle: the height will be the y-value -- and at point (x,0), the height is -2x + 8.

The area of the rectangle will be the length of its base (from point (-x,0) to (x,0) which is equal to 2x) times its height (which is -2x + 8).

Area = (2x)(-2x + 8)    --->   Area = -4x² + 16x

I'm assuming that you are taking calculus -- find the first derivative:  Area' = -8x + 16

To find a maximum (or a minimum, but this time it's a maximum), set the first derivative equal to zero and solve:

--->     -8x + 16 = 0     --->     -8x = -16     --->     x = 2

So, the total length of the base is 4.

The height is -2x + 8 = -2(2) + 8 = -4 + 8 = 4

So, the area is:  4 x 4  =  16  square units.

Question 4:  Use the slope-intercept form:  y - y₁  =  m(x - x₁)

m = 7  and  x₁= -4  and  y₁ = -3   --->    y - -3  =  7(x - -4)   --->   y + 3  =  7x + 28   --->   y  =  7x + 25

Question 5:  Lines are parallel when their slopes are parallel.

      Slope formula:  m  =  (y₂ - y₁) / (x₂ - x₁)

A:  (1,6) & (4,-3)   --->   m  =  (-3 - 6) / (4 - 1)   --->   m  =  -9/3  =  -3

     (6,4) & (9,-5)   --->   m  =  (-5 - 4) / (9 - 6)   --->   m  =  -9/3  =  -3      <---  These lines are parallel.

You should also check the rest ...

If Carolyn Tan's regular rate is 75.00, then her overtime rate is 75.00 x 1.5  =  112.50.

Her regular work week consists of 40 hours; this amount is 75.00 x 40  =  3000.00

If she worked 5 3/4 hours of overtime, this amount is (5 3/4) x 112.50  =  646.88.

The total amount is:  3000.00 + 646.88  =  3646.88.

How many primes less than 100 are divisible by 3?

One, just the number 3.

Any other number that is divisible by 3 can't be a prime!

Find the remainder when  f(x) = 2x^3 − x^2 + x + 1  is divided by  2x + 1.

The remainder theorem states that f(x) / (x - a)  will have a remainder of  f(a).

However, this doesn't apply if the coefficient of the x-term isn't 1.

So first, let's divide both  f(x)  and  2x + 1  by 2:

         ---   This is legal because  f(x) / (x - a)  =  [ f(x) / 2 ]  /  [ (x - a) / 2) ].

     f(x) / 2  =  (2x^3 − x^2 + x + 1) / 2  =  x^3 − (1/2)x^2 + (1/2)x + (1/2)

     (2x + 1) / 2  =  x + (1/2)

Now, the coefficient of the x-term is 1.

So, to find the remainder when  x^3 − (1/2)x^2 + (1/2)x + (1/2)  is divided by  x + (1/2),  replace  x  with  -(1/2):

     --->     (-1/2)^3 − (1/2)(-1/2)^2 + (1/2)(-1/2) + (1/2)

     --->     -1/8 - 1/8 - 1/4 + 12

     --->     -1/8 - 1/8 - 2/8 + 4/8

     --->     0

Therefore, the remainder will be  0.

A hint:  

1)  Draw the picture of the triangle and the circle; then, modify the circle to include only the area of the circle that is within the triangle

2)  Find the area of the triangle.

3)  Find the area of the circle; now modify this area to be only that part that is within the triangle.

4)  Divide the answer to step 3 by the answer to step 2.

To prove that 5¹⁸ - 25⁸ is divisible by 120, write 25 as 5² and:

     5¹⁸ - 25⁸  =  5¹⁸ - (5²)^{8  =}  5¹⁸ - 5¹⁶

Now, factor out 5¹⁶:

     5¹⁸ - 5¹⁶   =    5¹⁶(5² - 1)    =    5¹⁶(25​ - 1)    =     5¹⁶(24) 

Since  120  =  24 x 5:

    5¹⁶(24)  /  (24 x 5)    =    5¹⁵

| 1/x - 2/(x+1) |  =  2/x

Because the left-hand side is contained within absolute value bars, its value must be positive (or zero).

Thus, the right-hand side must also be positive; for  2/x  to be positive,  x  must be positive.

Simplifying the left-hand side:  | 1/x - 2/(x + 1) |  =  | 1(x + 1)/[x(x + 1)] - 2x/[x(x + 1)] |  

                                                                          =  | (x + 1 - 2x) / [ x(x + 1) ] |

                                                                          =  | (1 - x) / [ x(x + 1) ] |

Therefore:  | (1 - x) / [ x(x + 1) ] |  =  2/x

Since it is an absolute value equation, there are two possibilities:

     either     (1 - x) / [ x(x + 1) ]  =  2/x     or      (1 - x) / [ x(x + 1) ]  =  -2/x

If       (1 - x) / [ x(x + 1) ]  =  2/x

--->   (1 - x) / [ x(x + 1) ]  =  2(x + 1) / [ x(x + 1) ]

--->   1 - x  =  2(x + 1)

--->   1 - x  =  2x + 2

--->   -1  =  3x

--->   x  =  -1/3          <--- But this is impossible, because  x  must be positive.

If      (1 - x) / [ x(x + 1) ]  =  -2/x

--->  (1 - x) / [ x(x + 1) ]  =  -2(x + 1) / [x(x + 1) ]

--->   1 - x  =  -2(x + 1)

--->   1 - x  =  -2x - 2

--->   3  =  -x

--->   x  =  -1/3          <--- But this is also impossible.

So, I think that the correct answer is  d) it doesn't have any real roots.

Slope-intercept form of a straight line:  y  =  mx + b

Point-slope form of a straight line:        y - y₁  =  m(x - x₁)

A)  Through  (4,3 ) and  m  =  2/5

      --->  Use the point-slope form with  m = 2/5,  x₁ = 4, and  y₁  = 3:     y - 3  =  (2/5)(x - 4)

              --->   Cross-multiply:  5(y - 3)  =  2(x - 4)   --->   5y - 15  =  2x - 8   --->   5y  =  2x + 7    --->   y  =  (2/5)x + (7/5)

B)  Through  (-4,2)  and  (2,-5)

     --->  First:  find the slope:  m  =  (-5 - 2) / (2 - -4)  =  -7/6

     --->   Use the point-slope form (using either point):  y - 2  =  (-7/6)(x - -4)

     --->     6(y - 2)  =  -7(x + 4)   --->   6y - 12  =  -7x - 28   --->   6y  =  -7x - 16   --->   y  =  (-7/6)x - 8/ 3

C)  m = 3/5  and  y-intercept -4:

     --->  Use the slope-intercept form:  y  =  (3/5)x - 4

D)  x-int = 7  and  y-int = -2:

     --->  x-int = 7   --->   (7,0)

             y-int = -2   --->   (0,-2)

     --->  First:  find the slope:  m  =  (-2 - 0) / (0 - 7)  =  -2/-7  =  2/7

     --->   Use the slope-intercept form:  y  =  (2/7)x - 2

E)  Passes through  (-2,2)  and is parallel to  4x - 3y -7  =  0:

     --->   All parallel lines have the same x-coefficient and the same y-coefficient; so, all lines parallel to 4x - 3y -7  =  0                 have the form:  4x - 3y + k  =  0  (for some value of k).

     --->   To find the value of  k  replace x with -2  and  y with 2:

              --->     4(-2) - 3(2) + k  =  0   --->   -8 - 6 + k  =  0   --->   -14 + k  =  0   --->   k = 14

              --->  Equation:  4x - 3y - 14  =  0

What does  y  =  f(x)  =  1/2  mean?

This is a constant function, defined for every value of x, whose y-value is always 1/2.

When graphed, it is a horizontal line whose y-vaue is 1/2.