geno3141

Each side of the pyramid is a triangle and all these triangles are congruent.

We can find the lateral surface area by finding the area of one of the triangles and multiplying by the number of trianges.

The base of each triangle is 10 divided by 6, which gives us 5/3.

The height of each triangle is 20.

The area of each triangle is:  A  =  ½·base·height  =  ½·(5/3)·20  =  50/3.

The total surface area is:  6·(50/3)  =  100.

You might notice that another way to get your answer would be to use the formula:  A  =  ½·perimeter·height.

Divide the deck into three groups, the queens, the kings, and the rest of the deck.

You need to get two of the four queens  -- which occurs in  ₄C₂ = 6 ways.

You need to get two of the four kings -- which also occurs in  ₄C₂ = 6 ways.

You need to get none of the rest of the deck -- which occurs in  ₄₈C₀ = 1 way.

Multiplying these together gives you the various ways that you can get these four cards  = 6 x 6 x 1  =  36 ways.

To find the probability that this will occur, we need to find the number of ways that 4 cards can be chosen from 52 cards -- which occurs in  ₅₂C₄ = 270725 ways -- and divide 36 by 270725 to get the answer.

A bingo card contains 5 B-numbers in the range 1, 2, ..., 15;

                                    5 I-numbers in the range 16, 17, ..., 30;

                                    4 N-numbers in the range 31, 32, ..., 45;

                                    5 G-numbers in the range 46, 47, ..., 60;

                            and 5 O-numbers in the range 61, 62, ..., 75.

The diagonal going from top-left to bottom-right contains one B-number, one I-number, no N-number because of the FREE space, one G-number, and one O-number.

There are 15 x 15 x 15 x 15 possibilities.

These vertices form a right triangle with sides RS and RT; one will be the base, the other will be the height.

Find these two lengths and use the formula:  A  =  ½·base·height

Since you know the dimensions of the rectangular prism, find its volume:  V = l·w·h.

Then, multiply this value by its density to find the number of grams.

Step 1:  Since you know the three sides of the triangle, use the Law of Cosines to find the size of angle(B).

Step 2:  Now, using angle(B), AB and BM, use the Law of Cosines to find AM.

First:  find the y-value when x = 2:     y  =  16(2) - 12(2)² + 2  =  -14   --->   (2, -14)

Second:  in order to find the slope of the line, find the first derivative of y:

      y'  =  16 - 2·12x     --->     y'  =  16 - 24x

At x = 2   --->   y'  =  16 - 24·2  =  -32

You know have both the point and the slope at that point; use the point-slope form:

     y - -14  =  -32(x - 2)

Draw a diagram ...

Straight acroos from Ezra to Allie's flagpole is 15 feet.

From that point on Allie's flagpole up to Allie is 35 - 27 = 8 feet.

You have now drawn a right triangle with its two sides 15 feet and 8 feet; use the Pythagorean Theorem to find the hypotenuse.

4y - 4x²  =  1   --->   y - x²  =  1/4                      (note that y must be positive)

4x - 4y²  =  1   --->   x - y²  =  1/4                      (note that x must be positive)

Setting these two equations equal to each other:

      y - x²  =  x - y²

     y² + y  =  x² + x

Completing the square:

    y² + y + 1/4  =  x² + x + 1/4

        (y + 1/2)²  =  (x + 1/2)² 

Taking the square root of both sides (ignoring negative possibilities):

        y + 1/2  =  x + 1/2

                 y  =  x

Since  4x - 4y²  =  1   --->   4x - 4x²  =  1   --->   x² - x - 1/4  =  0

                                  --->   (x - 1/2)²  =  0   --->   x  =  1/2     and     y  =  1/2

x³ + y³  =  (1/2)³ + (1/2)³  =  1/4

1 / (1/4)  =  4

Drawing a sketch, placing Charles at point C, the base of the building immediately below Charlie at point Z, the car at point Y, and the truck at point X. 

Call the distance from Y to Z 'c', the distance from  to Z 'c + 800', and the height of the building 'h'.

Triangle CZY is a right triangle with Z the right angle and angle C = 66^o   --->   tan(66)  =  x / h

Triangle CZX is a right triangle with angle XCZ = 80^o   --->   tan(80)  =  (x + 800) / h

We now have:  tan(66)  =  x/h   --->   x  =  h·tan(66)

                         tan(80)  =  (x + 800)/h   --->   x + 800  =  h·tan(80)   --->   x  =  h·tan(80) - 800

Combining these two equations:  h·tan(66)  =  h·tan(80) - 800

                                   h·tan(66) -  h·tan(80)  =  -800

                                      h(tan(66) - tan(80))  =  -800

                                                                   h  =  -800 / (tan(66) - tan(80))

                                                                   h  =  233.6'