geno3141

The radius of the cylinder will be 8, 

So the lateral surface area of the cylinder will be:  pi · 8² · 12  =  768 pi

If two similar three-dimensional objects have corresponding areas of 16 ft² and 36 ft², 

their corresponding sides will be in the ratio as the square roots of these numbers:  4 ft and 6 ft

and their volumes will be in the ratio as the cubes of the sides:  64 ft³ and 216 ft³.

So, the smaller prism will have a volume of  (4 ft)³  =  64 ft³.

The height of the smaller prism is found by this formula:  V  =  (Area of Base) x (height)

                                                                                    64 ft³  =  16 ft² x height     --->     height  =  4 ft

The lateral area of the larger prism, which is a square 6 ft on each side, is 6 x 36 ft²  =  216 ft²

If one of the roots is 4, the quadratic could have (x - 4) as a factor.

If it does, the other factor would have to be (4x + 6), giving an equation of (x - 4)(4x + 6)  =  0

Multiplying this out:  4x² - 10x - 24  =  0.

Using a linear regression model, the r² value is 0.851.

Let the time of the faster one be "t"

then the time of the slower one is "t + 20".

Equation:                                                          [1/t] + [1/(t + 20)]  =  1/24

Multiplying through by  24(t)(t + 20)     --->     [24(t + 20)] + [24t]  =  [t(t + 20)]

                                                                            24t + 480 + 24t  =  t² + 20t

                                                                                t² - 28t - 480  =  0

                                                                              (t - 40)(t + 12)  =  0

         t = 40 minutes

t + 20  =  60 minutes

There are 7 red, 5 blue, and 6 green marbles, for a total of 18 marbles.

What is the probability of getting a red marble?     7/18

Replacing it, and getting a blue marble?                5/18

What is probability of doing both?                          7/18 x 5/18  =  35/324

A regular square pyramid has a base edge of 20 mm and total area of 1440 mm².

The area of the base is 20 mm x 20 mm  =  400 mm²

Subtracting this from the total area gives an area of 1440 mm² - 400 mm²  =  1040 mm² for the sides.

Since there are four congruent triangles for the sides, each triangle will have 1040 mm² / 4  =  260 mm².

The area of each of these triangles can be found by the formula:  Area  =  ½ · base · height.

Placing the values into this formula:  260 mm²  =  ½ · 20 mm · height     --->     height  =  26 mm.

This is the slant height of the pyramid.

To find the height of the pyramid, draw a right triangle from the peak down to the middle of the base (side a = unknown height) to tme middle of one of the sides (side b = 10) and back up to the peak (the hypotenuse = 26)

Using the Pythagorean Theorem:  a² + 10²  =  26²     --->     a  =  24 mm

This is the height of the pyramid.

Volume of the pyramid:  V  =  (1/3) · (Area of the Base) · (Height of the Pyramid)

                                       V  =  (1/3) · (400 mm²) · (24m)  =  3200 mm³  

I believe that the largest value will occur when a = b.

A right triangle with a = b and c the hypotenuse is a 45^o - 45^o - 90^o triangle.

The ratio of the sides of this triangle is sqrt(2) : sqrt(2) : 2

making  (a + b) / 2  =  ( sqrt(2) + sqrt(2) ) / 2  =  2·sqrt(2) / 2  =  sqrt(2) / 1  =  sqrt(2).

Since 2 students play both sports, 3 students play only basketball and 10 play only baseball.

There is a problem with the word "or". It has two separate meanings (each one ccan be correct).

There is an "exclusive or" that means either one or the other but not both.

In this case, there are 13 students who play basketball or baseball (but not both).

Probability:  13/27

There is an "inclusive or" that means one or the other or both.

In this case, there are 15 students who play basketball or baseball (and perhaps both).

Probability:  15/27.

Which answer is correct? Until the person who asks the question tells you which "or" they intend, you don't know.

With a probability of 0.500

2 out of 2 times occurs 1 our of every 4 times (  (1/2)²  =  1/4 )

while 10 out of 10 times occurs 1 out of every 1024 times ( (1/2)¹⁰  =  1/1024 ).

1 out of every 1024 times is 1/256^th as likely to occur as 1 out of every 4 times (because 1024 / 4  =  256).

Or, 1 out of every 2 times is 256 times as likely to occur as 1 out of every 1024 times.