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I'm pretty sure that \(49-9\ne\sqrt{40}\) but you have the right idea.

I am able to do all of them except the third one. Can you help me on that as well? 

Thanks!!

Arithmetic....the common difference between terms is  10 - 1 = 9

So

1, 10, 19, 28

Geometric...the common ratio between terms =  2nd term / 1st term  = 10 / 1  = 10

So....each successive term is 10 times the previous one

So

1, 10, 100, 1000

Neither

1, 10, 13, 15

We have no common difference or common rato

Well...this isn't too bad

Let the first growth factor = 2

The terms are

4 , 4(2), 4(2)^2, 4(2)^3  =

4, 8 , 16, 32

For the second, GM.....see if you can do it with a growth factor of 3

If you get stuck, let me know

Draw a perpedicular to the base from the top left vertex

Call the intersection point of the base and this perpendicular, E

Call the top left vertex D  and the bottom left vertex, F

So....we have right triangle DEF

EF [ 8 - 5]  = 3   = a leg

And the hypotenuse  FD  = 7

So...DE  is the missing leg = missing side

So...using the Pythagorean Theorem

DE  = √ (7^2-  3^2]  = √ [ 49 - 9 ]  = √40  =  √4 *√10  =  2√10  = missing side

Drop a perpendicular height from the right of the side length of 5 to the base of 8. This would create a right-triangle with a hypotenuse of 7 units and a measure of one of the legs, 3 units. Using the Pythagorean theorem, we have \(7^2-3^2=49-9=\sqrt{40}=2\sqrt{10}.\)

Draw a perpendicular line from the upper left corner to the base. 

That makes a right triangle with one side = 3 and the hypotenuse = 7.

Use Pythagoras' Theorem to determine the third side.

x² + 3²  = 7²

x² = 7² – 3²

x² = 49 – 9

x = square root of 40  =  2 sqrt(10)     A square root is plus and minus of course, but we disregard the negative answer.

.

In Triangle LMN, altitude LK is 12cm long. Through point J of LK a line is drawn parallel to MN, dividing the triangle into two regions with equal areas. find LJ>

Note that  the area of the large triangle must be twice that of the smaller triangle

Call the area of the large triangle L   and the area of the smaller triangle , S

Then    2S = L 

So.....

area of smaller triangle * (scale factor)^2   = area of large triangle

S * (scale factor)^2  = 2S       divide both sides by S

scale factor^2  = 2   take the square root of both sides

scale factor = √2

This means that every dimension of the larger triangle is √2 that of the smaller triangle

Or....put another way.....every dimension of the smaller triangle is 1 /√2 that of the larger triangle

So....since LK  = 12 = altitude of larger triangle

Then ....the altitude of the smaller triangle, LJ  = 12 / √2   = 6√2 units

Check

Area of large triangle  = (1/2)MN * 12  = 6MN

Base of smaller triangle = MN/√2 

Height of smaller triangle  6√2

So  area of smaller triangle = (1/2) (MN/√2)(6√2)  =  (1/2)(6)MN  = 3 MN  = 1/2 area of larger triangle

5.  The common ratio is 3   and the first term is 4

So

The 6th term is   4 (3) ^(6 - 1)  = 4 (3)^5  =  4 *243 = 972

6.  The common ratio is -4  and the first term is 5

So

The 9th term is    5 (-4)^(9 - 1)  = 5(-4)^8  = 327680

thanks!

I'm going to assume that the pairs aren't labeled, and that order within the pairs and triplet don't matter.

\(\text{Let's form the triplet first}\\ \text{There are }\dbinom{3}{1}=3 \text{ ways to choose the male }\\ \text{and }\dbinom{4}{2}=6 \text{ ways to choose the females for a total of 18 ways}\\ \text{next we form a pair. There are }\dbinom{2}{1}=2 \text{ ways to choose the male}\\ \text{and }\dbinom{2}{1}=2 \text{ ways to choose the female for a total of 4 ways}\\ \text{but the two pairs are unlabeled so we have to divide the total arrangments by }2!=2\\ \text{Thus there are }\dfrac{18\cdot 4}{2}=36 \text{ ways to arrange the people as required.}\)

This might be a VERY HARD permutations question

resolved

10. As shown in the diagram, a hemisphere sits on top of a cylinder with the same radius. We know the area of the bottom base of the cylinder is 1/6th of the surface area of the combined shape. What fraction of the volume of the combined shape is the volume of the cylinder? 

The combined surface area  =   base area of cylinder + lateral surface area of cylinder + surface area of hemisphere

(1/6)combined surface area = [ pi* r^2  + 2 pi * r * h   + 2 pi r^2] / 6  =  ( 3 pi r^2 + 2 pi *r * h) / 6 

Area of bottom base of cylinder =  pi  * r^2

So

pi r^2  =( 3pi r^2 + 2pi  * r *h) / 6

6 pi r^2  = 3pi r^2 + 2pi *r * h

3pi r^2  = 2pi r h

(3/2) r  =  h  =  height of cylinder

So....volume of cylinder  =

pi * r^2  * (3/2)r   =  (3/2) pi r^3    (1) 

Vol of combined shape =  volume of cylinder + volume of hemisphere  = (3/2)pi r^3 + (2/3)pi *r^3  =

(13/6) pi r^3       (2)

So  ratio  of (1) to (2)  =

(3/2) / ( 13/6)  =

(3/2) * (6/13)  =

18 /26  =

9 : 13

To be fair, the number of persons in the set must be a divisor of 28.

The divisors of 28 are {1, 2, 4, 7, 14, 28}

One of these divisors is a match on the multiple choice list.

GA

9.   A and B are two points on a unit sphere. We know the space distance between A and B is sqrt(2). What is the length of shortest path on the sphere that connects A to B.

Considering a cross-section....the space distance AB will serve as the hypoenuse of a right triangle will legs of 1

So....the central angle separating A and B wil = 90°

So....the shortest distance between A and B on the sphere wiil = 2 pi (r) * (90/360)=  2 ( 1/4) pi  = pi /2 units

8.   Sphere S is tangent to all 12 edges of a cube with edge length 6. Find the volume of the sphere.

The radius of the sphere =  (1/2) 6 √2  =    3 √2  units

So....the volume of the sphere =

(4/3)pi ( 3√2)^3  =

(4/3)pi (27)* 2√2  =

72√2 pi units^3

OK...I'll relent.....!!!!!

7.   The two bases of a cylinder are two parallel cross sections of a sphere. We know the radius of the sphere is 3 and the height of the cylinder is 4. Find the volume of the cylinder. 

Considering a cross-section.....

The radius of the sphere will be the hyptonuse of a right triangle  and 1/2 the height of the cylinder will be one of the legs

The radius^2 of the cylinder   =

r^2  = 3^2 - 2^2  =  5

So....the volume of the cylinder  =

pi * r^2 * h  =

pi * 5 * 4  =

20 pi units^3

32 was the correct answer! Thanks CPhill and Omi

5.   The surface area of a sphere is 1. What is the surface area (including the base area) of a hemisphere with the same radius? 

SA _sphere  =  4 pi r^2

1 = 4pi * r^2

1 /[ 4pi ]  = r^2

Area of base =  pi * r^2  =  pi * (1) /( 4 pi )  =  (1/4) units^2

So....surface area =    (1/2) of sphere surface area + area of base  = (1/2) +(1/4)  = 3/4 units^2