To find the number of handshakes, we can use the formula for the sum of the first n-1 positive integers, which is given by (n-1)n/2.
In this problem, n is equal to seven since there are seven people.
Using the formula, the number of handshakes will be (7-1) * 7/2 = 6 * 7/2 = 42/2 = 21.
Therefore, there will be 21 handshakes in total.
I am editing this answer in order to make it more readable.
It does look computer generated but I personally don't really care, except that where an answer comes from should ALWAYS be cited.
The answer appears to be a good one.
(all my changes are clearly marked)
Let’s assume the length of the rectangle is L and the width is W.
The area of the rectangle is given by A = L * W. The perimeter of the rectangle is given by P = 2L + 2W.
According to the problem, the number of square units in the area is four times the number of units in the perimeter. Mathe- matically, this can be represented as:
A = 4P
Substituting the expressions for A and P, we get:
L * W = 4(2L + 2W)
L * W = 8L + 8W
Rearranging the equation, we get:
L * W - 8L - 8W = 0
We can rewrite this equation as:
LW - 8L - 8W = 0
To make it easier to solve, let’s add 64 to both sides of the equation:
LW - 8L - 8W + 64 = 64
LW - 8L - 8W + 64 = 64
Now, we can factor the left side of the equation:
(L - 8)(W - 8) = 64
We are looking for the smallest possible perimeter, which means we want to minimize the sum of the length and width.
Therefore, we need to find the smallest possible values for L and W that satisfy the equation.
The factors of 64 are: 1, 2, 4, 8, 16, 32, 64.
We can try different combinations of L - 8 and W - 8 to see which ones give us integer values for L and W.
If L - 8 = 1 and W - 8 = 64, we get L = 9 and W = 72, which are not integer values. (I think you mean that they are integer values)
If L - 8 = 2 and W - 8 = 32, we get L = 10 and W = 40, which are integer values.
If L - 8 = 4 and W - 8 = 16, we get L = 12 and W = 24, which are integer values.
If L - 8 = 8 and W - 8 = 8, we get L = 16 and W = 16, which are integer values.
So far what you have said appears to make sense - But it also looks like it has been computer generated.
Therefore, the smallest possible perimeter of the rectangle is 2L + 2W = 2(10) + 2(40) = 20 + 80 = 100 units. You lose me here - Melody
Added by Melody.
I think this gives possible perimeters of
2(9+72) = 162
2(10+40) = 100
2(12+24) = 72
2(16+16) = 64 ( but this one is a square which is contrary to the conditions. )
So the smallest perimeter appears to be 72.
This happens when the length is 24 and the width is 12
Area = 288 u^2
Perimeter = 72
72* 4 = 288 excellent
To find out how much money Kenny saved, we need to analyze the given information step by step.
Let's assume Kenny's total amount of money is represented by "x."
According to the problem, Kenny spent 1/4 of his money on a bag. This means he spent (1/4)x on the bag.
Next, we are told that Kenny spent $120 on a belt. So, the total amount of money he spent on the bag and belt is (1/4)x + $120.
We also know that Kenny saved the rest of his money after buying the bag and belt. Therefore, the amount he saved can be calculated as x - [(1/4)x + $120].
The problem states that Kenny spent twice as much money as what he saved. So, we can set up an equation:
(1/4)x + $120 = 2 * [x - ((1/4)x + $120)]
Now, let's solve this equation to find the value of x, which represents Kenny's total amount of money:
(1/4)x + $120 = 2x - (1/2)x - $240
Multiplying through by 4 to eliminate fractions:
x + $480 = 8x - 2x - $960
Combining like terms:
x + $480 = 6x - $960
Subtracting x from both sides:
$480 = 5x - $960
Adding $960 to both sides:
$1440 = 5x
Dividing both sides by 5:
$288 = x
Therefore, Kenny had a total of $288.
To find out how much money Kenny saved, we substitute this value back into our earlier expression:
Amount saved = x - [(1/4)x + $120]
Amount saved = $288 - [(1/4) * $288 + $120]
Amount saved = $288 - [$72 + $120]
Amount saved = $288 - $192
Amount saved = $96
Hence, Kenny saved $96.
Diameter of the Semicircle is
= 4 + (3 + 2) + 3
radius of the semicircle = 6 in
Area of the semicircle = π(6)^2/2 sq in
= 18π sq inches
= 18 * 3.14 [give π = 3.14]
Area of the rectangle is (7 * 5) sq inches = 35 sq inches
Area of the square = (3 * 3) sq inches = 9 sq inches
Therefore area of the figure = (56.52 + 35 + 9)
= 100.52 sq inches
We have to find
a) How many different ways she can choose one book from each genre.
b) How many different ways can he choose two books for his summer reading.
c) How many different combinations of books can he select.
a) Hermione Granger
Hermione wants to borrow one book from each shelf, so she has 15 choices for the Potions shelf, 12 choices for the Charms shelf, and 18 choices for the Transfiguration shelf. Since the order in which she chooses the books matters, we need to multiply these numbers to find the total number of ways she can choose one book from each shelf.
15 * 12 * 18 = 32,400
Therefore, Hermione can choose one book from each shelf in 32,400 different ways.
b) Ron Weasley
Ron only wants to borrow books from Professor Phineas Crowley or Professor Theodore Travers. This means he has 15 choices for the Potions shelf and 18 choices for the Transfiguration shelf. Since the order in which he chooses the books matters, we need to multiply these numbers to find the total number of ways he can choose two books.
15 * 18 = 270
Therefore, Ron can choose two books from either the Potions or Transfiguration shelves in 270 different ways.
c) Harry Potter
Harry is planning to borrow a maximum of three books from any of the shelves. This means he could borrow 0, 1, 2, or 3 books.
If Harry borrows 0 books, then there is only 1 way he can do this.
If Harry borrows 1 book, then he has 3 choices for which shelf to borrow the book from.
If Harry borrows 2 books, then he has 3 choices for which shelf to borrow the first book from, and then 2 choices for which shelf to borrow the second book from. This gives a total of 6 ways to borrow 2 books.
If Harry borrows 3 books, then he has 3 choices for which shelf to borrow the first book from, 2 choices for which shelf to borrow the second book from, and 1 choice for which shelf to borrow the third book from. This gives a total of 6 ways to borrow 3 books.
Therefore, Harry can borrow a maximum of 3 books in 1 + 3 + 6 = 10 different ways.
(a) Hermione Granger can choose one book from each shelf in 32,400 different ways.
(b) Ron Weasley can choose two books from either the Potions or Transfiguration shelves in 270 different ways.
(c) Harry Potter can borrow a maximum of 3 books in 10 different ways.