BWStar

The circumference of a circle is given by the formula C = 2πr, where C is the circumference and r is the radius. In this case, we are given that the circumference is 32π inches. So:

32π = 2πr

Dividing both sides by 2π, we get:

r = 16

The diameter of the circle is twice the radius, so:

d = 2r = 2(16) = 32

Therefore, the diameter of the circle is 32 inches.

After 1 step, Franklin can take a step to the right, left, up, or down. So, there are 4 possible directions in which he can move. If Franklin moves one step to the right, he will end up at the point (1,0). If he moves one step to the left, he will end up at the point (-1,0). If he moves one step up, he will end up at the point (0,1). If he moves one step down, he will end up at the point (0,-1). Thus, there are 4 different points that Franklin could end up at after 1 step.

First, let's simplify the given equation:

x^3 - 2x + 5 = x^3 - x^2 + 9 x^2 - 2x - 4 = 0

We can use the quadratic formula to solve for x:

x = (2 ± √(2^2 - 4(1)(-4))) / 2 x = 1 ± √5

Since r is a root of x^3 - 2x + 5, we know that r^3 - 2r + 5 = 0.

Now, let's assume that r is irrational. Then we know that r can be expressed as a non-repeating, non-terminating decimal.

However, if we cube both sides of the equation r^3 - 2r + 5 = 0, we get:

r^3 = 2r - 5

Substituting in the decimal expression for r, we get:

(decimal expression for r)^3 = 2(decimal expression for r) - 5

This means that the cube of the decimal expression for r can be expressed as a finite sum of rational numbers (2 times the decimal expression for r, minus 5).

But this is a contradiction, since the cube of any non-terminating, non-repeating decimal is also a non-terminating, non-repeating decimal (and therefore cannot be expressed as a finite sum of rational numbers).

Therefore, r cannot be irrational.

Similarly, we can show that neither r^2 nor r^3 is irrational. If r^2 were irrational, then r^2 could be expressed as a non-repeating, non-terminating decimal. But then r^3 = (r^2)r would also be a non-terminating, non-repeating decimal, which contradicts our earlier conclusion.

Finally, if r^3 were irrational, then r^3 could be expressed as a non-repeating, non-terminating decimal. But then r = (r^3)/(2 - 5) would also be a non-terminating, non-repeating decimal, which again contradicts our earlier conclusion.

Therefore, none of r, r^2, or r^3 can be irrational.

Let's denote the side length of the square as s. Since PZXY is a square, we know that PZ = ZX = s, and QX = RY = 10 - s (since QX and RY are the remaining segments of length 10 after subtracting the side length of the square).

Using the Pythagorean theorem, we can set up two equations based on the right triangles PZQ and PXR:

s^2 + QX^2 = QP^2 (1) s^2 + RY^2 = RP^2 (2)

Substituting in the values we know:

s^2 + (10 - s)^2 = 100 (1) s^2 + (10 - s)^2 = 100 (2)

Expanding and simplifying:

2s^2 - 20s + 100 = 100 2s^2 - 20s = 0 2s(s - 10) = 0

This gives us two solutions: s = 0 (which doesn't make sense since the square has positive length) and s = 10.

Therefore, the side length of the square is 10.

If the problem is asking for exactly three:

There are 7*4 = 28 total cards
choose 3 colors -> 4c3

choose a number for the three colors to be -> 7c1

choose random other numbers to fill the rest of the hand -> 24c2

(4c3*7c1*24c2)/(28c5)

If asking for three or more: (basically > 3 cards that have the same number)

((ways of getting 3 card same) + (ways of getting 4 cards same)) / (ways to draw 5 cards)

= ((4c3*7c1*24c2)  + (4c4*7c1*24c1))/(24c5)

assuming the 1oo  is 1, the answer is 125

assuming that 1oo is 100, then 1 and 1/4 times.

Can you repost the question? the thread is hard to follow

We use trig for this.

in triangle ABC:

the largest side is opposite the largest angle.

If we use the law of cosine, and instead of using finding c, we find gamma. we set c to the largest side so that we can find c's gamma.

we then solve:

\(36=\sqrt{18^2+25^2-2\cdot 18\cdot 25\cos \left(y\right)}\)

and we get:

y is about 112.67

coffee for 2,

donuts for 1.5

c = coffee

d = donuts

lets make an equation

c*2 + (80-c)*1.5 = 141.5

from this we get:

 c= 43.

if c = 43, then d is 80-43 = 37.

they sold 43 coffees, and 37 donuts.

lets check the math

43*2+37*1.5 = 141.5. math checks out