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The smallest possible value of |w - z| is 2.

Let n be the number of tiles per side of the patio. Then the number of yellow tiles is n+(n−1)=2n−1. Since this is equal to 21, we have 2n−1=21, so n=11.

The number of gray tiles is then (n−2)^2 = (11−2)^2 = 9^2 = 81​.

Let n be the number of rows of tiles in the patio. Then the number of yellow tiles is n+(n−1)=2n−1. Since we know that this number is equal to 21, we have 2n−1=21, so n=11.

The number of gray tiles is the total number of tiles minus the number of yellow tiles, so it is n2=112=121​.

Find the coefficient of y^4 in the expansion of (2y-5+3y^2)^6.

That is way too much multiplication to do by hand.  

Fortunately, a useful tool has been provided for us, at  

https://www.symbolab.com/solver/expand-calculator

Per the "expand" tool on that page, the coefficient of y⁴ is 375.  

Would it be cheating, to use that website to avoid the drudgery of endless   

multiplications?  I do not consider it cheating to use any tool that has been  

made available.  It used to be slide rules, then calculators, now computers.   

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One ordered pair (a,b) satisfies the two equations ab^4 = 48 and ab = 72. What is the value of b in this ordered pair?    

To find b, consider                                                  ab⁴  =  48  

We will divide both sides by ab.  

Since ab=72, we will divide the left side  

by "ab" and the right side by its equal 72.  

                                                                              ab⁴         48  

                                                                             ——   =   ——  

                                                                              ab           72  

Note that ab⁴ = (ab) * (b³)  

Cancel ab out of the left side.  

Reduce 48/72 on the right side.  

                                                                               b³           2  

                                                                             ——   =   ——  

                                                                                1            3  

                                                                                 b   =   cube root of (2 / 3)  

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21-1 for the center, 

20/4 for the diagonals from the center to each corner. 5+5+1 for side length, 11^2 for total number of tiles, 121-21 yellow tiles for 100 gray tiles

the tank was 1/6 full, and after 2 gallons, is 1/2 full

that means 1/2 - 1/6 = 1/3 of the tank was filled

x: total capacity of the tank

x/3 = 2

x = 6

Let n be the number of rows of tiles in the patio. Then the number of yellow tiles is n+(n−1)=2n−1. Since there are 21 yellow tiles, we have 2n−1=21, so n=11.

The number of gray tiles is the total number of tiles minus the number of yellow tiles, which is n^2−(2n−1) = n^2−2n+1 = 112−2∗11+1 = 122​.

A + n = 4(B - n)

A - n = 8(B + n)

The following equations result from the above information. The rest is just clever manipulation of algebra.

A + n = 4(B - n)

A + n = 4B - 4n

A = 4B - 5n

A - n = 8(B + n)

A - n = 8B + 8n

A = 8B + 9n

4B - 5n = 8B + 9n

-4B = 14n

B = -7/2 * n

A = 4 * -7/2 * n - 5n

A = -14n - 5n

A = -19n

Since we seek the ratio, A/B = -19n/(-7/2) * n = 38/7

Just make sure I did not make any mistakes!

I think you are on the right track, but \(\sin 135^\circ \neq \frac{1}{2}\)

I think you are on the right track, but you made a miscalculation. \(\sin 135^\circ \neq \frac{1}{2}\)

The area of a triangle can be calculated using the following formula:

Area = (1/2) * base * height

In this case, we know that the base is BC = 8 and the angle between AB and BC is 135 degrees.

The height of the triangle is equal to the length of the altitude from A to BC. We can find the length of the altitude using the following formula:

altitude = AB * sin(angle ABC)

Plugging in the values we know, we get:

altitude = 6 * sin(135 degrees) = 6 * (1/2) = 3

Therefore, the area of triangle ABC is:

Area = (1/2) * 8 * 3 = 12

So the answer is 12​.

Here is my attempt at solving this.

I would solve this problem by rotating the triangle. Consider the following diagram: 

Now, notice how the height in this diagram is side length EB. The given info says that BC = 8, Since triangle BEC is a 45-45-90 triangle, we can determine that the height of the triangle is sqrt(18) or 3 * sqrt(2). We know that AB = 8, so we can find the area using the basic formula for the area of a triangle, Area = 1/2 * base * height.

Area = 1/2 * 8 * 3 * sqrt(2) = 12 * sqrt(2), so I believe this is the right answer.

I still need help...

will anyone please help me?

I've tried entering that answer,...but it's wrong...thanks for trying though!

The area of a triangle can be calculated using the following formula:

Area = (1/2) * base * height

In this case, we know that the base of the triangle is BC = 8, and the height of the triangle is the perpendicular distance from A to BC.

We can use the sine function to calculate the height of the triangle:

sin(angle ABC) = height / BC

sin(135 degrees) = height / 8

(1/2) = height / 8

height = 4

Now that we know the base and height of the triangle, we can calculate the area:

Area = (1/2) * 8 * 4 = 16

Therefore, the area of triangle ABC is 16.

HumenBeing is trying to cheat, as usual.

The area is 50.

I drove to the beach at a rate of 40 miles per hour.  If I had driven at a rate of 50 miles per hour instead, then I would have arrived 45 minutes later.  How many miles did I drive?  

You mean 45 minutes earlier.  Obviously, if you drive faster, you get there faster.  

This problem makes use of

the following relationship:                   Distance = Velocity x Time 

                                                           D  =  V • T  

case 1                                                 D  =  (40) • (T)  

case 2                                                 D  =  (50) • (T – 45)  

Since the Distance, D, is the  

same for both cases, let's set       

the "V•T"s equal to each other.             (50)(T – 45)  =  (40)(T)  

                                                               50T – 2250  =  40T  

Subtract 40T from both sides                  10T – 2250  =  0  

Add 2250 to both sides                                       10T  =  2250  

Divide both sides by 10                                           T  =  225   (this is in minutes)  

Divide minutes by 60 to get hours          225 minutes  =  3.75 hours  

Plug this T back into original equation                     D  =  (40 mi/hr) • (3.75 hr)  =  150 miles  

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