In order to combine like terms correctly, it is necessary to simplify every term and eliminate the parentheses.   Double negatives equal a positive sign.

Combine like terms.  The answer is:  

A difference is the result of a subtraction. The time difference between the two cities will be the result of subtracting the numbers along the top of the map for their respective time zones, and taking the absolute value of the difference. The numbers for Paris and Dallas, respectively, are 1 and , so the difference in hours between their times is .

The time difference between Winnipeg and the Azores is the result of subtracting the numbers along the top of their respective timie zones, which are and :

The Azores is five hours ahead of Winnipeg, so when the flight takes off, since

,

it is 10:00 PM in the Azores.

Add the seven hours flight time, and take the result modulo 12:

The time is 5:00 AM in the Azores when the plane lands there.

The only way to get a positive product from multiplication is to multiply two positive numbers together or two negative numbers together.  

If the first number was negative, the other number will also have to be negative.

Set  and use the formula for the surface area of a sphere:

The height of the submerged part of the cylinder is 27cm. 2πrh + πr² is equal to 270π + 25π = 295π

The general mechanics of this problem are simple. The lateral area of a right cylinder (excluding its top and bottom) is equal to the circumference of the top times the height of the cylinder. Therefore, the area of this can's surface is: 5π * 12 or 60π. If the cost per square inch is $0.00125, a single label will cost 0.00125 * 60π or $0.075π or approximately $0.24.

We have the following categories to consider:

<Aluminum Cost> = (<Area of the top and bottom of the can> + <Lateral area of the can>) * 0.0015

<Label Cost> = (<Area of Label>) * 0.0001

<Attachment cost> = 2 * 0.00125 = $0.0025

The area of ends of the can are each equal to π*5² or 25π. For two ends, that is 50π.

The lateral area of the can is equal to the circumference of the top times the height, or 2 * π * r * h = 2 * 5 * 12 * π = 120π.

Therefore, the total surface area of the aluminum can is 120π + 50π = 170π.  The cost is 170π * 0.0015 = 0.255π, or approximately $0.80.

The area of the label is NOT the same as the lateral area of the can. (Recall that it must be 2 inches longer than the circumference of the can.) Therefore, the area of the label is (2 + 2 * π * 5) * 12 = (2 + 10π) * 12 = 24 + 120π. Multiply this by 0.0001 to get 0.0024 + 0.012π = (approximately) $0.04.

Therefore, the total cost is approximately 0.80 + 0.04 + 0.0025 = $0.8425, or $0.84.

If broken down into parts, this is an easy problem. It is first necessary to isolate the dimensions of the walls. If the room is 9 ft high, we know 18 x 15 designates the area of the floor and ceiling. Based on this, we know that the room has the following dimensions for the walls: 18 x 9 and 15 x 9. Since there are two of each, we can calculate the total area of walls - ignoring doors and windows - by doubling the sum of these two areas:

2 * (18 * 9 + 15 * 9) = 2 * (162 + 135) = 2 * 297 = 594 ft²

Now, we merely need to calculate the area "taken out" of the walls:

For the door: 3 * 7 = 21 ft² 

For the windows: 2 * (2 * 5) = 20 ft²

The total wall space is therefore: 594 – 21 – 20 = 553 ft²

If broken down into parts, this is an easy problem. It is first necessary to isolate the dimensions of the walls. If the room is 10ft high, we know 23 x 17 designates the area of the floor and ceiling. Based on this, we know that the room has the following dimensions for the walls: 23 x 10 and 17 x 10. Since there are two of each, we can calculate the total area of walls - ignoring doors and windows - by doubling the sum of these two areas:

2 * (23 * 10 + 17 * 10) = 2 * (230 + 170) = 2 * 400 = 800 ft²

Now, we merely need to calculate the area "taken out" of the walls:

For the door: 2.5 * 8 = 20 ft² 

For the windows: 3 * 6 = 18 ft²

The total wall space is therefore: 800 – 20 – 18 = 762 ft²

Now, if one can of paint covers 57 ft², we calculate the number of cans necessary by dividing the total exposed area by 57: 762/57 = (approx.) 13.37.

Since we cannot buy partial cans, we must purchase 14 cans.