Could you maybe walk me through this Algebra word problem?

Here is the problem:

Anita wants to put a moat around her rectangular castle. The castle is 45 ft by 60 ft. Let m be the width of the moat.

1. Write a polynomial that represents the total area of the castle and moat.

2. Find the combined area if the moat is 10 feet wide.

3. If the moat is also 10 feet deep, calculate how many gallons of water are in the moat is 1 gallon (almost) = 0.134 ft (cubed). Round the answer to the gallon.

I'm just kinda lost. If you can show me how this problem is done, I'd be really grateful.

If the moat has width m,

castle area = 45*60 = 2700
castle+moat area = (45+2m)(60+2m)
since there is moat on both sides.
moat area = (45+2m)(60+2m)-2700

Now plug in your value for m, and recall that volume is area * depth

2500ft^2 x 1gal/(0.134ft^2)=~ 186567 gal. Right?

your answer is right, but your units are not The volume is

2500 ft^2 * 10 ft = 25000 ft^3

25000 ft^3 * 1gal/(0.134 ft^3) = 186567 gal

That's right.. sorry 'bout that. Thanks, Mate.

Of course! I'll walk you through each step of the problem.

1. To write a polynomial that represents the total area of the castle and moat, we need to add the areas of the castle and moat together.

The area of a rectangle is found by multiplying its length by its width. The castle has dimensions of 45 ft by 60 ft, so its area is 45 ft × 60 ft = 2700 ft^2.

To find the area of the moat, we need to consider that the moat surrounds the castle on all four sides. The moat's dimensions are determined by the width, which is represented by 'm'. Since the moat goes all the way around the castle, the length and width of the moat will be increased by 2m in each direction. So the length of the moat will be 45 ft + 2m, and the width will be 60 ft + 2m. Its area can be found by multiplying these dimensions: (45 ft + 2m) × (60 ft + 2m).

Combining the area of the castle and the area of the moat, we get the polynomial representation: 2700 ft^2 + (45 ft + 2m) × (60 ft + 2m).

2. To find the combined area if the moat is 10 feet wide, we substitute 'm' with 10 in the polynomial we obtained. So the expression becomes:

2700 ft^2 + (45 ft + 2(10 ft)) × (60 ft + 2(10 ft)).

Simplifying this, we have:

2700 ft^2 + (45 ft + 20 ft) × (60 ft + 20 ft).

2700 ft^2 + (65 ft) × (80 ft).

2700 ft^2 + 5200 ft^2.

Adding these areas together, we get:

7900 ft^2.

Thus, if the moat is 10 feet wide, the combined area of the castle and moat is 7900 ft^2.

3. To calculate the number of gallons of water in the moat, we need to find the volume of the moat first. The volume of a rectangular prism (the shape of the moat) is given by multiplying its length, width, and height. In this case, the length and width are 45 ft + 2m (which we'll use 45 ft + 2(10 ft) = 65 ft) and 60 ft + 2m (which we'll use 60 ft + 2(10 ft) = 80 ft), respectively.

Since the moat is also 10 feet deep, the height of the moat will be 10 ft.

Using these values, the volume of the moat is:

65 ft × 80 ft × 10 ft.

Converting cubic feet to gallons, we use the given conversion factor that 1 gallon is approximately equal to 0.134 ft^3.

So, the volume in gallons is calculated by dividing the volume in ft^3 by 0.134 ft^3/gallon:

(65 ft × 80 ft × 10 ft) / 0.134 ft^3/gallon.

By performing this calculation, we obtain the number of gallons of water in the moat.

Remember to round the answer to the nearest whole gallon, as specified in the question.

I hope this helps clarify the steps required to solve the problem. Let me know if you have any further questions!