help solve word problem with a polynomial answer

Question

Please help me solve this word problem.

 

"Conor wants to put a moat around his rectangle castle.  Let , m, = the width of the moat.

 

                                                                         ___________________________

                                                                          [                  60ft______ _   ]   ]

                                                                          [    [                                    ]   ]

                                                                          [    [                                    ]   ]    45ft vertical

                                                                          ["m"            Castle              ]   ]

                                                                          [    [                                    ]   ]

                                                                          [    [ __________________ _]   ]

                                                                          [______________________ _ _]

 

1.  Given the diagram above, how do I write a polynomial that represents the total area of tthe castle and the moat?

 

2.  How do I find the combined area if the moat is 10 feet wide?

 

3.  If the moat is also 10 feet deep, how do I calculate how many gallons of water are in the moat if 1 gallon = 0.134ft^3?  Round to the gallon.

 

A big thank you.

Answer 1

Hi L;

QUESTIONS 1 and 2...

TOTAL AREA OF CASTLE AND MOAT...

(m+60 ft)(m+45 ft)

m=10

(10 ft+60 ft)(10 ft+45 ft)

(70 ft)(55 ft)

3850 ft²

 

QUESTION 3...

[(m+60 ft)(m+45 ft)=area of castle and moat]

-[(60 ft)(45 ft)=area of castle]

area of moat

 

We must subtract the second equation from the first.

I am assuming that m=10 ft.

 

[(10 ft+60 ft)(10 ft+45 ft)=area of castle and moat]
-[(60 ft)(45 ft)=area of castle]

area of moat

 

[(70 ft)(55 ft)=area of castle and moat]

-[(60 ft)(45 ft)=area of castle]

area of moat

 

[(3850 ft²=area of castle and moat]

-[(2700 ft²)=area of castle]

1150 ft²=area of moat

 

(10 ft)(1150 ft²)=cubic feet of moat

11500 ft³=cubic feet of moat

1 gallon = 0.134 ft³

(1 gallon)/(0.134 ft³)=1

(11500 ft³)[(1 gallon)/(0.134 ft³)]

Let's cancel units.  Cubic feet, ft³, is in both the numerator and denominator...

(11500 ft³)[(1 gallon)/(0.134 ft³)]

(11500)[(1 gallon)/(0.134)]

The only unit remaining is gallon, which is what we want...

Now that our units are aligned, let's calculate...

85821 gallons

 

 

 

 

Answer 2

If the castle has dimensions of 60 ft and 45 ft, then the outer area would have dimensions of 2m +60 and 2m +45, This is because you have the m width of the moat all around the castle, contributing 2m to both width and length. Thus, for #1, area = (2m+60)(2m+45). For #2, substitute 10 ft in for m, and you get area = (80 ft)(65 ft) = 5200 ft^2 For #3, you need the volume for the larger perimeter with a 10 ft depth - the volume for the castle submerged 10 ft. Volume = length x width x height. V(total) = (2m+60)(2m+45)(10). V(submerged castle)= (60)(45)(10) Subtracting,(2m+60)(2m+45)-(27000 ft^3) Plugging in m = 10 ft, (80)(65)(10)-(27000 ft^3) = 52000 ft^3 - 27000 ft^3 = 25000 ft^3. Multiply by 1 gallon/0.134 ft^3 and you get the answer to #3, 186567.164 gallons.