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Luda [366]
3 years ago
7

A rectangle has a perimeter of 80 cm. If its width is x, express its length and area in terms of x, and find the maximum area.

Mathematics
1 answer:
Veseljchak [2.6K]3 years ago
7 0
Perimiter=2L+2W

w=x
P=80

80=2L+2x
diviide by 2
40=L+x

40-L=x
40-x=L


the area is found by legnth times width
LW or Lx
for a maximum area, try to get the legnths of the sides the same
therefor
L=W=x for max area
40=x+x
40=2x
divide 2
20=x
max area =legnth times width=20^2=400

note: all squares are rectangles, but not all rectangles are squares


L=40-x
A=(40-x)(x)

max area=400 cm^2

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A cylindrical tank has a base of diameter 12 ft and height 5 ft. The tank is full of water (of density 62.4 lb/ft3).(a) Write do
saw5 [17]

Answer:

a.  71884.8 π lb/ft-s²∫₀⁵(9 - y)dy

b.  23961.6 π lb/ft-s²∫₀⁵(5 - y)dy

c. 99840π lb/ft-s²∫₀⁶rdr

Step-by-step explanation:

.(a) Write down an integral for the work needed to pump all of the water to a point 4 feet above the tank.

The work done, W = ∫mgdy where m = mass of cylindrical tank = ρA([5 + 4] - y) where ρ = density of water = 62.4 lb/ft³, A = area of base of tank = πd²/4 where d = diameter of tank = 12 ft.( we add height of the tank + the height of point above the tank and subtract it from the vertical point above the base of the tank, y to get 5 + 4 - y) and g = acceleration due to gravity = 32 ft/s²

So,

W = ∫mgdy

W = ∫ρA([5 + 4] - y)gdy

W = ∫ρA(9 - y)gdy

W = ρgA∫(9 - y)dy

W = ρgπd²/4∫(9 - y)dy

we integrate W from  y from 0 to 5 which is the height of the tank

W = ρgπd²/4∫₀⁵(9 - y)dy

substituting the values of the other variables into the equation, we have

W = 62.4 lb/ft³π(12 ft)² (32 ft/s²)/4∫₀⁵(9 - y)dy

W = 71884.8 π lb/ft-s²∫₀⁵(9 - y)dy

.(b) Write down an integral for the fluid force on the side of the tank

Since force, F = ∫PdA where P = pressure = ρgh where h = (5 - y) since we are moving from h = 0 to h = 5. So, P = ρg(5 - y)

The differential area on the side of the tank is given by

dA = 2πrdy

So.  F = ∫PdA

F = ∫ρg(5 - y)2πrdy

Since we are integrating from y = 0 to y = 5, we have our integral as

F = ∫ρg2πr(5 - y)dy

F = ∫ρgπd(5 - y)dy    since d = 2r

substituting the values of the other variables into the equation, we have

F = ∫₀⁵62.4 lb/ft³π(12 ft) × 32 ft/s²(5 - y)dy

F = 23961.6 π lb/ft-s²∫₀⁵(5 - y)dy

.(c) How would your answer to part (a) change if the tank was on its side

The work done, W = ∫mgdr where m = mass of cylindrical tank = ρAh where ρ = density of water = 62.4 lb/ft³, A = curved surface area of cylindrical tank = 2πrh  where r = radius of tank, d = diameter of tank = 12 ft. and h =  height of the tank = 5 ft and g = acceleration due to gravity = 32 ft/s²

So,

W = ∫mgdr

W = ∫ρAhgdr

W = ∫ρ(2πrh)hgdr

W = ∫2ρπrh²gdr

W = 2ρπh²g∫rdr

we integrate from r = 0 to r = d/2 where d = diameter of cylindrical tank = 12 ft/2 = 6 ft

So,

W = 2ρπh²g∫₀⁶rdr

substituting the values of the other variables into the equation, we have

W = 2 × 62.4 lb/ft³π(5 ft)² × 32 ft/s²∫₀⁶rdr

W = 99840π lb/ft-s²∫₀⁶rdr

7 0
3 years ago
Rationalize the denominator of the following expression and simplify:
Mama L [17]
To "rationalize the denominator" is another way to say, getting rid of that pesky radical at the bottom.

we'll simply start by multiplying top and bottom by the "conjugate" of the denominator, recall difference of squares, anyhow, let's do so

\bf \cfrac{6+\sqrt{2}}{5-\sqrt{2}}\cdot \cfrac{5+\sqrt{2}}{5+\sqrt{2}}\implies \cfrac{(6+\sqrt{2})(5+\sqrt{2})}{(5-\sqrt{2})(5+\sqrt{2})}\implies \cfrac{(6+\sqrt{2})(5+\sqrt{2})}{5^2-(\sqrt{2})^2}
\\\\\\
\cfrac{30+6\sqrt{2}+5\sqrt{2}+(\sqrt{2})^2}{25-2}\implies \cfrac{30+11\sqrt{2}+2}{23}\implies \cfrac{32+11\sqrt{2}}{23}
4 0
3 years ago
Compare the volume of each shape. Then complete the statement.
stellarik [79]

Answer:

1/3

Step-by-step explanation:

The volume of a pyramid is always 1/3 of a prism and the volume of a prism is 3 times the volume of the pyramid

8 0
3 years ago
A gift box is 10 inches long and 6 inches wide. It is 3 inches tall. What is the volume of the box?
nikdorinn [45]

Answer:

180

Step-by-step explanation:

you multiply 10 times 6 times 3

7 0
3 years ago
Read 2 more answers
Twice a number is decreased by 7, and this quantity is multiplied by 3. the result is 9 less than 10 times the number. What is t
Lunna [17]

Answer:

(7-2x)3 =9-10x

x= -3

Step-by-step explanation:

we let the number to be x.

twice a number=2x is decreased by 7 that is to say 7-2x, is multiplied by 3, we have (7-2x)3.

the result is 9 less than 10times the number which is =9-10x.

so bringing all these together we have;

(7-2x)3=9-10x.

what is the number?

solving, we have:

(7-2x)3=9-10x

21-6x=9-10x

21-9= -10x+6x

12= -4x

dividing both sides by -4 we have our number

x= -3.

for proof we can substitute x=-3 into the equation.

21-6(-3) =9-10(-3)

21+18=9+30

39=39

5 0
3 years ago
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