Answer:
a) [tex]x =\dfrac{a+b -\sqrt{(a + b)^2 - 3\cdot a \cdot b} }{6}[/tex]
b) The volume of the box is maximized when 'x' is approximately 1.585 feet
Step-by-step explanation:
a) Let 'a' represent the length of the rectangular sheet of paper and let 'b' represent the with of the rectangular sheet of paper, we have;
The length of the side of the square cut from the corners of the rectangular sheet of paper = x
The length of the box formed after cutting the square sides are;
The length of the box, l = a - 2·x
The width of the box, w = b - 2·x
The height of the box, h = x
The volume of the box, V = l × w × h = (a - 2·x)·(b - 2·x)·x = 4·x³ - 2·a·x² - 2·b·x² + a·b·x
The value of 'x' that gives the maximum volume of the box is given by finding the extremum of the volume, 'V', of the box as follows;
At the extremum, dV/dx = 0
∴ d(4·x³ - 2·a·x² - 2·b·x² + a·b·x)/dx = 12·x² - 4·a·x - 4·b·x + a·b = 0
12·x² - 4·(a + b)·x + a·b = 0
[tex]x = \dfrac{4\cdot (a + b) \pm \sqrt{(-4\cdot (a + b) )^2-4 \times12\times a \cdot b } }{2 \times 24} =\dfrac{a+b\pm\sqrt{(a + b)^2 - 3\cdot a \cdot b} }{6}[/tex]
Found from part (b), the value of 'x' (in terms of 'a' and 'b') that maximizes the volume of the box is [tex]x =\dfrac{a+b -\sqrt{(a + b)^2 - 3\cdot a \cdot b} }{6}[/tex]
b), When a = 8.5 inches and b = 11 inches, we have;
x = (8.5 + 11 +√((8.5 + 11)² - 3×8.5×11))/6 = 4.914
x = (8.5 + 11 -√((8.5 + 11)² - 3×8.5×11))/6 ≈ 1.585
We get;
When x = 1.585, V = 4·1.585³ - 2·8.5·1.585² - 2·11·1.585² + 8.5·11·1.585 ≈ 66.15
When x = 4.914, V = 4·4.914³ - 2·8.5·4.914² - 2·11·4.914² + 8.5·11·4.914 ≈ -7.65
Therefore, the value of 'x' that gives the maximum volume of the box is x = (8.5 + 11 -√((8.5 + 11)² - 3×8.5×11))/6 ≈ 1.585
(The value of 'x' in terms of 'a' and 'b' is therefore;
[tex]x =\dfrac{a+b -\sqrt{(a + b)^2 - 3\cdot a \cdot b} }{6}[/tex])
The volume of the box is maximized when, x ≈ 1.585 feet
Ron took a multiple choice test. The test was 80 points and each question was worth 2 points. Write an equation to model the number of points Ron earned, y, if he got x questions incorrect.
Answer:
the equation is y=80-x
hope it helps
If the if a cylinder is 7 in tall and has a volume of 63 Pi inches what is the area of the cross section it goes right down the middle
Answer: [tex]9\pi \ in.^2[/tex]
Step-by-step explanation:
Given
The height of the cylinder is [tex]h=7\ in.[/tex]
The volume of the cylinder is [tex]V=63\pi \ in.^3[/tex]
The volume of the cylinder is the product of area and height
[tex]\Rightarrow V=\pi r^2h\\[/tex]
Insert the values
[tex]\Rightarrow 63\pi =A\times 7\\\\\Rightarrow A=\dfrac{63\pi}{7}\\\\\Rightarrow A=9\pi\ in.^2[/tex]
Thus, the area of the cross-section is [tex]9\pi \ in.^2[/tex]
Which side lengths form a right triangle?
Answer:
a is the answer
hope this will help you
18ft times 12ft long
Answer:
216 ft
Step-by-step explanation:
18 x 12 = 216 so ya ezzzzzz :)))))
Whch stuction represents a proportional relationshp?
A) The cost of purchasing dog food for $1.20 per pound and
having to pay $6 n shipping
B) The cost of purchasing gift cards at $20 per card and
having to pay a $2 och nation fee per card
C) The cost of purchasing oranges of So yo per orange and
having a $1 off your total purchase coupon
D) None of the above represent a proportional relationshp
Can you guys help me please
Xavier needs to bike a minimum of 90 miles this week for his training. If he has already biked 18 miles this week, then how many miles should he bike on each of the 6 remaining days this week?
Answer:
It's 12.
Step-by-step explanation:
90-18=72
72/6 =12
If f(x) = 2(3^x) + 1, what is the value of f(2)?
(1) 13
(3) 37
(2) 19
(4) 54
Answer:
Given
F ( x ) = 3 x 2 + 1 , to find F ( − 2 ) , we have to just put
x = − 2 in the formula for F ( x ) .
therefore , F ( − 2 ) = 3 ( − 2 ) 2 + 1 = 3 ( 4 ) + 1 = 13 .
How many feet of fencing is needed to enclose a rectangular garden with a 9 foot long side and a 15 foot long diagonal
Answer:
42 ft
Step-by-step explanation:
To find the unknown side:
[tex]\sqrt{15^2-9^2} =12[/tex]
Total perimeter: 2(12 + 9) = 42 ft
Ill give for 32, plz solve them. :)
Answer:
t>=7
Step-by-step explanation:
"The sum of negative two and nine times a number is less than ninety four."
Answer:
I believe you would write the answer is 7 * x is less than 94 (use actual less than symbol for answer).
Step-by-step explanation:
"The sum of negative two and nine" is saying -2 + 9.
"times a number" is saying the answer of -2 + 9 is multiplied by a number, or x.
"is less than ninety four." is saying that -2 + 9 * x is less than (use symbol) 94.
So the equation looks like this:
-2 + 9 * x is less than 94.
Now you can simplify the equation:
-2 + 9 = 7
7 * x is less than(use actual less than symbol) 94.
Could I get brainliest please?
Graph ATUV with vertices T(-3, 2), UC-1, 0), and (4,-2) and its image after the glide reflection.
Answer:
T (2, 2)
U (4 , 0)
V (1, -2)
Step-by-step explanation:
The translation is (x + 5, y), meaning 5 is being added to the X-values and Y stays the same.
I need help with this work !!!
Answer:
16. B
17. C
18. D
19. B
20. D
The perimeter of a square is 54 cm.Find the length of a diagonal, rounding to the nearest tenth
Answer:
19.09 = 19.1
Step-by-step explanation:
we can split the square into two right triangles. Becasue the perimeter of the square is 54cm, we know the each side is 13.5 cm (54cm / 4). Using the pythagoras theorem, the diagonal is 19.1 cm.
If a cylinder has a radius of 10 cm and a surface area of 1800 cm2, what is its height?
ALSO
Why do you need to know the height of a cylinder to figure out its surface area?
Answer:
18.65 cm
Step-by-step explanation:
the surface area of a cylinder is the combination of the two circles (1 at the top, 1 at the bottom) and the area of the side wall or "mantle".
so, two equally sized circle areas and the side area.
this side area is nothing else than a rectangle that is bent around and connected to the circumference of both circles. therefore, the side lengths of this rectangle must be the cylinder height and the length of the circumference of the circles.
that is also why we need the cylinder height to get its surface area (and also volume by the way).
circle area : pi × r²
rectangle area : length × width
cylinder surface area = 2 circle areas + side rectangle area
=> 1800 = 2 × (pi×r²) + (2×pi×r)×h = 2×pi×100 + 2×pi×10×h
=> 1800 = 200×pi + 20×pi×h
=> (1800 - 200×pi) = 20×pi×h
=> (1800 - 200×pi) / (20×pi) = h = 18.65 cm
One gallon of gasoline in Buffalo,new York costs $2.29 .Across the border in Toronto,Canada, one liter of gallon costs $0.91
Complete question :
One gallon of gasoline in buffalo, New York costs $2.29. In Toronto, Canada, one liter of gas costs $0.91. There are 3.8 liters in one gallon. How much does one gallon of gas cost in Toronto?
Answer:
$3.458
Step-by-step explanation:
Given that :
Cost per gallon in Buffalo, New York = $2.29
Cost per Litre in Toronto, Canada = $0.91
Number of litres in 1 gallon = 3.8
The cost per gallon in Toronto, Canada will be :
3.8 * cost of a Litre
3.8 * $0.91 = $3.458
Hence, cost of a gallon in Toronto, Canada = $2.73
Reuben and Jim made the most points in the basketball game. Reuben made twelve points. Jim made twice as many points as Reuben. How many points did Jim make?
A. 36
B. 24
c. 18
D. 12
E. 6
Answer:24
Step-by-step explanation:
12x2=24
there are 6 pizzas to be divided for a baseball team wich has 15 players how much of each pizza will each player receive
Answer:
0.4 or 2/5
Step-by-step explanation:
The flagpole at school casts a shadow that is 18 feet long. A student nearby is 5.5 feet tall and casts a shadow that is 3 feet long. How tall is the flagpole?
Answer:
33 feet.
Step-by-step explanation:
If the ratio of the student to their shadow and flagpole to its shadow are equal, then you can use the proportion
5.5/3 = x/18 and cross multiply to find x.
5.5 × 18 = 99
3 × x = 3x
3x/3 = x, 99/3 = 33.
x = 33.
Help me pls pls pls pls
Anybody know the answer please?
Answer:
D) 80.5 cm sq
Step-by-step explanation:
square 10x7 = 70
Triangle (7x3) /2 = 10.5
70+ 10.5 = 80.5
Volume (look at the picture)
Answer:
108
Step-by-step explanation:
You start by finding the area of the trapezoid, which is ((3+5)/2) 3 = 12, then you multiply 12 by 9 which is 12*9 = 108
Solve for v.
–9,45 − 6,1v = –2,8v + 13,98
Answer:
V would equal -7.1.
Step-by-step explanation:
It appears you have set the decimal points as commas. Let's remove that and replace it with decimal points. (AkA ".")
First, we move every term with "V" in this problem to the left.
-9.45 - 3.3v = 13.98
Next, we add -9.45 to 13.98 which equals 23.43.
We then make 3.3v negative.
-3.3v = 23.43
We then divide 23.43 and 3.3v by 3.3.
We cancel out 3.3v to leave 23.43/3.3.
We can now divide 23.43 by 3.3 to get -7.1.
V = -7.1
Answer:
V would equal -7.1.
Step-by-step explanation:
The area of the
rhombus is 52 m2
What is the base of
the rhombus?
i have explained through pictures
Jane is buying a jacket that regularly costs $54. However, the jacket is on sale for 25% off the regular price. A sales tax of 6% will be added to the final purchase price. How much will Jane pay for the jacket, including tax?
Answer:
42.93
Step-by-step explanation:
54 x 0.25 = 13.50
54 - 13.5 = 40.50
40.5 x 0.06 = 2.43
40.50 + 2.43 = 42.93
Answer:
$42.93 I received 100%
Step-by-step explanation:
Which measure describes the standard error of the mean?
A.
the central limit theorem
B.
the confidence interval
C.
the sample mean
D.
the standard deviation of the means
Answer: the standard deviation of the means (D)
Step-by-step explanation:
Please, real answers.
My teacher only taught this to 2 kids, please help me so I don't snap from stress
Answer:
1332.8 ft^3
Step-by-step explanation:
separate the two figures into a flat cylinder and a flat box
Volume of a cylinder= [tex]\pi r^{2} h[/tex]
where radius= 3ft and height= 3ft
therefore
[tex]=\pi (3)^{2} (3) \\= 84.8 ft^{3}[/tex]
now for the flat box:
volume of box= [tex]w*l*h[/tex]
where width= 6ft, length= 16ft, height= 13ft
therefore
[tex]=(6)(16)(13)\\=1248 ft^{3}[/tex]
now add the volume of the cylinder and box together
[tex]84.8 + 1248\\= 1332.8 ft^{3}[/tex]
What is the slope of a line that
crosses through the points (-1,-8)
and (2, 1)?
Answer:There's the answer
Just open it
Step-by-step explanation:
One number is 2 more than another number. If the product of the two numbers is 35, what are the numbers? Please show the work
Answer: 5 and 7.
Step-by-step explanation: I’m sorry I don’t know what kind of equation to do but I’ll show how I got the answer if that’s what you mean.
First I made a list of numbers that when multiplied equal 35
Then I went through the list and took out all the options that were not one number being 2 more than the other.
The only option left is 5 x 7 (5*7)
what kind of equation do you want? Like 5*7 = 35 and 5 + 2 = 7?
The numbers are 5 and 7.
What is equation?A mathematical statement consisting of an equal symbol between two algebraic expressions that have the same value.
let the first number be 'x'
another number = x+2
According to question,
x(x+2)= 35
x²+2x-35=0
x²+7x-5x-35=0
x(x+7)-5(x+7)=0
(x-5)(x+7)=0
x= +5, -7
Hence, the numbers are 5 (x) and 7 (x+2).
Learn more about equation here:
https://brainly.com/question/2263981
#SPJ2
What is 4^4 in algerbra mathematics?
Answer:
256
Step-by-step explanation:
4 to the exponent of 4 = 256