A rectangular box, closed at the top, with a square base, is to have a volume of 4000 cm^ 3 . W What must be its dimensions (length, width, height ) if the box is to require the least possible material?
Answers
Solution
Area of square base of sides x is
[tex]Area=x^2[/tex]Volume = 4000cm^3
[tex]\begin{gathered} Volume=Bh \\ B=Base\text{ }Area \\ h=height \end{gathered}[/tex]Thus,
[tex]\begin{gathered} Volume=Bh \\ 4000=x^2h \\ \\ h=\frac{4000}{x^2} \end{gathered}[/tex]For the box to require the least possible material, is to simply minimize the surface area of the rectangular box
The surface Area is given as
[tex]\begin{gathered} Area=2(lw+wh+lh) \\ Since,\text{ it is a square base} \\ l=x \\ w=x \\ \\ Area=2(x^2+xh+xh) \\ Area=2(x^2+2xh) \\ Area=2(x^2+2x(\frac{4000}{x^2})) \\ \\ Area=\frac{16000}{x}+2x^2 \end{gathered}[/tex]Now, we differentiate
[tex]\begin{gathered} Area=\frac{16,000}{x}+2x^{2} \\ A=16000x^{-1}+2x^2 \\ By\text{ differentiating} \\ \frac{dA}{dx}=-16000x^{-2}+4x \\ \\ At\text{ minimum area, }\frac{dA}{dx}=0 \\ 4x=16000x^{-2} \\ x^3=4000 \\ x=10\sqrt[3]{4} \end{gathered}[/tex]Now, to find h
[tex]\begin{gathered} h=\frac{4000}{x^2} \\ h=\frac{4000}{100(4)^{\frac{2}{3}}} \\ h=4^{\frac{1}{3}}\times10 \end{gathered}[/tex]Therefore,
[tex]\begin{gathered} Length=10\sqrt[3]{4}cm=15.874cm\text{ \lparen to three decimal places\rparen} \\ Width=10\sqrt[3]{4}cm=15.874cm\text{ \lparen to three decimal places\rparen} \\ height=15.874cm\text{ \lparen to three decimal places\rparen} \end{gathered}[/tex]Related Questions
Goran rented a truck for one day. There was a base fee of $20.99, and there was an additional charge of 92 cents for each mile driven. Goran had to pay $252.82 when he returned the truck. For how many miles did he drive the truck?
Answers
Let x be the miles driven, and y be the cost, then we can write the following relationship:
[tex]y=0.92x+20.99[/tex]In our case, we know that the final cost was y=252.82 and we need to find the miles given by x, then, we have
[tex]252.82=0.92x+20.99[/tex]By moving 20.99 to the left hand side, we have
[tex]\begin{gathered} 252.82-20.99=0.92x \\ 231.83=0.92x \end{gathered}[/tex]then, x is given by
[tex]\begin{gathered} x=\frac{231.83}{0.92} \\ x=251.98\text{ miles} \end{gathered}[/tex]then, the answer is 251.98 miles
Identify the units you would expect for the given quantity.The price of a bottle of French perfume, found by multiplyingthe unit price of the perfume in euros per milliliter by thevolume of the bottle in milliliters.
Answers
Given:
The price of a bottle of French perfume, found by multiplying
the unit price of the perfume in euros per milliliter by the volume of the bottle in milliliters.
the unit of the unit price is: euros per milliliter
And the unit of the volume is: milliliters
By multiplying the two units: the unit of the result will be euros
fill in the table using the function rule y=3x+5
Answers
Fill in the table using the function rule y= 3x + 5
________________________________________
Replace each value of x in the expression
____________________________________
x= -4
y = 3* (-4)+5
= -12 +5
= -7
_______________
x= -2
y = 3* (-2)+5
= -6 +5
= -1
_____________
x= 0
y = 3* (0)+5
= 0 +5
= 5
_____________
x= 2
y = 3* (2)+5
= 6 +5
= 11
______________________________
Answer
_________________________
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Stacy made a square tablecloth with a side length of 3.5 feet. She put lace along each side of the tablecloth.How much lace did Stacy need for the tablecloth?
Answers
The side of the square is 3.5 feet
The perimeter is
[tex]4\times3.5=14ft[/tex]Stacy need 14 feet lace
PO Triangle ABC is similar to triangular DEF. What is the value of x?
Answers
If triangle ABC is similar to DEF then the ratio between the sides will be a constant so we can get the expression:
[tex]\frac{15}{30}=\frac{x}{36}[/tex]and we solve for x so:
[tex]\begin{gathered} x=\frac{36\cdot15}{30} \\ x=18 \end{gathered}[/tex]Determine the area of the base of a cone with a volume of 36 cubic inches and a height of 9 inches?
Answers
the are of a base cone, while we have the volume and the height
V= pi*r^2*(h/3)
36=pi*r^2*3
we can find out now the r, radius
r=sqrt(3*36/pi*9)=1.95
Area, A= pi*r^2=
we can look at the Volume formula
36=pi*r^2*3
and if we divide both sides by 3 we get
12=pi*r^2, which is area
A= 12
Write the equation of the parabola in vertex form given the vertex (–2, 3) and point (0, 1).
Answers
the equation is
[tex]y=-\frac{1}{2}(x+2)^2+3[/tex]A shipment of computers arrived at. a warehouse.If each computers is valued at$995. What is the total value of the shipping?
Answers
Total number of computers = 20
Cost per a computer = $995
Total cost of 20 computers = 20 x $995
= $19,900
What are the coordinates of point w? 5 Z 3 2 Y 3 0 2 2. 4 -1 -2 W -3 -5 Х
Answers
the point W is positioned 3 units to the right (the x-coordinate is 3) and 2 units down (the y-coordinate is -2), thus the coordinates of W are (3,-2)
If we chose a theater and movie to attend at random, what probability would we have of seeing anything other than a romantic or sci-fi movie ?
Answers
we have that
The probability of seeing a romantic or sci-fi movie is
P=0.203+0.12=0.323
therefore
the probability of seeing anything other than a romantic or sci-fi movie is
P=1-0.323
P=0.677
the answer is option BYou bought a magazine for $7 and some notepads for $5 each. You spend a total of $27 how many notepads did you buy.
Answers
From the information available, some magazines (m) and some notepads (n) were bought.
If you bought a magazine for $7, then mathematically that would be;
[tex]\begin{gathered} 1\times m=7 \\ m=7 \end{gathered}[/tex]Also, if you bought some notepads for $5 each, that would be expressed as;
[tex]5\times n=5n[/tex]This represents 5 dollars times every given number of notepads, hence we would have;
[tex]\begin{gathered} m+5n=27 \\ \end{gathered}[/tex]Note that one magazine was bought which is why we have m = 7. We shall now substitute for thw value of m;
[tex]\begin{gathered} m+5n=27 \\ 7+5n=27 \\ \text{Subtract 7 from both sides;} \\ 7-7+5n=27-7 \\ 5n=20 \\ \text{Divide both sides by 5;} \\ \frac{5n}{5}=\frac{20}{5} \\ n=4 \end{gathered}[/tex]The value of n = 4
ANSWER:
This means you bought 4 notepads
Use a venn diagram to represent this problemJar A contains numbers that are less than 26 and evenly divisible by 2, Jar B contains numbers that are less than 20 and evenly divisible by 4.
Answers
Given:
Jar A contains numbers that are less than 26 and evenly divisible by 2.
The number less than 6 and divisible by 2 are,
[tex]A=\mleft\lbrace2,4,6,8,10,12,14,16,18,20,22,24\mright\rbrace[/tex]Jar B contains numbers that are less than 20 and evenly divisible by 4.
The set is,
[tex]B=\mleft\lbrace4,8,12,16\mright\rbrace[/tex]The Venn diagram is,
Dave started at the black dot and traveled the distance shown on the map on his bike. The length of the section in red is not known.
About how far did Dave travel on his bike?
Answers
Answer: 9,00
Step-by-step explanation:
I am right.
Answer:
1,402
Step-by-step explanation
the red line is further than the 1,396
Convert 15 gal to quarts
Answers
The conversion rate for a gallon to quart is given as
[tex]1\text{ ga }\to4\text{ quart}[/tex]This means that to get the number of quarts in a given measure in gallons, we multiply the number by 4.
The question asks us to convert 15 gallons to quarts.
This can be calculated as
[tex]\begin{gathered} 15\times4 \\ =60\text{ quarts} \end{gathered}[/tex]Therefore,
[tex]15\text{ gal }\to60\text{ quarts}[/tex]In AFGHFH.GF +40, HF 3x - 20, and GH find the value of 21 20
Answers
You have the next triangle:
As the triangle has two angles that are congruent, then it is a isisceles triangle. The opposite sides of the equal angles have the same measure.
Sides FG and GH have the same measure:
[tex]FG\cong GH[/tex][tex]x+40=2x+20[/tex]Use this equation to find the value of x:
- Substract in both sides of the equation 2x:
[tex]\begin{gathered} x-2x+40=2x-2x+20 \\ -x+40=20 \end{gathered}[/tex]- Substract in both sides of the equation 40:
[tex]\begin{gathered} -x+40-40=20-40 \\ -x=-20 \end{gathered}[/tex]- Multiply both sides of the equation by -1:
[tex]\begin{gathered} (-1)(-x)=(-1)(-20) \\ \\ x=20 \end{gathered}[/tex]Answer x=20Question 4 please . Using a graphing utility (geogebra) to graph the function
Answers
Problem N 4
we have the function
[tex]f\mleft(x\mright)=x^4-3x^2+2x-1[/tex]Interval (-2,2)
using a graphing tool
Local minimum value at (1,-1)
Local maximum value at (0.37,-0.65)
Increasing functionIntervals (-1.37,0.37) U (1, infinite)
Decreasing functionIntervals (-infinite, -1.37) U (0.37,1)
find the first five terms of the recursive sequence. aₙ = -6aₙ₋₁ where a₁ = 45
Answers
The first terms is
[tex]undefined[/tex]Substitute 2, 3, 4, and 5 for n in the equation to find first four next terms.
URGENT!! ILL GIVE
BRAINLIEST!!!! AND 100 POINTS!!!!!
Answers
Answer: A 143 degrees because angle 5 is a congruent to angle one
Step-by-step explanation:
The vertices of ABCDE are A(-6,0), B(-3,0), C(0, -3), D(-3,-6), and E(-6, -3). Find the vertices of the image after a translation using the rule (x,y) - (x + 9, y-6) and a dilation with a scale factor of 4s centered at the origin.
Answers
Vertices of new image: A' = (12, -24)
B' = (24, -24)
C' = (36, -36)
D' = (24, -48)
E = (12, -36)
Explanation:A(-6,0), B(-3,0), C(0, -3), D(-3,-6), and E(-6, -3)
A translation of (x + 9, y-6)
A becomes A'
A' = (-6 + 9, 0 - 6) = (3, -6)
B(-3,0)
B becomes B'
B' = (-3+9, 0 - 6) = B' (6, -6)
C(0, -3)
C becomes C'
C' = (0+9, -3-6) = C' (9, -9)
D(-3,-6)
D becomes D'
D' = (-3+9, -6-6) = (6, -12)
E(-6, -3)
E becomes E'
E' = (-6+9, -3-6) = (3, -9)
A dilation with scale factor of 4, we multiply the cooordinates of the alphabeths with prime with 4.
Vertices of new image:
A' = 4(3, -6) = (12, -24)
B' = 4(6, -6) = (24, -24)
C' = 4(9, -9) = (36, -36)
D' = 4(6, -12) = (24, -48)
E = 4(3, -9) = (12, -36)
6 At a 33-foot depth underwater, the pressure is 29.55 pounds per square inch (psi). At a depth of 66 feet, the pressure reaches 44.4 psi. At what rate is the pressure increasing? ur answer on ouatlon for the
Answers
To solve the exercise you can apply the following formula:
[tex]rate=\frac{\text{difference in pressure}}{\text{difference in depth}}[/tex]In this case, based on the information given in the exercise, you can identify that:
[tex]\begin{gathered} \text{difference in pressure= (44.4.}-29.55)psi=14.85\text{ psi} \\ \text{difference in dept}=(66-33)ft=33\text{ ft} \end{gathered}[/tex]Substituting values, you get:
[tex]rate=\frac{14.85\text{ ps i}}{33\text{ ft}}=0.45\text{ }\frac{ps\text{ i}}{ft}[/tex]Therefore, The pressure is increasing at 0.45 psi per foot.
Which of the following is equal to ? 1/5^-2
Answers
25
1) Let's evaluate this expression making use of one exponent property:
[tex](\frac{1}{5})^{-2}=(\frac{5}{1})^2=(5)^2=25[/tex]2) Note that negative exponents reciprocate the base.
3) Hence, the answer is 25
Express the following as an algebraic function of x.sin(sin-'(x) – cos-'(x))
Answers
Hello there. To solve this question, we'll have to remember some properties about inverse trigonometric functions.
Given the expression:
[tex]\sin (\sin ^{-1}(x)-\cos ^{-1}(x))[/tex]We want to express it as an algebraic function of x.
For this, imagine the following triangle:
Now, we find the missing leg of the triangle applying Pythagoras theorem:
[tex]\begin{gathered} 1^2=x^2+?^2 \\ 1-x^2=?^2 \\ ?^{}=\sqrt[]{1-x^2} \end{gathered}[/tex]Now, finding the sine and cosine of the angles alpha and beta, we get:
First, remember the sine of an angle is equal to the ratio between the opposite side to the angle and the hypotenuse of the triangle. The cosine of the same angle is equal to the ratio between the adjacent side to the angle and the hypotenuse of the triangle.
Therefore, we have:
[tex]\begin{gathered} \sin (\alpha)=\frac{x}{1}=x \\ \cos (\beta)=\frac{x}{1}=x \\ \sin (\beta)=\frac{\sqrt[]{1-x^2}}{1}=\sqrt[]{1-x^2} \\ \cos (\alpha)=\frac{\sqrt[]{1-x^2}}{1}=\sqrt[]{1-x^2} \end{gathered}[/tex]This means that:
[tex]\begin{gathered} \alpha=\sin ^{-1}(x) \\ \beta=\cos ^{-1}(x) \end{gathered}[/tex]Now, the expression we had earlier turns into:
[tex]\sin (\alpha-\beta)[/tex]For this, we'll use the angle sum formula:
[tex]\sin (u-v)=\sin (u)\cos (v)-\sin (v)\cos (u)[/tex]Which gives us:
[tex]\sin (\alpha)\cos (\beta)-\sin (\beta)\cos (\alpha)_{}[/tex]Plugging the results we got earlier, this is simply:
[tex]x\cdot x-\sqrt[]{1-x^2}\cdot\sqrt[]{1-x^2}[/tex]As x > 0, because we're using it as a triangle side (but it could be negative considering inverse sine and cosine as functions), we get:
[tex]\begin{gathered} x^2-(1-x^2) \\ x^2-1+x^2 \\ 2x^2-1 \end{gathered}[/tex]Type the correct answer in the box. Use numbers instead of words.
The number 392,000 is divided by 10.
What is the value of the digit 2 in the quotient?
Answers
The value of 2 in the quotient is 200.
What is division?
One of the four fundamental operations of arithmetic, or how to mix numbers to create new ones, is division. Addition, subtraction, and multiplication are the other operations.
The opposite of multiplication is division. When you multiply three groups of four to produce twelve, you get four in each group when you split twelve into three equal groups.
Let, the number 392,000 is divided by 10.
That means, [tex]\frac{392000}{10}[/tex]
Here, 392000 is the numerator and 10 is the base.
Simplifying the fraction, we get 39200.
39200 is the quotient.
Here 2 is on the hundredth place.
Therefore, the value of 2 in the quotient is 200.
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When graphing unit rates, the unit rate begins at (0, 0), and then at (1, y). What will the 2nd and third points be? (2, _) and (3, _)
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When graphing unit rates, the unit rate begins at (0, 0), and then at (1, y).
So, it will represents a line passes through the given points
The slope of the line will be y
So,
The second point will be ( 2, 2y)
And the third point will be ( 3, 3y)
Evaluate: sin(30 degrees)cos(60 degrees)=(See attached image to assist with these 2 problems)
Answers
According to the figure we need to evaluate the sin(30°) and the cos(60°). Remember the trigonometric relations defined over the rectangle triangles as follows, suppose we have an angle called "alpha"
[tex]\begin{gathered} \sin(\alpha)=\frac{oc}{h}, \\ \\ cos(\alpha)=\frac{ac}{h}, \\ \\ tan(\alpha)=\frac{co}{ca} \\ \\ where\text{ }h:Hypotenuse,\text{ }ac:Adjacent\text{ }cathetus\text{ and }oc:Opposite\text{ }cathetus \end{gathered}[/tex]Now, according to the figure, we have that for the angle of 60 degrees:
[tex]\begin{gathered} h=2x,ac=x,oc=\sqrt{3}x \\ \\ \sin(60°)=\frac{oc}{h}=\frac{\sqrt{3}x}{2x}=\frac{\sqrt{3}}{2} \\ \\ \cos(60^{\circ})=\frac{ac}{h}=\frac{x}{2x}=\frac{1}{2} \end{gathered}[/tex]And for the angle of 30 degrees we get the following
[tex]\begin{gathered} h=2x,oc=x,ac=\sqrt{3}x \\ \\ \sin(30°)=\frac{oc}{h}=\frac{x}{2x}=\frac{1}{2}=\cos(60°) \\ \\ \cos(30^{\circ})=\frac{ac}{h}=\frac{\sqrt{3}x}{2x}=\frac{\sqrt{3}}{2}=\cos(60^{\circ}) \end{gathered}[/tex]So, your answer is: sin(30°)=1/2=cos(60°).
How to solve 2(x+7)=-4x+14
Answers
Let's solve the following equation
[tex]\begin{gathered} 2x+14=-4x+14 \\ 2x+4x=-14+14 \\ 6x=0 \\ x=0 \end{gathered}[/tex]The answer would be x = 0
A lumber yard has fixed costs of $7259.40 per day and variable costs of $0.07 per board-foot produced. Lumbar sells for $1.87 per board-foot. How many board-feet must be produced and sold daily to break even?
Answers
The number of board-feet that must be produced and sold daily to break even is:4033.
How to find the number of board-feet to breakeven?Let the number of board-feet produced per day = x
Let the lumber yard's costs each day be: C =$7259.40+ 0.07x
Let the lumber yard's profits be: P = 1.87x
Let P - C = 0
Hence,
1.87x - ($7259.40 + $0.07x) = 0
1.87x - $7259.40 - 0.07x = 0
Combine like terms
1.8x - $7259.40 = 0
1.8x = $7259.40
Divide both side by 1.8x
x = $7259.40/ 1.8
x = 4033
Therefore the breakeven is 4033.
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Find the prime factorization of the following number write any repeated racists using exponents
Answers
Prime factorization refers to the process of decomposing a given number into a product of prime numbers that can be repeated or not.
Let's remember that prime numbers are numbers that can only be divided by 1 or itself.v Having this in mind we have that the prime factorization of 66 is:
[tex]66=(2)(3)(11)[/tex]What is the Y intercept of 8X plus 4Y equals -48
Answers
the given equation is,
[tex]8x+4y=-48[/tex]Now, we will solve it further,
[tex]\begin{gathered} 8x+4y=-48 \\ 4y=-8x-48 \end{gathered}[/tex][tex]\begin{gathered} y=\frac{-8x-48}{4} \\ y=-2x-12 \end{gathered}[/tex]Now, we will put x=0 to get the intercept of the line on the Y axis,
[tex]\begin{gathered} y=-2x-12 \\ y=-2\times0-12 \\ y=-12 \end{gathered}[/tex]So, the Y-intercept of the given line equation is -12.
Which of the following is the graph of f(x)= x² +3x-4?
Answers
Given the function
[tex]f(x)=x^2+3x-4[/tex]To determine which graph corresponds to this function you have to determine the coordinates of the vertex and the roots of the function.
Vertex
To determine the coordinates of the vertex you have to calculate the x-coordinate using the formula:
[tex]x=-\frac{b}{2a}[/tex]a is the coefficient of the quadratic term
b is the coefficient of the x-term
The term of the quadratic term, in this case, is a=1 and the term of the x-term is b=3
[tex]\begin{gathered} x=-\frac{3}{2\cdot1} \\ x=-\frac{3}{2}=-1.5 \end{gathered}[/tex]Replace the x-coordinate in the function to calculate the corresponding value of f(x):
[tex]\begin{gathered} f(x)=x^2+6x-4 \\ f(-3)=(-\frac{3}{2})^2+3\cdot(-\frac{3}{2})-4 \\ f(-3)=\frac{9}{4}-\frac{9}{2}-4 \\ f(-3)=-\frac{25}{4}=-6.25 \end{gathered}[/tex]The coordinates of the vertex are (-1.5,-6.25)
Roots of the function
To determine the roots of the function you have to use the quadratic formula:
[tex]x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}[/tex]a is the coefficient of the quadratic term
b is the coefficient of the x-term
c is the constant of the function
For a=1, b=3, and c=-4
[tex]\begin{gathered} x=\frac{-3\pm\sqrt[]{3^2-4\cdot1\cdot(-4)}}{2\cdot1} \\ x=\frac{-3\pm\sqrt[]{9+16}}{2} \\ x=\frac{-3\pm\sqrt[]{25}}{2} \\ x=\frac{-3\pm5}{2} \end{gathered}[/tex]Solve the addition and the subtraction separately
Addition
[tex]\begin{gathered} x=\frac{-3+5}{2} \\ x=\frac{2}{2} \\ x=1 \end{gathered}[/tex]Subtraction
[tex]\begin{gathered} x=\frac{-3-5}{2} \\ x=\frac{-8}{2} \\ x=-4 \end{gathered}[/tex]The roots of the function are (1,0) and (-4,0)
The graph that corresponds to this function is