The dimensions that will minimize the cost, obtained from the cost function are;
Width = [tex]\sqrt[3]{\frac{45}{11} }[/tex] feet
Length = [tex]2\cdot \sqrt[3]{\frac{45}{11} }[/tex] feet
Height = [tex]\dfrac{20}{\left(\sqrt[3]{\frac{45}{11} }\right)^2 }[/tex]
What is a function?A function is a rule, or law or expression that provides the definition between an input variable and an output variable.
Base length of the box = 2 × The base width of the box
Cost of the material used to build the top and the bottom = $11 per square feet
Cost of the material used to build the sides = $3 per square feet
Volume of the box = 40 cubic feet
Let x represent the width of the base of the box, and let y represent the height of the box
The information from the question indicates;
The length of the base of the box = 2·x
The volume of the box, V = x × 2·x × y = 2·x²·y = 40
V = 2·x²·y = 40
[tex]y = \dfrac{40}{2\cdot x^2} = \dfrac{20}{x^2}[/tex]
The cost of the box which can be obtained from the surface area of the box can be found as follows;
Cost of the top and the bottom = 2 × x × 2·x × 11 = 44·x²
Cost of the material for the sides = (2 × x × y + 2 × 2·x × y) × 3 = 18·x·y
Cost of the box, C = 44·x² + 18·x·y
[tex]C = 44\cdot x^2 + 18\cdot x \times \dfrac{20}{x^2} = \dfrac{44\cdot x^2+360}{x}[/tex]
To minimize the cost of the box, we get;
[tex]C' =\dfrac{d}{dx}\left( \dfrac{44\cdot x^2+360}{x}\right) = 0[/tex]
[tex]\dfrac{d}{dx}\left( \dfrac{44\cdot x^3+360}{x}\right) = \dfrac{x\times 44\times 3 \times x^2 - \left(44\times x^3+360\right)}{x^2} = \dfrac{88\cdot x^3-360}{x^2} =0[/tex]
At the minimum cost, [tex]\dfrac{88\cdot x^3-360}{x^2} =0[/tex]
Therefore; 88·x³ - 360 = 0
88·x³ = 360
[tex]x = \sqrt[3]{\dfrac{360}{88} } =\sqrt[3]{\dfrac{45}{11} }[/tex]
The width of the base of the box that minimizes the cost is; [tex]x =\sqrt[3]{\dfrac{45}{11}}[/tex]
The length of the base of the box that minimizes the cost is; [tex]2\cdot x =2\times \sqrt[3]{\dfrac{45}{11}}[/tex]
The height of the box that minimizes the cost is; [tex]\dfrac{20}{x^2} = \dfrac{20}{\sqrt[3]{\dfrac{45}{11} } }[/tex]
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consider functions f and g defined by the graph and table
The correct statements regarding the functions are given as follows:
Function g has a greater average rate of change on interval [0,2] than function f.Function g is exponential and eventually exceeds function f, which is linear.Function f has a greater y-intercept than function g.What are the classifications of the functions?Regarding the type of function, we have that:
f(x) is a linear function, as it's graph is a straight line.f(x) is an exponential function, as when x increases by one, y is multiplied by 1.5.The y-intercepts are the values of y for which x = 0, hence they are given as follows:
f(x) = 15.g(x) = 8.Hence function g(x) starts less than f(x), but it eventually exceeds function f(x), as it is exponential, meaning that it has a greater average rate of change.
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You move left 1 unit and right 8 units. You end at (5, 5). Where did you start?
The coordinates of the starting point is (12, 5)
What are ordered pairs?Ordered pairs refers to the arrangement of 2 numbers in the form (a, b)
The rule to move ordered pair one unit left 1 will be subtracted in the x coordinates
(5, 5) → one unit left → ( 5 - 1, 5 ) = (4, 5)
The rule to move ordered pair 8 units left, 8 will be added in the x coordinates
(4, 5) → one unit left → ( 4 + 8, 5 ) = (12, 5)
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Carmen is drawing a card from a standard deck of playing cards. Find each probability below.
Answer: How many cards are there tho?
Step-by-step explanation:
Value=x-y=3 questions is y-x
The value of (y-x) when (x-y) = 3 is -3.
According to the question,
We have the following equation:
x-y = 3
Now, we have to find the value of (y-x). We can easily do this by multiplying both sides of this equation by -1.
We know that the multiplication of two negative integers always gives us the positive results and the multiplication of one negative and one positive integer give us the negative results.
So, we will have the following expression:
-x+y = -3
It can rewritten as follows:
y-x = -3
Hence, the value of (y-x) when (x-y) = 3 is -3.
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Texas Roadhouse Corporate has decided to increase the price of
fried pickles by 15%. What will be the new price of the pickles?
The pickles are $3.99
Answer: The pickles are $3.99
Step-by-step explanation:For percents,this is always done by simply dividing the percent (in this case 15%) by 100%. So, the conversational term "15%" becomes 15% / 100% = 0.15 in terms of a real mathematical number. 0.15 x $152.00=$3.99
[tex]\frac{5x^{6}5x^{-2}125x2x^{4}2x^{-5} }{2x^{5}5x^{-3}2x^{6\\} }[/tex]
[tex]\cfrac{5x^6 5x^{-2}125x 2x^4 2x^{-5}}{2x^5 5x^{-3} 2x^6}\implies \cfrac{5\cdot 5\cdot 125\cdot 2\cdot 2}{2\cdot 5\cdot 2}\cdot \cfrac{x^6 x^{-2}x x^4 x^{-5}}{x^5 x^{-3} x^6} \\\\\\ \cfrac{ 5\cdot 125}{1}\cdot \cfrac{x^{6-2+1+4-5}}{x^{5-3+6}}\implies 125\cdot \cfrac{x^4}{x^8}\implies 125\cdot \cfrac{1}{x^8 x^{-4}}\implies \cfrac{125}{x^{8-4}}\implies {\Large \begin{array}{llll} \cfrac{125}{x^4} \end{array}}[/tex]
Graph the equation below
Y=1/3x-1
Answer:
Step-by-step explanation:
Find the equation of a parabola with a focus at (–2,–3) and a directrix at y = –5.
A)
y = 1∕4(x + 2)2 – 4
B)
y = –1∕16(x + 2)2 – 4
C)
y = 1∕4(x – 2)2 – 4
D)
y = 1∕4(x + 2)2 – 8
The vertex form of Parabola is y= 1/4(x + 2)² -4 .
What is Vertex form of Equation?The general equation of a parabola is given by y = a(x – h)² + k or
x = a(y – k)² +h. Here (h, k) is vertex.
Given:
focus at (–2,–3) and a directrix at y = –5.
We know, the standard form is (x - h)² = 4p (y - k),
where the focus is (h, k + p) and the directrix is y = k - p
So, (h, k + p) = (-2, -3)
h = -2, k + p = -3
Directrix is y = k - p:
k - p = -5
k + p = -3
so, 2k = -8
k = -4
and, p = -3+ 4 = 1
Thus, the vertex is at (-2, -4) and p = 1, the equation is:
(x + 2)² = 4(y+ 4)
1/4(x + 2)² = y+4
y= 1/4(x + 2)² -4
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2. When x = 0, y = 3 and z = 5, what is the value
of 3xy³ + 2xy³z + xyz + xz?
The value of 3xy³ + 2xy³z + xyz + xz = 0
Substituting given value in an Equation:
To find the value of a given equation or expression replaces each variable with the given value.
For example, a = 1 and b = 3 then find 3a + 2b.
=> 3(1) + 2(3) = 3 + 6 = 9
Therefore, the value of 3a + 2b = 9
Multiplication with zeroMultiplying any number with zero will give zero i.e 3 × 0 = 0. No matter how large the number is when we multiply it by zero the result will be zero.
Here we have
x = 0, y = 3 and z = 5
We need to find the value of 3xy³ + 2xy³z + xyz + xz
Substitute given values in the given expression
=> 3(0)(y)³ + 2(0)y³z + (0)yz + (0)z [ x × 0 = 0 ]
=> 0 + 0 + 0 + 0
=> 0
Therefore,
The value of 3xy³ + 2xy³z + xyz + xz = 0
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Write the equation of a line in
y = mx + b form that passes
through (5,-5) and (-5, -3)
The equation of a line in y = mx + b form that passes through (5,-5) and (-5, -3) is y = (-1/5)x - 4.
The formula to calculate the slope between two points is equal to
[tex]m = y2-y1/(x2-x1)[/tex]
Find the slope of Line a
We have the points
(5,-5) and (-5, -3)
Substituting the value in the formula
[tex]m = -3 - (-5)/(-5-5)\\m = (-3 + 5)/(-10)\\m = 2/-10\\m = -1/5[/tex]
The slope is -1/5.
Finding the y-intercept from the below-given equation:
y = mx + b ...... (1)
Putting value of y = - 5, x = 5, m = -1/5
-5 = (-1/5) x (5) + b
-5 = - 1 + b
=> b = - 4
Putting the value of b = - 4 and m = -1/5 in equation (1). We get,
y = (-1/5)x - 4
Hence, the equation of the line is y = (-1/5)x - 4.
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can you find the the mistake
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
a boy ate three fifthof an icicle. he gave away one thirdof it. how much icicle was eaten
In terms of fractions, 2/5 of the icicle was eaten by the boy.
If a boy ate three fifths of an icicle, then he would have two fifths, or two-fifths, left. One third of two-fifths is two-fifths of three, or two-fifths of three fifths, which is two-fifths. Therefore, the boy ate two-fifths of the icicle.In mathematical term:
A boy ate three fifth of an icicle = 3/5
A boy gave = 1/3
So, Icicle eaten by the boy = 3/5 - 1/3
= (9 - 3) / 15
= 6/15
= 2/5
Hence, In terms of fractions 2/5 of the icicle was eaten by the boy.
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You have four sweaters, five pairs of pants, and three pairs of shoes. How many different combinations can you make, wearing one of each? Explain your answer. a. 4 times 5 + 3 = 23 b. 4 times 5 = 20 c. 4 times 5 times 3 = 60 d. 4 + 5 + 3 = 12 Please select the best answer from the choices provided A B C D
There are total 4*5*3 = 60 ways of wearing one each.
What is Fundamental Counting Principle?
A rule used to count the entire number of alternative outcomes in a circumstance is known as the fundamental counting principle.
Solution:
Let,
for three events A, B, and C, number of outcomes be a,b, and c respectively.
So, the total number of outcomes is given by the product of outcomes of all the events i.e. by using Fundamental Counting Principle.
In the question mentioned,
a = 4
b = 5
c = 3
So, total outcomes = 4*5*3 = 60 ways
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Help Please............
The statement that is true for the function, h is h(2) = 16
It is given that there is a function h
The range of function, h is [tex]1\leq h(x)\leq 25[/tex]
The domain of the function, h is [tex]-3 \leq x\leq 11[/tex]
And it is also given that h(8) = 19 and h(-2) = 2
Now,
The 1st option is h(-3) = -1
But the range of h is between 1 to 25, hence, this option is not true.
The 2nd option is h(8) = 21
But it is already given that h08) = 19, thus this option is not true.
The 3rd option is h(13) = 18
But the domain of h is between -3 and 11, So, this option is not true as well.
The fourth option is h(2) = 16
In this the values of range and domain are correct.
Thus, h(2) = 16.
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Which of the following is an example of the associative property of multiplication?
Answer:
The associative property of multiplication states that the product of three or more numbers remains the same regardless of how the numbers are grouped. For example, 3 × (5 × 6) = (3 × 5) × 6.
Step-by-step explanation:
A parallelogram has been found to have diagonals which are NOT congruent and slopes which are negative reciprocals.
What kind of quadrilateral is it?
It is a rhombus kind of quadrilateral.
What is quadrilateral?
A quadrilateral has four sides, is two-dimensional (flat), closed (the lines join up), and has straight sides.
A rhombus is a four-sided quadrilateral with all of the features of a parallelogram.
A rhombus has three more characteristics: consecutive sides that are congruent; diagonals that bisect pairs of opposite angles; and diagonals that are perpendicular to one another.
Moving clockwise, beginning at point A in the top left corner of a hypothetical rhombus ABCD, you can see that side AB is congruent to side BC and side CD is congruent to side DA.
Additionally, you can see that diagonals AC and DB are perpendicular to one another and that the rhombus' diagonals split pairs of opposing angles.
Hence, It is a rhombus kind of quadrilateral.
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Which expression is equivalent to 1/3(x+6)−4
The expression that must be equivalent to the one give must be 1/3x - 2
Equivalent ExpressionEquivalent expressions are expressions that work the same even though they look different. If two algebraic expressions are equivalent, then the two expressions have the same value when we plug in the same value for the variable. Two expressions are said to be equivalent if they have the same value irrespective of the value of the variable(s) in them.
In this problem, we just need to simply the expression to get the answer to this.
1/3(x + 6) - 4
Step 1 : Expand or open the bracket
1/3x + 2 - 4
Step 2: Simplify the expression.
1/3x + 2 - 4 = 1/3x - 2
The equivalent expression must be 1/3x - 2
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what is the volume of the recycling box shown in the diagram , 20cm, 40cm, 60cm
Answer:
48000
Step-by-step explanation:
I multiplied the length times width time height (20x40x60) to get 48000.
write the equation of the line, in standard form, that passes through the point (4, -8) and is parallel to x - 4y=8
Answer: x - 2y = -10.
Step-by-step explanation:
What is 2(5+777x60 ÷6[8-9]
Answer:
-15530
Step-by-step explanation:
plug it in to a calculator
why so random numbers
Answer:
Step-by-step explanation:
2(5+777x60/6x(-1))
2(5-7770)
2x(-7765)
-15530
A giant tortoise can travel 0.15 miles in 1 hour. At this rate, how long would it take the tortoise to travel 1 mile
It would take the tortoise approximately ____
Answer:
6 [tex]\frac{2}{3}[/tex] Hours
Step-by-step explanation:
Distance = rate times time
Let t = time
1 = .15t Divide both sides by .15
6.666... = t The 6 is repeating so as a fraction it would be
6 [tex]\frac{2}{3}[/tex]
The cellular phone service for a business executive is $35 per month plus $0.40 per minute of phone use over 900 min. For a month in which the executive's cellular phone bill was $87.80, how many minutes did the executive use the phone?
The number of minutes that the executive use the phone is 131 minutes.
How to calculate the number of minutes?From the information, the cellular phone service for a business executive is $35 per month plus $0.40 per minute of phone use.
Let the number of minutes be m.
This will be illustrated as:
35 + 0.40m = 87.80
Collect like terms
0.40m = 87.80 - 35
0.40m = 52.80
Divide
m = 52.40 / 0.40
m = 131
The number of minute is 131 minutes.
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What is the factored form of 8x24-27y?
o (8x8-27y2) (2x¹6+xy+3y¹)
○ (2x-3y²) (4x¹6 - 6x³y² +9y4)
16
16
o(2x8-3y2) (4x¹6 +6x8y² +9y4)
O (8x8-27y2) (2x¹6-6xy+3y4)
The factored form of 8x²⁴-27⁶ is (2x⁸-3y²)(4x¹⁶+6x⁸y²+9y⁴)
What is a factored form?Equations may be solved and x-intercepts can be found very quickly when a quadratic expression is in factored form and equal to 0. You may also find vertices and maximum and minimum values of the expression.
The four main methods of factoring are the Grouping method, Difference in Two Squares, Sum or Difference in Cubes, and Greatest Common Factor (GCF).
It is referred to as being in factored form when a quadratic expression is represented as the sum of two linear factors plus a constant. Examples of factors include ((5x + 2)(3x-1)) and ((2(x-1)(x+3)).
Given,
8x²⁴-27⁶
=(2x⁸)³-(3y²)³
=(2x⁸-3y²)((2x⁸)²+2x⁸×3y²+(3y²)²
=(2x⁸-3y²)(4x¹⁶+6x⁸y²+9y⁴)
Therefore, the factored form of 8x²⁴-27⁶ is (2x⁸-3y²)(4x¹⁶+6x⁸y²+9y⁴)
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Make x the subject of m=n+x/p
The value of x is x=p(m-n).
What is the solution to an equation?
In order to make the equation's equality true, the unknown variables must be given values as a solution. In other words, the definition of a solution is a value or set of values (one for each unknown) that, when used as a replacement for the unknowns, transforms the equation into equality.
Given expression: m=n+(x/p)
To find: Make x the subject.
To make x the subject we have to separate the x to one side.
Expression m=n+(x/p)
Take n to another side,
(x/p)=m-n
Cross multiply,
x=p(m-n)
Therefore, the value of x is x=p(m-n).
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Limit x approaches 0 (tan^2x-sin) divide by x4
[tex]\displaystyle \lim_{\theta \to 0}\cfrac{\tan^2(\theta )-\sin(\theta )}{\theta 4}\implies \stackrel{\textit{\Large L'Hopital's rule}}{\lim_{\theta \to 0}\cfrac{\frac{d}{d\theta }[\tan^2(\theta )-\sin(\theta )]}{\frac{d}{d\theta }[\theta 4]}} \\\\[-0.35em] ~\dotfill[/tex]
[tex]\cfrac{d}{d\theta }[\tan^2(\theta )-\sin(\theta )]\implies \cfrac{d}{d\theta }[[\tan(\theta )]^2-\sin(\theta )]\implies \stackrel{chain~rule}{2\tan(\theta )\cdot \sec^2(\theta )} ~~ -\cos(\theta ) \\\\\\ 2\tan(\theta )\cdot \cfrac{1}{\cos^2(\theta )}-\cos(\theta )\implies \cfrac{~~ \frac{2\sin(\theta) }{\cos(\theta) } ~~}{\cos^2(\theta )}~~ -\cos(\theta ) \\\\\\ \cfrac{2\sin(\theta) }{\cos^3(\theta) }~~ -\cos(\theta )\implies \cfrac{2\sin(\theta)-\cos^4(\theta)}{\cos^3(\theta)} \\\\[-0.35em] ~\dotfill[/tex]
[tex]\cfrac{d}{d\theta}[4\theta]\implies 4 \\\\[-0.35em] ~\dotfill[/tex]
[tex]\displaystyle \lim_{\theta \to 0}\cfrac{\tan^2(\theta )-\sin(\theta )}{\theta 4}\implies \lim_{\theta \to 0}\cfrac{ ~~ \frac{2\sin(\theta)-\cos^4(\theta)}{\cos^3(\theta)} ~~ }{4}\implies \lim_{\theta \to 0}\cfrac{2\sin(\theta)-\cos^4(\theta)}{4\cos^3(\theta)} \\\\\\ \displaystyle \lim_{\theta \to 0}\cfrac{2\sin(0)-\cos^4(0)}{4\cos^3(0)}\implies \lim_{\theta \to 0}\cfrac{2(0)-1^4}{4(1^3)}\implies {\Large \begin{array}{llll} -\cfrac{1}{4} \end{array}}[/tex]
Find the equation (in slope-intercept form) of the line passing through the points with the given coordinates.
(1,5) , (3,−2)
The required equation is y = - 3.5x + 8.5 (in slope-intercept form) of the line passing through the points with the given coordinates.
What is the slope of the line?The slope of a line is defined as the gradient of the line.
Given that the line that passes through two points (1,5), and (3,−2)
Let the required equation of the line will be as :
y - 5 = [(-2 - 5)/(3-1)](x - 1)
y - 5 = (-7/2)(x - 1)
y - 5 = - 3.5(x - 1)
y - 5 = - 3.5x + 3.5
y = - 3.5x + 3.5 + 5
y = - 3.5x + 8.5
Thus, the required equation is y = - 3.5x + 8.5 (in slope-intercept form) of the line passing through the points with the given coordinates.
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the line segment from (0,0) to (6,18) is revolved about the x-axis to form a cone. what is the volume of this cone?
Answer:
Step-by-step explanation:
do you happen to know the formula for finding the area of a cone?
V=π*[tex]r^{2}[/tex]*h/3
does that formula look familiar?
we know the radius , r, is 18 and the height, h is 6 :) can you figure this out now? just plug those into the formula above :)
Solve for x. Round to the nearest tenth, if necessary.
Value of x rounded to nearest tenth is 25.5 units.
Given in question,
Right angled triangle CDE
θ = 20°
CD = perpendicular = 24 units
CE = Hypotenuse = x
Therefore,
cosθ = Perpendicular/Hypotenuse
cos 20 = 24/x
0.9397 = 24/x
x = 24/0.9397
= 25.54
Hence, value of x rounded to nearest tenth is 25.5 units.
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An ostrich farmer wants to enclose a rectangular area and then divide it into three pens with fencing parallel to one side of the
rectangle (see the figure below). There are 700 feet of fencing available to complete the job. What is the largest possible total
area of the three pens?
The largest possible total area of the three pens is 15312.5sq.units.
What is a rectangular area?
The quantity of space occupied by a flat surface with a specific shape is referred to as the area.
Area of a Rectangle = (Length × Breadth) square units.
Here,
We let x be the length and y be the breadth of a rectangle area.
Given equation : 2y + 4x = 700
y + 2x = 350
y = 350 - 2x .......(1)
Area of rectangular field (A)= xy
A = x(350 - 2x)
A = 350x - 2x²
Differentiating w.r.t x we get,
A' = 350 - 4x
Again, differentiating w.r.t x we get,
A" = -4
Let A' = 0
350 - 4x = 0
4x = 350
x = 87.5
A is maximum at 87.5feets.
Hence, the largest possible total area of the three pens = XY = 87.5×175 = 15312.5 sq. units.
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A solve the first question
Answer:
A or C
Step-by-step explanation: