Answers
Lets make a picture of our problem:
where h denotes the height of the box.
We know that the volume of a rectangular prism is
[tex]\begin{gathered} V=(4x)(x)(h) \\ V=4x^2h \end{gathered}[/tex]Since the volume must be 8 cubic centimeters, we have
[tex]4x^2h=48[/tex]Then, the height function is equal to
[tex]h=\frac{48}{4x^2}=\frac{12}{x^2}[/tex]On the other hand, the function cost C is given by
[tex]C=1.80A_{\text{bottom}}+1.80A_{\text{top}}+2\times3.60A_{\text{side}1}+2\times3.60A_{\text{side}2}[/tex]that is,
[tex]\begin{gathered} C=1.80\times4x^2+1.80\times4x^2+3.60(8xh+2xh) \\ C=3.60\times4x^2+3.60\times10xh \end{gathered}[/tex]which gives
[tex]C=3.60(4x^2+10xh)[/tex]By substituting the height result from above, we have
[tex]C=3.60(4x^2+10x(\frac{12}{x^2}))[/tex]which gives
[tex]C=3.60(4x^2+\frac{120}{x})[/tex]Now, in order to find minum cost, we need to find the first derivative of the function cost and equate it to zero. It yields,
[tex]\frac{dC}{dx}=3.60(8x-\frac{120}{x^2})=0[/tex]which is equivalent to
[tex]\begin{gathered} 8x-\frac{120}{x^2}=0 \\ \text{then} \\ 8x=\frac{120}{x^2} \end{gathered}[/tex]by moving x squared to the left hand side and the number 8 to the right hand side, we have
[tex]\begin{gathered} x^3=\frac{120}{8} \\ x^3=15 \\ \text{then} \\ x=\sqrt[3]{15} \\ x=2.4662 \end{gathered}[/tex]Therefore, by substituting this value in the height function, we get
[tex]h=\frac{12}{2.4662^2}=1.9729[/tex]therefore, by rounding to the neastest hundredth, the height which minimize the cost is equal to 1.97 cm
Related Questions
Solve. Your answer should be in simplest form. (2 1/6)(1 1/3) HELP!!!!
Answers
The simplified form of the expression (2 1/6 ) × (1 1/3) is 26/9.
What is the simplified form of the given expression?Given the expression in the question;
(2 1/6 ) × (1 1/3)
To simplify, first convert from mixed to improper fraction.
(2 1/6 ) × (1 1/3)
( (2×6 + 1)/6 ) × (1 1/3)
( (12 + 1)/6 ) × (1 1/3)
( 13/6 ) × (1 1/3)
( 13/6 ) × (1×3 + 1/3)
( 13/6 ) × (3 + 1/3)
( 13/6 ) × (4/3)
Now, cancel the common factor 2.
13/6 × 4/3
13/3 × 2/3
( 13 × 2 ) / ( 3 × 3 )
( 26 ) / ( 9 )
26/9
Therefore, the simplified form is 26/9.
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determine the missing angle measures in each triangle
Answers
ANSWER:
50°
STEP-BY-STEP EXPLANATION:
We can calculate the value of the missing angle, since there is a right angle (that is, 90°) and the other is 40 °, we apply the property that says that the sum of all the internal angles of a triangle is equal to 180°, Thus:
[tex]180=90+40+x[/tex]Solving for x:
[tex]\begin{gathered} x=180-90-40 \\ x=50\text{\degree} \end{gathered}[/tex]Joseph owns a 50 inch TV and it measures 50 inch on the diagonal. if the television is 40 inches across the bottom find the height of the TV
Answers
Let's draw the tv with the given values.
Note that we will form a right triangle with heigh of h, base of 40 and a hypotenuse of 50.
The Pythagorean Theorem is :
[tex]c^2=a^2+b^2[/tex]where c is the hypotenuse, a and b are the legs of the triangle.
Using the formula above. we will have :
[tex]\begin{gathered} 50^2=40^2+h^2 \\ 2500=1600+h^2 \\ h^2=2500-1600 \\ h^2=900 \\ \sqrt[]{h^2}=\sqrt[]{900} \\ h=30 \end{gathered}[/tex]The answer is 30 inches
First find the circumference. Do you need to divided by two? Find X. Then show all work to calculate the composite perimeter.
Answers
We are given the radius of the circle =5
then the circumference is given by
[tex]C=2\pi *r[/tex][tex]C=2\pi *5[/tex][tex]C=10\pi[/tex]then the cicumference of the semicircle is
[tex]\frac{C}{2}=\frac{10\pi}{2}=5\pi[/tex]Now let's find X
given the radius=5
the diameter = 2r = 5*2 = 10 in
then X is given by
[tex]X=4+10+4.5[/tex][tex]X=18.5[/tex]now the lateral side of the rectangle is given by
12-5= 7 in
then
the composite perimeter is
[tex]P=\frac{C}{2}+4.5+7+X+7+4[/tex][tex]P=5\pi+4.5+7+18.5+7+4[/tex][tex]P=5\pi+41[/tex][tex]P=56.70\text{ in}[/tex]then the composite perimeter is 56.7 in
Consider these functions:
ƒ(x) = 1/3 x² + 4
g(x)=9x - 12
What is the value of g(f(x))?
Answers
The value of the composite function g(f(x)) is 1 / 3 (81x² - 216x + 144) + 4
How to solve composite function?A composite function is a function that depends on another function. In a composite function, the output of one function becomes the input.
Therefore, let's solve the function as follows;
f(x) = 1 / 3 x² + 4
g(x) = 9x - 12
The value of g(f(x)) can be found as follows:
To find g(f(x)) we have to substitute the f(x) in g(x).
Therefore,
g(f(x)) = 1 / 3 (9x - 12)² + 4
(9x - 12)(9x - 12) = 81x² - 108x - 108x + 144
(9x - 12)(9x - 12) = 81x² - 216x + 144
Therefore,
g(f(x)) = 1 / 3 (81x² - 216x + 144) + 4
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Ava graphs the function h(x) = x^2 + 4. Victor graphs the function g(x) = (x + 4)^2. Which statements are true regarding the two graphs? Select three options.Ava’s graph is a vertical translation of f(x) = x^2.Victor’s graph is a vertical translation of f(x) = x^2.Ava’s graph moved 4 units from f(x) = x^2 in a positive direction.Victor’s graph moved 4 units from f(x) = x^2 in a positive direction.Ava’s graph has a y-intercept of 4.
Answers
Given,
Ava graphs the function h(x) = x^2 + 4.
Victor graphs the function g(x) = (x + 4)^2.
Required:
Check the correct statement about graph.
The graph of Ava and vector function is:
Here, victor graph was represented by blue curve and ava graph by green curve.
For first statement,
Ava’s graph is a vertical translated by 4 units.
Hence, statement is true.
For second statement,
The graph of victor is not vertically translated.
Hence, statement is false.
For statement three,
The curve of the Ava graph is moved 4 unit up in the positive direction. It is in y axis. Hence, statement is true.
For statement forth,
The curve of the victor graph is moved to negative direction not positive. Hence, statement is false.
For statement fifth,
The graph of Ava has the y intercept at 4. So, statement is correct.
Hence, option A (Ava’s graph is a vertical translation of f(x) = x^2), option C (Ava’s graph moved 4 units from f(x) = x^2 in a positive direction) and option E (Ava’s graph has a y-intercept of 4.) is true.
How do I tell if a parabola has a minimum or a maximum?
Answers
We can write the equation of a parabola in two different ways:
The standard form:
[tex]\begin{gathered} y=ax^2+bx+c \\ a\ne0 \end{gathered}[/tex]And the vertex form:
[tex]y=a(x-h)^2+k[/tex]If the parabola has a minimum or a maximum depends on the leading coefficient (the coefficient of x²) or in both cases the coefficient a.
Let's see the cases:
[tex]a>0_{\text{ }}(a_{\text{ }}is_{\text{ }}positive)[/tex]If a is positive, the parabola opens upwards, so the parabola has a minimum.
[tex]a<0_{\text{ }}(a_{\text{ }}is_{\text{ }}negative)[/tex]If a is negative, the parabola opens downwards, so the parabola has a maximum
Susan has a job selling cars, and earns 1.25% commission on the first $100,000 in sales,The commission increases to 4.95% on the next $200,000. Last month her total sales were$387,000. How much was her commission if she received 7.25% for any sales over $300,000
Answers
Solution:
Susan earns a commission based on car sales made.
Given:
Total sales made for the month = $387,000
Her commission is calculated based on levels and commision rate for each level.
On the first $100,000, she earns 1.25% commission.
Total sales at this level is $100,000
[tex]\begin{gathered} \text{Commision made on the first \$100,000 is;} \\ \frac{1.25}{100}\times100000=\text{ \$1,250} \\ =\text{ \$1,250} \end{gathered}[/tex]On the next $200,000, she earns 4.95% commission.
Total sales at this level is $300,000
[tex]\begin{gathered} \text{Commision made on the next \$200,000 is;} \\ \frac{4.95}{100}\times200000=\text{ \$9,900} \\ =\text{ \$9,900} \end{gathered}[/tex]On the next $87,000, total sales at this level is $387,000. She earns 7.25% commission for sales above $300,000.
[tex]\begin{gathered} \text{Commision made on the next \$87,000 is;} \\ \frac{7.25}{100}\times87000=\text{ \$6,}307.50 \\ =\text{ \$6,}307.50 \end{gathered}[/tex]Therefore, Susan's total commission received for the month is;
[tex]\begin{gathered} \text{ \$1250 + \$9900 + \$6307.50} \\ =\text{ \$17,457.50} \end{gathered}[/tex]Hence, her commission received in total for the sales made is $17,457.50
Hurry and answer this pls Bc this have to be turned in
Answers
Write an addition equation and a subtraction equation
to represent the problem using? for the unknown. Then solve.
There are 30 actors in a school play. There are 10 actors from second grade. The rest are from third grade. How many actors are from third grade?
a. Equations:
b. Solve:
Answers
The Equation is 10 + x= 30 and 20 actors are from third grade.
What is Equation?Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side. It demonstrates the equality of the relationship between the expressions printed on the left and right sides. LHS = RHS is a common mathematical formula.
Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.
Given:
There are 30 actors in a school play.
There are 10 actors from second grade.
The rest are from third grade.
let the actors in third grade is x.
Equation is:
Actors from second grade + Actors from third grade = Total actors
10 + x= 30
Now, solving
Subtract 10 from both side
10 +x - 10 = 30 - 10
x = 20
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what is the domain and range of {(1,0), (2,0), (3,0) (4,0), (5,0)}
Answers
We have the following:
The domain is the input values or the values of x and the range is the output values or the values of y
Therefore:
[tex]\begin{gathered} D=\mleft\lbrace{}1,2,3,4,5\mright\rbrace \\ R=\mleft\lbrace0\mright\rbrace \end{gathered}[/tex]URGENT!! ILL GIVE
BRAINLIEST!!!! AND 100 POINTS!!!!
Answers
Answer:
A. reflection over the y-axis
B. translation 3 units right
C. translation 4 down
D. reflection over the x-axis
What is the probability that a randomly chosen marble is red or small?
Answers
We have the next formula
[tex]P\mleft(RorS\mright)=P\mleft(R\mright)+P\mleft(S\mright)-P\mleft(RandS\mright)[/tex]P(R)=0.7
P(S)=0.9
P(RandS)=0.6
The probability that randomly chosen marbñe is red or small is
[tex]\begin{gathered} \\ P(RorS)=0.7+0.9-0.6=1 \end{gathered}[/tex]Is 7.787887888... a rational number?Highlight the correct answer below.a) Yes; it has a pattern which is repeatingb) Yes; it has a pattern which isterminatingc) No; it has a pattern which isterminatingd) No; it has a pattern which is repeating
Answers
A)
If This number 7.787887888... could be written as a ratio
[tex]\frac{a}{b}[/tex]Then it is called rational.
Since it has 7.78788788788... is an infinite number, with a repeating pattern notice it in bold. Then the only possible answer is:
Yes, it as a rational number, with a repeating pattern.
A.
May I please get help with numbers (4), (5), and (6). I have tried multiple times to find the correct answers for each of them but still could not get the accurate or at least correct answers for each of them. I would appreciate it so much if I could get help
Answers
EF =21
4) Let's find out the measure of the line segment EF, using the Trapezoid Midsegment Formula:
[tex]M=\frac{B+b}{2}[/tex]4.1) We can plug into that the lengths of AD and BC:
[tex]M=\frac{18+24}{2}=\frac{42}{2}=21[/tex]Note that the Midsegment is the average of the bases of a trapezoid.
4.3) Hence, the answer is 21
Select the statement that accurately describes the following pair oftriangles.
Answers
In any pair of similar triangles, (side side side )
Each correspondent side has the same ratio so let's examine
ΔCDE and ΔFGH
Write a quadratic equation in standard form with the given roots. a. Write a quadratic equation with a double root of -5.
Answers
a quadratic function has any root when replacing that number the equation is equal to zero
so
[tex](x+5)(x+5)[/tex]now solve the multiplication
[tex]\begin{gathered} (x\times x)+(x\times5)+(5\times x)+(5\times5) \\ x^2+5x+5x+25 \\ x^2+10x+25 \end{gathered}[/tex]What type of number is {-4}{2}
−4/2
start fraction, minus, 4, divided by, 2, end fraction?
Choose all answers that apply:
(Choice A)
Whole number
(Choice B)
Integer
(Choice C)
Rational
(Choice D)
Irrational
Answers
Answer:
B and C
Step-by-step explanation:
Whole numbers are:
0, 1, 2, 3, 4, 5, 6...
The number we are looking at is -4/2, which is -2. Whole numbers aren't negative. So not choice A.
Integers are:
...-3, -2, -1, 0, 1, 2, 3...
Positives and negatives are included (just no fractions or decimals) So, -4/2 which is -2 IS an integer.
Rational numbers can be written like a ratio (like a fraction) So -4/2 totally IS a rational number.
Irrational numbers are decimal numbers that go on forever without repeating, like pi and sqrt2 and sqrt5. -4/2 is NOT irrational.
If he paints 1/2 of the wall blue, hoy many square feet will be blue?
Answers
To determine how much of the wall is blue we first need to find its area. The wall is a rectangle, then its area is the product of its height and length:
[tex]\begin{gathered} A=(8\frac{2}{5})(16\frac{2}{3}) \\ A=(\frac{42}{5})(\frac{50}{3}) \\ A=(14)(10) \\ A=140 \end{gathered}[/tex]Hence, the area of the wall is 140 square ft. To determine how much of the wall is already painted we multiply this by 1/2, then:
[tex](140)(\frac{1}{2})=70[/tex]Therefore, 70 square ft are blue.
NEED HELP ASAP
What is the value of X? Justify each step
Answers
The value of x = 3 ,where ,
AC = 32 , AB = 2x , BC = 6x + 8 .
Solution:Here given,
AB = 2x
BC = 6x + 8
AC = 32
AC = (AB + BC) (Rule of addition).
So ,
2x + 6x + 8 = 32 (by applying substitution rule) .
In the equation AB + BC = AC, substitute for AB, BC, and AC.
Simplifying,
8x + 8 = 32
2x + 6x + 8 = 32 (when simplified by incorporating similar terms).
8x = 24
8x = 32 - 8
8x = 24
On dividing both sides by 8
8x / 8 = 24/8
x = 3
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How do I know which score is the highest frequency? how do I figure the scores had a frequency of 2?
Answers
To determine the score which presents the highest frequency, we need to check the last column, the frequency one, and find the highest value among them. The score which is in the same row that this value will be the score with the highest frequency.
In the present problem, there are values of frequency equal to 1, 2, 3, and 4. The one with frequency 4 is the one with the highest frequency. (8th row). And the Score related to it is Score 8.Once we check the frequency column once again, we see that the 2nd, the 4th, and the 7th rows have a frequency equal to 2.
Checking the Scores of the related rows, we are able to say that the scores with frequency 2 are: 2, 4, and 7.For which equation will x=-2 make the equation true? Equation 1: 2.4x+ 2.6 = 17 Equation 2: 16 = -8(-6 - 2x) X - 4 Equation 3: 3 Equation 4: -6 = -5x - 3+ Óx
Answers
We need to check each of the options to determine which equation is true
x = -2
equation1: 2.4x + 2.6 = 17
2.4(-2) + 2.6 = 17
-4.8 + 2.6 = 17
-2.2 = 17
equation2: 16 = -8(-6 - 2x)
16 = -8(-6 -2(-2))
16 = -8(-6+4)
16 = -8(-2)
16 = 16
equation 3: 3
1. Jessica finishes her book in 2 1
3
hours. Eric takes 11
2
times longer than
Jessica to finish his book.
This model represents the amount of time Jessica takes to finish her
book. It has a width of 1 and a length of 2 1
3
. The model is 2 1/3 out of 3
Answers
The time taken for Eric to finish the book is 3 1/2 hours.
What is a fraction?A fraction simply means a part of a whole. It van also refer to any number of equal parts.
The information illustrated that Jessica finishes her book in 2 1 / 3 hours and that Eric takes 1 1 / 2
times longer than Jessica to finish his book.
In this case, the time that was used by Eric will be the multiplication of the fractions given. This will be illustrated as:
= 2 1/3 × 1 1/2
Change to improper fraction
= 7/3 × 3/2
= 7 / 2
= 3 1 / 2
Eric used 3 1/2 hours.
This illustrates the concept of multiplication of fractions.
The complete question is written below.
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Jessica finishes her book in 2 1/3.hours. Eric takes 1 1/2 times longer than Jessica to finish his book. How long did Eric take yo finish the book?
Each year, a scientist measures the water level of a local lake. Negative numbers indicatethat the water level is below its historical average. Which list shows the water levels in orderfrom highest to lowest?0.7, 0.38, 0.09, – 0.41, – 0.60.7, 0.38, 0.09,-0.6,- 0.41-0.6, 0.38, – 0.41, 0.09, 0.70.38, 0.09, 0.7,– 0.6 – 0.410.38, 0.7, 0.09 – 0.41, -0.6
Answers
Answer:
-0.6, -0.41, 0.09, 0.39, 0.7
Step-by-step explanation:
Negative numbers: The higher the absolute number, the lower it is. For example, -2 is lower than -1.
Positive numbers: The lower the absolute number, the lower it is. For example, 1 is lower than 2.
In this question:
We have these following values:
0.7, 0.39, 0.09, -0.41, -0.6
Ranking from lowest to highest, it is:
-0.6, -0.41, 0.09, 0.39, 0.7
The formula log in a natural logarithm can be written as?
Answers
Solution:
Given the logarithmic expression:
[tex]\log_545[/tex]According to the change of base formula,
[tex]\log_BA=\frac{\ln A}{\ln\text{ B}}[/tex]Thus, expressing the logarithm expression in a natural logarithm,
[tex]\log_545=\frac{\ln45}{\ln5}[/tex]Hence, we have
[tex]\frac{\ln45}{\ln5}[/tex]Clarence is saving money to buy a skateboard that costs $97.50. He has $15.05 already saved and plans to save $5.50 each week from his allowance. He also earns $15.60 every two weeks for walking dogs. Suppose Clarence wants to spend some of the money from walking dogs on other things. To the nearest dollar, how much would he need to save from walking dogs each week in order to buy the skateboard 4 weeks earlier than if he just saves his allowance?
Please explain. Thanks!
Answers
Answer:88
Step-by-step explanation:
Select the correct answer from the drop-down menu.Find the polynomial,{4'" is the solution set of
Answers
Let P(x) be the polynomial such that the given set is its solution set.
Now notice that:
[tex]\begin{gathered} x=-\frac{1}{3}\Rightarrow x+\frac{1}{3}=0\Rightarrow3x+1=0, \\ x=4\Rightarrow x-4=0. \end{gathered}[/tex]Therefore (x-4) and (3x+1) divide to P(x), then:
[tex]\begin{gathered} Exists\text{ k such that:} \\ P(x)=k(x-4)(3x+1). \end{gathered}[/tex]Simplifying the above result we get:
[tex]P(x)=k(3x^2-11x-4).[/tex]Setting k=1 we get that:
[tex]P(x)=3x^2-11x-4.[/tex]Answer: Second option.
Line / contains the points (-4, -1) and (1, 1) asshown below.432-10x-212-1P-2--3-Line m will be drawn perpendicular to line I andcontaining point P. Identify the coordinates ofanother point on line m.(-1, 4)O (1,3)(-2,-4)(5,3)
Answers
Let's begin by listing out the information given to us:
Line m is perpendicular to line P
Line P: (x, y) = (-4, -1), (1, 1)
We will proceed to calculate for the slope of the line P (as shown below):
Slope (m) = Δy/Δx
Slope (m) = (1 - - 1)/(1 - - 4) = 2/5
Slope (m) = 2/5
The slope of a parallel line is the negative reciprocal of the slope of the line.
Line m: slope (m) = -1/(2/5) = -5/2
Line m: slope (m) = -5/2
We calculate for the equation of the line using the point-slope equation. We have
y - y1 = m(x - x1) ⇒
(x1, y1) = (1, 1)
y - 1 = 2/5 (x - 1) ⇒ y - 1 = 2/5x - 2/5
y = 2/5x - 2/5 + 1
y = 2/5x + 3/5
We will proceed to put the value of the new slope into the equation. We have:
y = -5/2x + b ; (x, y) = (1, 1) ⇒ 1 = -5/2(1) + b
⇒ b = 5/2
Substitute the value of b into the point-slope equation, and we obtain the equation of line m. We have:
y = -5/2x + 5/2
What is the Y intercept of the graph below? A. (0,-2)B. (0,-4) C. (0, 2) D. (0,4)
Answers
Recall that the y-intercept of a graph is the point where the graph intersects the y-axis.
From the given graph we get that the line intersects the y-axis at (0,2).
Answer: Option C.
If we have a system of two linear equations in two variables that has no solution, what would we see on the graph?
Answers
Answer:
The graph will have two lines which will never intersect