A rectangular page is to contain square inches of print. The margins on each side are to be inches. Find the dimensions of the page such that the least amount of paper is used.
Answers
Let l represent the length of the printed rectangular region of the page.
Given that the area of the printed rectangular region is 36, then
width of the printed portion or region = 36/l
The margin left on both sides is 1.5 inches. Thus,
length of page = l + 1.5(2) = l + 3
width of page = 36/l + 1.5(2) = 36/l + 3
Area = length x width
Area = (l + 3)(36/l + 3)
Area = 36 + 3l + 108/l + 9
Area = 36 + 9 + 3l + 108/l
A(l) = 45 + 3(l + 36/l)
We would minimise A(l)
For Amin, A'(l) = 0
3(1 - 36/l^2) = 0
3 = 0 or 1 - 36/l^2 = 0
1 = 36/l^2
l^2 = 36
l = ±√36
l = ±6
Also,
A''(l) > 0
A''(l) = 3(0 - 36(-2)l^-3) = 72/l^3
Substituting l = 6,
72/6^3 > 0
Thus,
l = 6 gives Amin
The dimensions would be
length = l = 6 + 3 = 9
width = 36/6 + 3 = 6 + 3 = 9
Length = 9 inches
width = 9 inches
Related Questions
How many minutes are in 584 miles?
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mark me best !!!
[tex]1/2 \: of \: 4 \: \: 2/9[/tex]
Answers
The given expression is:
[tex]\frac{1}{2}\;of\;4\frac{2}{9}[/tex]The term 'of' in algebra means multiplication. So the expression becomes:
[tex]\frac{1}{2}\times4\frac{2}{9}[/tex]Write the mixed fraction as an improper fraction:
[tex]\frac{1}{2}\times\frac{38}{9}[/tex]Simplify the expression:
[tex]\frac{1}{\cancel{2}}\times\frac{\cancel{38}19}{9}=\frac{19}{9}=2\frac{1}{9}[/tex]The answer is 2 1/9.
Is the sequence 0.5, 2, 8, 32, 128 an arithmetic or geometric sequence? What number does the sequence have a common difference or ratio of?
Answers
a geometric sequence
The sequence have a common ratio of 4
Explanation:Sequence: 0.5, 2, 8, 32, 128
An arithmetic sequence is gotten using a common difference
common difference = next term - previous term
common difference = 2 - 0.5 = 1.5
common difference = 8 - 2 = 6
common difference is not the same. Hence, it is not arithmetic sequence
A geometric sequence is determined using a common ratio
common ratio = next term/previous term
common ratio = 2/0.5 = 4
common ratio = 8/2 = 4
common ratio is the same. Hence, it is a geometric sequence
The sequence have a common ratio of 4
B(-3, 2) is dilated by a factor of 2. The image is located at (_____),. (_______)
Answers
B(-3, 2) dilated by a factor of 2
To find the dilation, we multiply each value by 2:
B(-3, 2) Becomes (-6, 4)
Answer:
-6, 4
Select whether each statement is true or false about the time spent waiting.
Answers
Answer:
• False
,• False
,• True
,• False
Explanation:
Statement 1
The average time-actual time spent waiting at Table A = 2½
[tex]\begin{gathered} 25-\text{Actual Time=2}\frac{1}{2} \\ \text{Actual Time at Table A =}25-\text{2}\frac{1}{2}=22\frac{1}{2} \end{gathered}[/tex]The average time-actual time spent waiting at Table B = -3½
[tex]\begin{gathered} 25-\text{Actual Time=-3}\frac{1}{2} \\ \text{Actual Time at Table B =}25+3\frac{1}{2}=28\frac{1}{2} \end{gathered}[/tex]The first statement is False.
Statement 2
[tex]\begin{gathered} \text{Actual Time at Table C=}25-4.5=20.5 \\ \text{Actual Time at Table D=}25+5.75=30.75 \end{gathered}[/tex]The second statement is False.
Statement 3
[tex]\begin{gathered} \text{Actual Time at Table C=}20.5 \\ \text{Actual Time at Table A=}22.5 \end{gathered}[/tex]The third statement is True.
Statement 4
The customer at Table D spent the most time waiting.
The 4th statement is False.
If u = 2i – j; v = -5i + 4j and w = j find -2u. (v + w).
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To find -2u. (v + w).
First replace the values and then use the order of operations:
-2u. (v + w)
-2(2i – j)(( -5i + 4j)+( j)=
-2(2i – j)(-5i +5j)
-4i +5j(-5i +5j)
20i² -20ij - 25ij + 25j²
20i² -45ij + 25j²
Explain what other information is needed to prove these triangles congruent using the given reasons. (1 pt. each) a) SSS b) SAS c) AAS d) ASA e) HL
Answers
Solution
For this case we can see that :
AB = DE
We also know that :
m < ABC = m < EDF
For this case we have:
a) SSS
We need to have the hypothenuse congruent or the other side congruent in both triangles
b) SAS
We just need to have BC= DF
c) AAS
We need to satisfy:
m< BCA = m< DFE
d) ASA
We need to satisfy:
me) HL
We need to satisfy:
AC = EF
Write the slope-intercept for of the equation of the line through the given point with thegiven slope.through (2,1) and slope = 3
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The equation of a line in slope-intercept form, is:
[tex]y=mx+b[/tex]Let X represent the number of tires with low air pressure on a randomly chosen car.
Answers
Answer:
[tex]\text{ P\lparen Greater than or equal to zero\rparen = 1}[/tex]Explanation:
Here, we want to get the probability of X greater than or equal to zero
Mathematically, what we have to do here is to add the probabilities of values for which we have X equals zero or greater
We have this as:
[tex]\begin{gathered} \text{ P\lparen Greater than or equal to zero\rparen = P\lparen0\rparen + P\lparen1\rparen + P\lparen2\rparen + P\lparen3\rparen + P\lparen4\rparen} \\ =\text{ 0.1 + 0.3 + 0.2 + 0.1 + 0.3 = 1.0} \end{gathered}[/tex]Question 6 (1 pointDetermine the value of x for the triangle below if K is the incenter.EG9x16X =DKH184x + 9
Answers
Solution:
To find the value of x
Using SAS congruency rule
[tex]9x-16=4x+9[/tex]Solving for x
[tex][/tex]Solve the following problems by writing an equation a) what is 80% of 9?b) 16 is what percent of 400?
Answers
a) 7.2 is 80% of 9
b) 16 is 4% of 400
Here, we want to solve the problems by writing an equation
a) Let x be the 80% of 9
Thus, we have it that;
[tex]\begin{gathered} \frac{80}{100}\times9\text{ = x} \\ \\ \frac{720}{100}\text{ = x} \\ \\ 100\times\text{ x = 720} \\ 100x\text{ = 720} \\ \text{ x = }\frac{720}{100} \\ x\text{ = 7.2} \end{gathered}[/tex]80% of 9 is 7.2
b) Let the percentage of 400 that is 16 be x
Thus, mathematically;
[tex]\begin{gathered} \frac{16}{400}\times100\text{ = x} \\ \\ 16\times100\text{ = 400}\times x \\ 1600\text{ = 400x} \\ x\text{ = }\frac{1600}{400} \\ x\text{ = 4\%} \end{gathered}[/tex]4x+4y+z=242x-4y+z=05x-4y-5z=12
Answers
Which expression is equivalent to 7(5a +4) +8?
Answers
7(5a +4) +8
expand
= 35a + 28 + 8
= 35a + 36
write a system of equations that could be used to determine the number of bracelets made and the number of necklaces made. Define the variables that you used to write the system.
Answers
Step 1: Problem
write a system of equations that could be used to determine the number of bracelets made and the number of necklaces made. Define the variables that you used to write the system.
Step 2: Concept
Form a system of linear equations
Step 3: Method
Let m = number of bracelet
Let n = number of necklace
6 grams of gold in each bracelet and 16 grams of gold in each necklace.
Total weight of gold = 178 grams
6m + 16n = 178 .................................. 1
There are 7 necklaces more than the bracelet
n - m = 7 ..................................... 2
Step 4: Final answer
Let m = number of bracelet
Let n = number of necklace
System of equations
6m + 16n = 178
n - m = 7
You draw one card from a 52-card deck. Then the card is replaced in the deck and the deck isshuffled, and you draw again. Find the probability of drawing a nine the first time and a diamond thesecond time.The probability of drawing a nine the first time and a diamond the second time is(Type an integer or a simplified fraction.)
Answers
GIVEN:
You draw one card from a 52-card deck. Then the card is replaced in the deck and the deck is shuffled, and you draw again.
Required;
Find the probability of drawing a nine the first time and a diamond the
second time.
Step-by-step solution;
To solve this math problem, we take note of the following;
A standard 52-card deck contains,
[tex]\begin{gathered} 13\text{ }Hearts \\ \\ 13\text{ }Diamonds \\ \\ 13\text{ }Clubs \\ \\ 13\text{ }Spades \end{gathered}[/tex]Each of the suits (that is, hearts, diamonds, clubs and spades) has a 9 card each which means there are four 9s in the entire 52-card deck.
Hence, the probabilty of drawing a 9 is given as follows;
[tex]\begin{gathered} Probability\text{ }of\text{ }[event]=\frac{number\text{ }of\text{ }required\text{ }outcomes\text{ }}{number\text{ }of\text{ }all\text{ }possible\text{ }outcomes} \\ \\ Probability\text{ }[9]=\frac{4}{52} \\ \\ Probability\text{ }[9]=\frac{1}{13} \end{gathered}[/tex]Note that the card is replaced before the commencement of the next experiment.
This means for the second draw, we have a complete 52-card deck.
Therefore, for the probability of drawing a diamond, note that we have 13 diamonds in all. Hence,
[tex]\begin{gathered} Probability\text{ }[diamond]=\frac{13}{52} \\ \\ Probability\text{ }[diamond]=\frac{1}{4} \end{gathered}[/tex]The probability of event A and event B is the product of probabilities. This means;
[tex]Probability\text{ }of\text{ }A\text{ }and\text{ }Probability\text{ }of\text{ }B=P[A]\text{ }\times\text{ }P[B][/tex]Therefore, the probability of drawing a nine the first time and a diamond the second time is given as;
[tex]\begin{gathered} P[9]\text{ }and\text{ }P[diamond]=\frac{1}{13}\times\frac{1}{4} \\ \\ P[9]\text{ }and\text{ }P[diamond]=\frac{1}{52} \end{gathered}[/tex]ANSWER:
[tex]\frac{1}{52}[/tex]Solve for the indicated variable. Include all of your work in your answer. Submit your solution.A=_h_(a+b); for h 2
Answers
Given
[tex]\begin{gathered} A=\frac{h}{2}(a+b) \\ \frac{A}{(a+b)}=\frac{h}{2} \\ h=\frac{2A}{a+b} \end{gathered}[/tex]
The answer would be h = 2A/(a+b)
You pick 4 cards from a deck without replacing the card each time before picking the next card.What is the probability that all 4 cards are Jacks?Leave your answer as a fraction.
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We know that in a deck of cards there is 52 cards, 4 been jacks, so if we want to take to take all four jacks in a row, the probability is:
[tex]p(\text{jack and jack and jack and jack)=}\frac{4}{52}\times\frac{3}{51}\times\frac{2}{50}\times\frac{1}{49}=\frac{24}{6497400}[/tex]I need help solving thisIt’s from my ACT prep guide
Answers
Given the expression:
[tex]\frac{\tan(\frac{-2\pi}{3})}{\sin(\frac{7\pi}{4})}-\sec (-\pi)\text{ =}[/tex][tex]\frac{\sqrt[]{3}}{\frac{-\sqrt[]{2}}{2}}-(-1)\text{ =}[/tex][tex]-\frac{\sqrt[]{3}}{\frac{\sqrt[]{2}}{2}}+1\text{ =}[/tex][tex]-\frac{2\sqrt[]{3}}{\sqrt[]{2}}+1=[/tex][tex]-\sqrt[]{6}+1[/tex]Using the net below, find the surface area of the rectangular prism. 7 cm 3 cm 7cm 5 cm 15cm 13cm 3 cm 3 cm Surface Area Il 1 [?] cm
Answers
In this case, we have a rectangular prism with three pairs of equal faces, equal faces are located on opposite sides like the front and back faces or top and bottom faces. Then in order to calculate the total surface area of the prism we just have to find the area of each different face, multiply it by 2 so we take into account its equal pair and then sum them all up, like this:
Total surface area = 2×(7×5) + 2×(3×5) + 2×(7×3) = 2×35 + 2×15 + 2×21 = 70 + 30 + 42 = 142
Then, the total surface area of the given prism is 142 cm²
translate the following phrase to a mathematical expression and then
Answers
1) Since we have " For n=5, the quotient of twenty and a number."
Given that Quotient is yielded by the division we can state after examining the options:
20 ÷ n, 4
What is the order of rotational symmetry of a figure out the right
Answers
ANSWER
4
EXPLANATION
The order of rotational symmetry is the number of times a figure fits into itself in one complete rotation. In this case,
Hence, the order of rotational symmetry is 4, because the figure fits 4 times into itself in one complete rotation.
In how many different orders can you line up 3 cards ona table?
Answers
The number of cards is 3. the number of orders in which the cards can be lined up is 3!
3! = 3 * 2 * 1 = 6
6 different orders
Assuming the cards are A, B and C. The orders are ABC, ACB, BCA, CBA, CAB, BAC. That is 6 different orders rent orders
15y + 25 What is the Greatest Common Factor?Expand the expression.
Answers
ANSWER
5(3y + 5)
EXPLANATION
Both terms of this expression have constants: 15 and 25. Let's write the multiples of each,
[tex]\begin{gathered} 15\colon3\times5 \\ 25\colon5\times5 \end{gathered}[/tex]The greatest common factor between these two is 5. To take it out as a common factor, we have to divide the two constants from the expression by 5,
[tex]15y+25=5(3y+5)[/tex]Im confused on how to do these questions. Transformation of functions.
Answers
Given a parent function below
[tex]f(x)=|x|[/tex]Find the absolute value vertex. In this case, the vertex for
[tex]y=|x|[/tex]To find the x coordinate of the vertex, set the inside of the absolute value x equal to 0. In this case x = 0
Replace the variable x with 0 in the expression
y = |0|
The absolute value is the distance between a number and zero. The distance between 0 and 0 is 0
y = 0
The absolute value vertex is (0,0)
The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.
Interval Notation:
[tex](-\infty,\infty)[/tex]For each x value, there is one y value. Select few x values from the domain. It would be more useful to select the values so that they are around the x value of the absolute value vertex.
[tex]\begin{gathered} y=|x|,\text{ i.e} \\ x=-3,y=3 \\ x=-2,y=2 \end{gathered}[/tex](b) The domain of a graph consists of all the input values shown on the x-axis which are -3, -2, -1, 0, 1, 2, 3
(c) The range is the set of possible output values, which are shown on the y-axis which are 3, 2, 1, 0, 1, 2, 3
-44 ∘C to Fahrenheit. Express your answer using two significant figures.
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The relationship between the Celcius scale and the Fahrenheit scale is given by the equation:
[tex]^oF=1.8^0C+32[/tex]Thus,
[tex]\begin{gathered} ^0F=1.8(-44)+32^{} \\ ^0F=-79.2+32 \\ ^0F=-47.2^0F \end{gathered}[/tex]Find the reference angle to 5 pi and full in the calculations.
Answers
The angle given is in the second quadrant, this comes from the fact that is more than pi/2 but less than pi.
Then the reference angle is given by:
[tex]\alpha=\pi-\frac{5\pi}{8}[/tex]And the angles is:
[tex]\alpha=\frac{3\pi}{8}[/tex]Determine whether each table is a linear quadratic or exponential function
Answers
From top to bottom:
1) Linear
2) Quadratic
3) Exponential
1) Examining the tables, let's begin by focusing on the y-column:
As we can see the difference from each output is the same, namely: -3
So, we can tell this is a Linear Equation.
2) In this one, let's start out counting the first difference. Note that the 1st difference is not constant. Now, let's do the second difference that yields a constant result:
Since the 2nd difference is the same then we can tell that this is a Quadratic function.
3) Finally, let's examine the last table (from top to bottom):
Focusing on the y-values column we can see a ratio of 2 between each value. Each value is twice the preceding one.
So, this is an exponential function.
Answer:
First table models Linear, second quadratic and third one exponential functions.
Step-by-step explanation:
To find whether the given table models a linear function there should be a constant change in y values with the constant change in x values of the table.
We take the example of first table written as linear.
Here change in x values is
6 - 5 = 1
7 - 6 = 1
8 - 7 = 1
Similarly change in y values is
1 - 4 = -3
-2 - 1 = -3
-5 -(-2) = -3
There is a common difference in y values = -3
So the given table models linear function.
We take the second table.
For quadratic function with the constant change in x values, difference of difference in y values is constant.
Change in x - values
6 - 5 = 1
7 - 6 = 1
8 - 7 = 1
Difference in y values
1 - 0 = 1
4 - 1 = 3
9 - 4 = 5
Now difference in difference of y values
3 - 1 = 2
5 - 3 = 2
Here, difference in difference of y values is 2
So the given table models a quadratic equation.
Now we take the third table.
For exponential function in the form of there should be a common ratio in the terms of y values.
So there is a common ratio of 2 in each term.
Therefore, the given table models exponential equation.
First table models Linear, second quadratic and third one exponential functions.
PLS MARK ME AS BRAINLIEST.
2. Solve for x(1/3)(5+7x) = 12 - (1/2)(3-2x)
Answers
To solve the equation below:
[tex]\frac{1}{3}(5+7x)=12-\frac{1}{2}(3-2x)[/tex]eliminate the denominators by multiplying each term on both sides of the equation by the least common denominator or the LCD. In this case, since the least common multiple of 3 and 2 is 6, the LCD must be 6.
[tex]\begin{gathered} 6\lbrack\frac{1}{3}(5+7x)\rbrack=6\lbrack12-\frac{1}{2}(3-2x)\rbrack \\ 6\lbrack\frac{1}{3}(5+7x)\rbrack=6\lbrack12\rbrack-6\lbrack\frac{1}{2}(3-2x)\rbrack \\ 2(5+7x)=72-3(3-2x) \end{gathered}[/tex]Eliminate the parentheses by distributing the numerical values outside the parentheses. In this case, we distribute the 2 to 5 and 7x and then distribute -3 to 3 and -2x.
[tex]\begin{gathered} 2(5)+2(7x)=72-3(3)-3(-2x) \\ 10+14x=72-9+6x \end{gathered}[/tex]Simplify both sides of the equation by combining like terms. Like terms are the terms with the same literal coefficients.
[tex]10+14x=63+6x[/tex]Isolate the variable terms by subtracting 6x and 10 from both sides of the equation.
[tex]\begin{gathered} 10+14x-6x-10=63+6x-6x-10 \\ 8x=53 \end{gathered}[/tex]Solve for x by dividing both sides of the equation by 8.
[tex]\begin{gathered} \frac{8x}{8}=\frac{53}{8} \\ x=\frac{53}{8} \end{gathered}[/tex]Therefore, the value of x must be 53/8.
I’m having trouble understanding how to solve this problem….A company wants to construct an open rectangular box with a volume of 375in^3 so that the length of its base is 3 times the with. Express the surface area, S, of the box as a function of the with w. Also find the domain.The answer I got is in the picture
Answers
We have:
Volume = V = 375in^3
Length = 3w
Width = w
Area = S
Height = h
Then, the formula of the volume is given by:
[tex]V=length\times width\text{ }\times height[/tex]Substitute the values:
[tex]375=3w\times w\times h[/tex]Solve for h:
[tex]\begin{gathered} 375=3w^2h \\ \frac{375}{3w^2}=\frac{3w^2h}{3w^2} \\ h=\frac{125}{w^2} \end{gathered}[/tex]Next, the surface area, S, of the box is:
[tex]\begin{gathered} S=area\text{ of base}+2area\text{ vertical side + 2area other vertical side} \\ S=3w(w)+2(3w)(h)+2(w)(h) \end{gathered}[/tex]Simplify:
[tex]S=3w^2+6wh+2wh=3w^2+8wh[/tex]Substitute the value of h:
[tex]\begin{gathered} S=3w^2+8w(\frac{125}{w^2}) \\ S=3w^2+\frac{1000}{w} \end{gathered}[/tex]Answer:
[tex]S(w)=3w^2+\frac{1000}{w}[/tex]10 is 4% of what number
Answers
EXPLANATION
Dividing 10 by 4 (4%=0.04) and multiplying by 100 give us the corresponding 100% number as shown as follows:
[tex]10\cdot\frac{100}{4}=250[/tex]The answer is 250