f(x) = 4x
To find the inverse of a function, you have to replace every x with an y, as follows:
f(x) = 4y
then, replace f(x) with an x,
x = 4y
and now, isolate y,
x/4 = y
Finally, replace y whit h(x),
h(x) = 1/4x
an open-top box is to be constructed from a sheet of tin that measures 22 inches by 14 inches by cutting out squares from each corner. let V(x) denote the volume of the resulting box. step 1 of 2: write V(x) as a product of linear factorsstep 2 of 2: among the values of x for which V(x)=0, which are physically possible?
It is given that an open-top box is to be constructed from a sheet of tin that measures 22 inches by 14 inches by cutting out squares from each corner
Let x be the measure of the side of the square.
Length of the resulting box =22-2x
Width of the resulting box=14-2x
Height of the resulting box=x
The volume of the box is
[tex]V=\text{length }\times width\times height[/tex]Substitute values, we get
[tex]V(x)=(22-2x)(14-2x)x[/tex][tex]=(22-2x)(14x-2x^2)[/tex][tex]=22\mleft(14x-2x^2\mright)-2x\mleft(14x-2x^2\mright)[/tex][tex]=22\times14x-22\times2x^2-2x\times14x-(-2x)2x^2[/tex][tex]=308x-44x^2-28x^2+4x^3[/tex][tex]V(x)=4x^3-72x^2+308x[/tex]Putting V(x)=0, we get
[tex]4x^3-72x^2+308x=0[/tex][tex]4x(x^2-18x+77)=0[/tex][tex]4x=0,(x^2-18x+77)=0[/tex]Here x is not zero
[tex]x^2-18x+77=0[/tex][tex]x^2-11x-7x+77=0[/tex][tex]x(x^{}-11)-7(x-11)=0[/tex][tex](x^{}-11)(x-7)=0[/tex][tex](x^{}-11)=0\text{ or }\mleft(x-7\mright)=0[/tex][tex]x^{}=11\text{ or }x=7[/tex]The height of the box is 11 or 7
If the height is 11 inches, substitute x=11 in the length equation, we get
[tex]\text{length =22=2x=22-2}\times11=22-22=0[/tex]we get a length is 0, so it is not possible to make the box.
Setting x=7, the height of the box is 7 inches.
[tex]\text{Length =22-2x=22-2}\times7=22-14=8inches[/tex][tex]\text{width =14-2}\times7=14-14=0[/tex]we get a width is 0, so it is not possible to make the box.
Hence among the values of x for which V(x)=0 is not physically possible.
Tilusorativ dhernatcs 8. Here is a graph of the equation 3x-2y = 12. 2 Select all coordinate pairs that represent a solution to the equation. O A. (2,-3) B. (4, 0) C. (5,-1) D. (0, -6) E. (2, 3)
Let's evaluate every pair:
(2,-3):
[tex]3(2)-2(-3)=6+6=12=12[/tex](2,-3) represent a solution
-----------------------------------------------------------------------------------------------------
(4,0):
[tex]3(4)-2(0)=12-0=12=12[/tex](4,0) represent a solution
---------------------------------------------------------------------------------------------------------------
(5,-1):
[tex]3(5)-2(-1)=15+2=17\ne12[/tex](5,-1) Don't represent a solution
-----------------------------------------------------------------------------------------------------------
(0,-6):
[tex]3(0)-2(-6)=0+12=12=12[/tex](0,-6) Represent a solution
--------------------------------------------------------------------------------------------------
(2,3):
[tex]3(2)-2(3)=6-6=0\ne12[/tex](2,3) Don't represent a solution
WZ = 32, YZ = 6, and X is the midpoint of WY. Find WX.
We are given the length of two segments:
WZ = 32
YZ = 6
and we are told that x is the midpoint of the segment WY
We are asked to find the length of the segment WX
Notice that the total length of the segment WZ is 32. from the point Y to the point Z we have 6 units. therefore, between W and Y there is 32 - 6 = 26 units.
SInce X is the midpoint of the distance between W and Y, then it has to cut the segment WY (26 units long) in two equal parts, each of length 13 units (half of 26).
Therefore, WX must be of length 13 units.
Content attributionQUESTION 5.1 POINTTranslate and solve: 6 greater than b is greater than 84.Give your answer in interval notation.Provide your answer below:
6 greater than b is
[tex]=b+6[/tex]6 greater than b is greater than 84. is represented as
[tex]b+6>84[/tex]Step :Subtract 6 from both sides
[tex]\begin{gathered} b+6>84 \\ b+6-6>84-6 \\ b>78 \\ \end{gathered}[/tex]Therefore,
[tex]\begin{bmatrix}\mathrm{Solution\colon}\: & \: b>78\: \\ \: \mathrm{Interval\: Notation\colon} & \: \mleft(78,\: \infty\: \mright)\end{bmatrix}[/tex]Hence,
The interval notation is (78,∞)
Which scenario has more arrangements?:2:• 5 letter arrangements using the letters from the word CHAMPION.• 4 letter arrangements using the letters from the word ABRUTPING.. The total number of ways the word EDMONTON can be arranged.Prepare your work on paper, take an image and post in the answer box provided.s:ParagraphVB1UAVLato (Recom19pxVEa5 с:
This is a simple question to solve. Let's first calculate all the arrangements for the first case to understand the logic:
As we can see above, once we have 8 letters, and we need to calculate the numbers of arrangements with 5 letters, for the first letter we have 8 possible letters, for the second letters we have 7 possible letters once one letter was used for the first one. For the third letter we have 6 possible letters, for the fourth, 5 possible letters and for the fifth, 4 possible letters. So, we just multiply 8*7*6*5*4 = 6720 possible arrangements.
For the second situation we can follow the same logic:
And finally for the third situation we have:
As we can see above, the third scenario has more arrangements.
Question 1 Business Analytics
The responses to the linear optimization questions are;
Question 1
The optimal daily profit is $380
Question 2
The combination of x and y that yield the optimal value is the option;
x = 0, y = 3
What is linear optimization or optimization?Linear programming is a method by which the optimal solution can be obtained from a model represented mathematically and in which the constraints of the model have linear relationships.
Question 1
Let B represent the number of bear claws, C the almond-filled croissant, F represent the flour, Y represent the amount of yeast and A represent the number of almonds.
The amount of ingredient per each produce is therefore;
B = 6·F + 1·Y + 2·A
C = 3·F + 1·Y + 4A
The amount of ingredient available for the days production is as follows;
Ingredient available; 6600·F + 1400·Y + 4800·A
The constraints are therefore
6·B + 3·C ≤ 6,600
B + C ≤ 1,400
2·B + 4·C ≤ 4800
The maximizing function is therefore;
Profit = 0.2·B + 0.3·F
The equations of the lines are therefore;
B = 1,100 - 0.5·C
B = 1400 - C
B = 2400 - 2·C
The vertices of the feasible region are;
(0, 1100), (600, 800), (1000, 400), 1200, 0)
The values of the maximizing function at the vertices of the feasible region are;
Profit, P = 0.2×1100 + 3×0 = 220
At point (600, 800), P = 0.2×800 + 0.3×600 = 340
At point (1000, 400), P = 0.2×400 + 0.3×1000 = 380
At point (1200, 0), P = 0.2×0 + 0.3×1200 = 360
The maximum profit is $380, obtained when 400 Bear claws and 1000 almond filled croissants are producedQuestion 2
Maximize $3·x + $15·y
Subject to the following constraints;
2·x + 4·y ≤ 12
5·x + 2·y ≤ 10
x, y ≥ 10
The equations are therefore;
4·y ≤ 12 - 2·x
y ≤ 3 - 0.5·x...(1)
5·x + 2·y ≤ 10
2·y ≤ 10 - 5·x
y ≤ 5 - 2.5·x...(2)
x ≥ 10, y ≥ 10
The coordinates of the vertices of the feasible region are;
(0, 3), (1, 2.5), and (2, 0)
The values of the maximizing function are therefore;
At (0, 3), M = $3 × 0 + $15 × 3 = $45
At (1, 2.5), M = $3 × 1 + $15 × 2.5 = $40.5
At (2, 0), M = $3 × 2 + $15 × 0 = $6
The combination of x and y that yield the optimum is therefore;
(x, y) = (0, 3)
x = 0, and y = 3
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1 + z/3 + 2w. Which part of the expression is a product of two factors? Describe it's part e form quotient of two factors? Describe its parts.
2w is the part of two factors.
The part of the expression is a product of two factors.
The expression we have is:
[tex]1 + \frac{z}{3}+2w[/tex]
Let's analyze the parts of this expression.
The first term of the expression is a constant term: 1.
1 is not a product of two factors, so this is not the answer.
The second term of the expression is: z/3.
This part of the expression is a division or quotient between z and 3. Thus, since it is a division and not a product, this is also not the answer we are looking for.
The third term of the expression is: 2w
In this case, the term 2w is a product between "2" and "w". Thus, 2w is a product of two factors. The parts of this product are 2 and which when multiplied result in 2w.
Hence the answer is 2w is the part of two factors.
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you Owens 15 books before Christmas,but after Christmas you now own 21 books. is this a decrease or increase explain.find the percent of change
Let's begin by listing out the given information:
Before Christmas: 15 books
After Christmas: 21 books
This is an increase
The percentage increase is given by % increase = Increase ÷ Original Number × 100:
[tex]\begin{gathered} \text{\%}increase=Increase\div OriginalNumber\times100\text{\%} \\ \text{\%}increase=\frac{21-15}{15}\times100\text{\%} \\ \text{\%}increase=\frac{6}{15}\times100\text{\%}=40\text{\%} \\ \text{\%}increase=40\text{\%} \end{gathered}[/tex]Don Stone obtained an $8.500 installment loan at 14% for 42 months. The loan's balance after 26 payments is 3.733.55. What is the interest for payment 27?
Given:
The unpaid balance after the 26 payments is $3,733.55.
Therefore, the interest for payment 27 will be
[tex]14\text{ \% of \$3733.55}[/tex]Evaluating
[tex]\frac{14}{100}\times3733.55=0.14\times3733.55=522.697\approx522.70(nearest\text{ cent)}[/tex]Hence, the interest for payment 27 is $522.70.
7:20 A.M to 9:49 A.M
We can add the minutes from 7:20 AM to the next hour (8:00 AM), that is 40 minutes.
Then, from 8:00 AM to 9:00 AM we have 60 more minutes.
Then, from 9:00 AM to 9:49 AM we have 49 additional minutes.
We add all three segments as:
[tex]40+60+49=149[/tex]As 60 minutes is 1 hour, 120 minutes is 2 hours. Then, 149 minutes are 2 hours and 29 minutes.
Answer: the time elapsed us 149 minutes (or 2 hours and 29 minutes)
I have an advanced trig equation it's a word problem about non-right triangles it's just for practice not for a graded homework or a quiz. it is a word problem and a picture is included.
Using trigonometric equations we calculate the length of the guy wire from the tower is approximately 1306.5 feet .
The given information about the Tower are :
ED = 175 feet
∠DAB = 57°
∠CED = 30°
Now in the triangle ADB we have ∠ABD = 90° (refer to diagram below)
Therefore ∠ADB = 180° - (57° + 90°) = 33°
Now ∠ EDC = 180° - 33° = 147 °
Hence in triangle EDC ,
∠ECD = 180° - (147°+ 30°) = 3°
Now we will use the law of sines to find the height of the tower.
We know from the law of sines that in ΔEDC
[tex]\frac{ED}{sin\angle C} =\frac{CD}{sin\angle E} =\frac{CE}{sin\angle D}[/tex]
now we will use this to find the height of the tower which is CD
∴ CD = sin °30 × 175 ÷ sin 3°
CD = 1671.8907...
CD ≈ 1671.9 feet.
length of the guy wire = CE
∴CE = sin 157° × 175 ÷ sin 3°
CE = 1306.5195...
CE ≈ 1306.5 feet
Hence the height of the tower is 1671.9 feet and the length of the guy wire is 1306.5 feet.
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Using the data in the image could you help with this question State some possible causes of the error in your measured value. What techniques could be used to correct it?
Answer:
Step-by-step explanation:
State what additional information is required in order to know that the triangles are congruentfor the reason given.11) SSS12) SAS
11) SSS
The Side Side Side (SSS) theorem states that if three given sides of one triangle are equal to the three sides of another triangle, both triangles are congruent.
Since we are given two equal sides (VC = DC) and one common side(CE), to prove this SSS theorem, the additional information required is to indicate the third side of both triangles are equal (VE = DE).
We have the figure below:
Where:
VC = DC
CE = CE
The additional information required is:
VE = DE
ANSWER:
VE = DE
Simplify the expression. 2m - 8 - 2m - 1
Rewrite each equation in slope intercept form, if necessary. Then determine whether the lines are parallel. Explain3x+4y = 86x+3y = 6Are these lines parallel?A.B.C.D.(look at image for answer choices)
We can rewrite the next equations in the slope-intercept form:
The first equation:
[tex]3x+4y=8\Rightarrow4y=8-3x\Rightarrow y=\frac{8}{4}-\frac{3}{4}x\Rightarrow y=2-\frac{3}{4}x\Rightarrow y=-\frac{3}{4}x+2[/tex]The second equation:
[tex]6x+3y=6\Rightarrow3y=6-6x\Rightarrow y=\frac{6}{3}-\frac{6}{3}x\Rightarrow y=2-2x\Rightarrow y=-2x+2_{}[/tex]As we can see, the slope of the first line is m = -3/4, and the slope of the second line is m = -2. Then, since the slope is different, these lines are not parallel (Option C).
Given a triangle ABC at points A = ( - 6, 3 ) B = ( - 4, 7 ) C = ( - 2, 3 ), and a first transformation of up 2 and right 3, and a second transformation of down 1 and left 6, what would be the location of the final point B'' ?
B'' = (-7, 8)
Explanation:The points of the triangle are:
A = ( - 6, 3 ) B = ( - 4, 7 ) C = ( - 2, 3 )
The first transformation:
2 units up and 3 units right
B' = (-4+3, 7+2)
B' = (-1, 9)
Second transformation:
1 unit down, 6 units left
B'' = (-1-6, 9-1)
B'' = (-7, 8)
Find the missing the side length leave the answer as radical form. Question 3.
In order to calculate the value of x, we can use the cosine relation of the angle 60°.
The cosine is equal to the length of the adjacent leg to the angle over the length of the hypotenuse.
So we have:
[tex]\begin{gathered} \cos(60°)=\frac{2}{x}\\ \\ \frac{1}{2}=\frac{2}{x}\\ \\ x=4 \end{gathered}[/tex]Now, to calculate the value of y, we can use the tangent relation of the angle 60°.
The tangent is equal to the length of the opposite leg to the angle over the length of the adjacent leg to the angle.
So we have:
[tex]\begin{gathered} \tan(60°)=\frac{y}{2}\\ \\ \sqrt{3}=\frac{y}{2}\\ \\ y=2\sqrt{3} \end{gathered}[/tex]2. Ashley purchased a new television for$2400 and a surround sound for $980.The sales tax is 7%. Find the totalamount of money that Ashley will payfor her two items including tax.
Ashley has to pay $2400 + $980 = $3380 for both items.
We need to calculate the 7% of this amount to find how much she has to pay in taxes.
[tex]3380\cdot\frac{7}{100}=236.6[/tex]Finally, the total amount she has to pay is $3380 + $236.6 = $3616.6
At the independent record company where Gwen works, the vinyl format has been experiencing a resurgence in popularity. Record sales are increasing by 11% each year. If 19,360 records were sold this year, what will annual sales be in 2 years?If necessary, round your answer to the nearest whole number.
Step 1:
Write the given data
r = 11% = 0.11
P = 19360
t = 2
Step 2:
Apply exponential increase or growth formula
[tex]\begin{gathered} A=P(1+r)^t \\ A\text{ = future value} \\ P\text{ = present value} \\ r\text{ = rate} \\ t\text{ = time} \end{gathered}[/tex]Step 3:
Substitute in the formula
[tex]\begin{gathered} A\text{ = 19360 }\times(1+0.11)^2 \\ A\text{ = 19360 }\times1.11^2 \\ A\text{ = }23853.456 \end{gathered}[/tex]Final answer
23853
During 7 1/2 months of hibernation, a black bear experienced a weight loss of 64.4 lbs. on average what was the bears weight change per month. Round to the nearest tenth.
Hibernation time: 7 1/2 months = 15/2 months
Weight loss: 64.4 lbs
We can calculate the average weight change per month using the equation:
average_weight_loss = weight_loss / time
We know that:
weight_loss = 64.4 lbs
time = 15/2 months = 7.5 months
Then, using the equation above:
average_weight_loss = 64.4 lbs / 7.5 months
average_weight_loss = 8.5867 lbs/month
To the nearest tenth, the average monthly weight loss of the black bear was 8.6 lbs/month.
the line with the slope of 1/5 and passing through the point D(2,2)
Answer:
Y = 1/5 x + 8/5
Explanation:
The equation of a line in slope intercept form is
[tex]y=mx+b[/tex]we are told that the slope of the equation is 1/5; therefore,
[tex]y=\frac{1}{5}x+b[/tex]Furthermore, we are also told that the line passes through (2,2), meaning it should satisfy the condition when y = 2, x = 2
Putting in x =2 and y = 2 in the above equation gives
[tex]2=\frac{1}{5}(2)+b[/tex][tex]2=\frac{2}{5}+b[/tex]subtracting 2/5 from both sides gives
[tex]2-\frac{2}{5}=b[/tex][tex]b=\frac{8}{5}=1.6[/tex]Hence the equation of the line is
[tex]y=\frac{1}{5}x+\frac{8}{5}[/tex]The graph of the equation is given below.
Solve using the elimination method:4x + y + 5z = -40-3x + 2y + 4z = 10x - y - 2z = -2
Let's take
4x + y + 5z = -40 (Eq1)
-3x + 2y + 4z = 10 (Eq2)
x - y - 2z = -2 (Eq3)
Now create a new system using elimination
-2* (4x + y + 5z = -40) (Eq1)
1* (-3x + 2y + 4z = 10) (Eq2)
----------------------------------
-11x - 6z = 90 (Eq4)
Use elimination again
1* (-3x + 2y + 4z = 10) (Eq2)
2* (x - y - 2z = -2) (Eq3)
---------------------------------------
-x = 6 (Eq5)
From Equation 5 we have that
x = -6
Replace the value of x in Equation 4 and clear z
-11(-6) - 6z = 90
-6z = 90 - 66
-6z=24
z = 24/-6
z = -4
Replace x and z in equation 3 and clear y
-6 - y - 2*(-4) = -2
-y + 8= -2 +6
-y = 4 - 8
-y= -4
y = 4
a sea turtle can swim at rate of 20 miles per hour. How many feet per hour can a sea turtle swim
The rate at which turtle can swim is 20 miles per hour or 20 miles in one hour.
For conversion, 1 mile is equal to 5280 foot.
Convert 20 miles per hour in foot per hour.
[tex]\begin{gathered} 20\text{ mile per hour=20 miles per hour}\times\frac{5280\text{ foot per hour}}{1\text{ mile per hour}} \\ =20\cdot5280\text{ foot per hour} \\ =105600\text{ foot per hour} \end{gathered}[/tex]So answer is 105600 foot per hour.
write (2 to the power of -1) to the power of 3 with the same base but one exponent
Explanation
Step 1
[tex](2^{-1})^3[/tex]remember
[tex]\begin{gathered} a^{-n}=\frac{1}{a^n} \\ (a^n)^m=a^{n\cdot m} \\ (\frac{a}{b})^m=\frac{a^m}{b^m} \end{gathered}[/tex]Step 2
solve
[tex]\begin{gathered} (2^{-1})^3 \\ (2^{-1})^3=(\frac{1}{2^1})^3=(\frac{1^3}{2^3})=\frac{1}{8} \end{gathered}[/tex].The 9th-grade students are sellingI chocolate bars for a fundraiser.Each student is encouraged tosell at least 12 chocolate bars.Pam sells 3 bars on Monday and4 bars on Tuesday. Write andsolve an inequality to find the remainingpossible number of bars Pam cansell to reach the goal.
Answer:
The possible number of bars Pam can sell to reach the goal must be at least 5 bars.
Explanation;
Let the remaining number of bars Pam can sell to reach the goal be "x"
If Pam sells 3 bars on Monday and 4 bars on Tuesday, the total number of bars sold will be 3 + 4 = 7bars
Also if each student is encouraged to sell at least 12 chocolate bars, the required inequality expression to solve will be:
[tex]\begin{gathered} 4+3+x\ge12 \\ 7+x\ge12 \\ x\ge12-7 \\ x\ge5 \\ \end{gathered}[/tex]This shows that the possible number of bars Pam can sell to reach the goal must be at least 5 bars.
how do i findFind the domain of f ∘ g in the equation
First, we need to find the composite function of f ∘ g.
We need to write the function f(x) in terms of g(x).
Then:
[tex]\begin{gathered} (fog)(x)=\frac{6}{g(x)+7}=\frac{6}{x+5+7} \\ =\frac{6}{x+12} \end{gathered}[/tex]Now, to find the domain we need to look at the x values that the function can take.
The function is a rational function, then the domain is given using the denominator because it can be equal to zero.
x+12 = 0
x=-12
Therefore, the domain is the interval (-∞.-12)U(-12,∞)
Sharon Nguyen has $25,000 to invest and believes that she can earn 8% compounded semiannually. Find the amount if she invests for 10 years
Solution:
Given:
[tex]\begin{gathered} P=\text{ \$25,000} \\ r=8\text{ \%}=\frac{8}{100}=0.08 \\ t=10\text{years} \\ n=\text{twice a year(semiannually),}n=2 \end{gathered}[/tex]
To get the amount, we use the compound interest formula;
[tex]A=P(1+\frac{r}{n})^{nt}[/tex]
Substituting the given values into the formula,
[tex]\begin{gathered} A=P(1+\frac{r}{n})^{nt} \\ A=25000(1+\frac{0.08}{2})^{2\times10} \\ A=25000(1+0.04)^{20} \\ A=25000(1.04)^{20} \\ A=25000\times1.04^{20} \\ A=\text{ \$54,778.08} \end{gathered}[/tex]
Therefore, the amount after 10 years is $54,778.08
If Ellen's gross pay for a two-week period is $1680.00, what is her net pay?O $1606.92O $168.00O $1341.48O $1478.40
It's important to know that the gross pay refers to money before taxes, while the net pay refers to money after deductions.
Hence, the net payment must be less than $1,680.
Hence, the answer is $1,606.92.Points W, X, and Y are collinear. WY = 25 andthe ratio of WX to XY is 2:3. Find WX.wY
WX is 10
Explanation goes as follows:
WY = 25 from the question given
adding the ratios together, we will have 2+3= 5
WX : XY = 2: 3
To find WX, we will simply say;
WX = 2/5 multiplied by 25
WX = 2/5 x 25
WX = 50/5
WX=10
Like-wise to find WY
we will simply say;
WY = 3/5 multiplied by 25
WY = 3/5 x 25
WY = 75/5
WY =15
Suppose that f is an even function whose domain is the set of all real numbers. Then which of the following can we claim to be true?
If f is an even function, then f can't have an inverse, because even functions don't have inverses. Therefore the correct answer is A.