The arithmetic sequence follows
[tex]a_n=-30+(n-1)5[/tex]Lets see if the 10th term is 15 by replacing the n for 10
n=10
[tex]a_{10}=-30+(10-1)5[/tex][tex]a_{10}=-30+(9)5=-30+45=15[/tex]Now, Lets see if the 14th term is 35 by replacing the n for 14
n=14
[tex]a_{14}=-30+(14-1)5[/tex][tex]a_{14}=-30+(13)5=-30+65=35[/tex]Then so, the sequence follows the equation an =-30+(n-1)5.The perimeter of a geometric figure is the sum of the lengthsof its sides. If the perimeter of the pentagon (five-sided figure)to the right is 140 meters, find the length of each side.
Given figure is a regular pentagon, it means it will have 5 sides of equal length.
Let the length of each side of the polygon be 'x' meters.
The perimeter of the pentagon will be the sum of all its 5 sides,
[tex]\begin{gathered} P=x+x+x+x+x \\ P=5x \end{gathered}[/tex]Given that the perimeter of the pentagon is 140 meters,
[tex]P=140[/tex]Substitute the value in the equation,
[tex]\begin{gathered} 140=5x \\ x=\frac{140}{5} \\ x=28 \end{gathered}[/tex]Thus, the length of each side of the regular pentagon is 28 meters.
Ed spends 21 hours practicing drums over a fortnight. At that rate, how much time will he spend on practicing drums over a 7 week period?
This problem can be solve with the simple rule of three:
[tex]\begin{gathered} 1\text{ fortnight}\to21\text{hours of practice} \\ 7\text{ weeks}\to x \\ x=\frac{7weeks\cdot21\text{ hours of practice}}{1fortnight} \\ 7\text{ we}eks=7\cdot7days=49days \\ 1\text{ fortnight=15days} \\ x=\frac{49days\cdot21\text{ hours of practice}}{15\text{ days}}=68.6\text{ hours of practice} \end{gathered}[/tex]In 7 weeks Ed practice 68.6 hours.
Note we convert week and fortnight to days.
The table below, from the Statistical Abstract of the United States, showsamusement park attendance at the top 15 amusement parks for given years. Findthe equation of the line that best fit the data.Year (x) | ,Amusement Park Attendance at Top 15 Amusement Parks (in thousands) (y)2009 | 107,3482010 | 109,3212011 | 112,5092012 | 116,4202013 | 119,951
Year (x) | Attendance (in thousands) (y)
2009 | 107,348
2010 | 109,321
2011 | 112,509
2012 | 116,420
2013 | 119,951
Using Excel the next equation is obtained:
[tex]y=3230.5x-6\cdot10^6[/tex]Use calculus to find the dimensions of a rectangle with area of 196 square-feet that has the smallest perimeter.
Explanation
In the question, we are given that the area of the rectangle is;
[tex]\text{Area}=196\text{ square fe}et[/tex]Recall that the area and perimeter of a rectangle are given by the formulas below.
[tex]\begin{gathered} \text{Area = Lenth x Width = L}\times W \\ \text{Perimeter = 2(L+W)} \end{gathered}[/tex]From the area of the rectangle, we can isolate the variable of the width.
[tex]\begin{gathered} \text{Area}=\text{ L x W} \\ W=\frac{\text{Area}}{L} \\ W=\frac{196}{L} \end{gathered}[/tex]Therefore, the formula for the perimeter is transformed to give;
[tex]\begin{gathered} \text{Perimeter = 2( L + }\frac{\text{196}}{L}) \\ \text{Simplifying the expression gives;} \\ P=2(\frac{L^2+196}{L}) \\ P=\frac{2L^2+392^{}}{L} \\ P=2L+392L^{-1} \end{gathered}[/tex]Recall, via the rules of differentiation
[tex]\begin{gathered} \text{for y = x}^n \\ \frac{dy}{dx}=nx^{n-1} \end{gathered}[/tex]Therefore,
[tex]\begin{gathered} \frac{dP}{dL}=2-392L^{-2}^{} \\ \text{But }\frac{dP}{dL}=0 \\ 0=2-392L^{-2} \\ 392^{}L^{-2}=2 \\ \frac{1}{L^2}=\frac{2}{392}^{} \\ L^2=\frac{392}{2} \\ L^2=196 \\ L=\sqrt[]{196} \\ L=14 \end{gathered}[/tex]Since
[tex]W=\frac{196}{L}=\frac{196}{14}=14[/tex]Answer: Length = 14 and Width = 14
The area of a rectangle is x^2+14x+24. What is the length when the width is x+2?
we have that:
[tex]\frac{x^2+14x+24}{x+2}=\frac{(x+2)(x+12)}{x+2}=x+12[/tex]so the length is x+12
Explain why we need math? (Need three sentences)
1. Mathematics is necessary for engineering. Without them we could not have the technological advances that make our lives easy.
2. Mathematics is necessary just as art is necessary. For its beauty. They have a spiritual and aesthetic value.
3. Mathematics is necessary for our daily life. From calculating the cost of things or calculating how we can carry out our dreams.
4. Mathematics is fundamental for the intellectual development of human beings: that is, because it helps them to be logical, to reason in an orderly manner and to have a mind prepared for thought, criticism and abstraction
3.If F, G, and H are the midpoints of the sidesof AJKL, FG = 37, KL = 48, and GH = 30,find each measure.Ka) FH=Gb) JL =F>Lc) KJ =H||Jd) FJ =
3.If F, G, and H are the midpoints of the sides of AJKL, FG = 37, KL = 48, and GH = 30,
Find each measure.
a) FH = 1/2 KL; FH= 24
b) JL = 2* FG = 2* 37
c) KJ = 2* GH = 2*30 =
d) FJ = 1/2 KJ = GH = 30
__________________________________________________
I'm working on your image, please give me a few minutes
What is 1584 in terms of pi?
Use the graph to find the appropriate solutions to the equation
The first thing that you should notice is that you have two functions:
[tex]f(x)=\text{ -}2|x\text{ -}3|+1;\text{ and }g(x)=\text{ -}2\sqrt{x\text{ -}1}[/tex]Then their respective graphs are given by the blue and the red functions in your picture. Now, the main idea, is to know the points where they are equal and in that way, you will have the initial equation as follows:
[tex]f(x)=\text{ -}2|x\text{ - }3|+1=\text{ -}2\sqrt{x\text{ - }1}\text{ }=g(x)[/tex]The equation will be satisfied in terms of their graphs at the points where they intersect themselves, at the x-coordinate.
The intersection points A and B at the x-coordinate are the answer for the equation. Then
[tex]x\approx1.7\text{ or }x\approx5.7[/tex]Question 5Points 3A model rocket is projected straight upward from the ground level according to theheight equation h =-16f2 + 144t, t> 0, where h is the height in feet and t is the time inseconds. At what time is the height of the rocket maximum and what is that height?
Solution
Step 1:
Write the equation:
[tex]h\text{ = -16t}^2\text{ + 144t}[/tex]Step 2
[tex]At\text{ maximum height, }\frac{dh}{dt}\text{ = 0}[/tex]Step 3:
[tex]\begin{gathered} h\text{ = -16t}^2\text{ + 144t} \\ \\ \frac{dh}{dt}\text{ = -32t + 144} \\ \\ 32t\text{ = 144} \\ t\text{ = }\frac{144}{32} \\ t\text{ = 4.5} \end{gathered}[/tex]Step 4
Substitute t = 4.5 into the height equation.
[tex]\begin{gathered} h\text{ = -16 }\times\text{ 4.5}^2\text{ + 144 }\times\text{ 4.5} \\ h\text{ = -324 + 648} \\ \text{h = 324} \end{gathered}[/tex]Find the slope of the line.
Find the volume of a cylinder with a height of 3 m and a diameter of 3 . Round to the nearest tenth
EXPLANATION
The volume of a cylinder will be:
Volume= π*(d/2)^2*h
Substituting:
Volume= π*(3/2)^2*3 = 21.2 m^2
The answer is 21.2 m^2
Nancy needs at least 1000 gigabytes of storage to take pictures and videos on her upcoming vacation. She checks and finds that she has 105 GB available on her phone. She plans on buying additional memory cards to get the rest of the storage she needs. The cheapest memory cards she can find each hold 256 GB and cost $10. She wants to spend as little money as possible and still get the storage she needs. Let C represent the number of memory cards that Nancy buys. 1) Which inequality describes this scenario? Choose 1 answer: 105 + 100 < 1000 105 + 100 > 1000 105 + 2560 < 1000 105 + 256C 1000
The total GB is 1000
The available GB is 105 GB.
Each 256 GB and cost $10.
Let C be the number of memory cards, then we have,
[tex]105+256C\ge1000[/tex]Thus, option D is correct.
Is the following pair of vectors Parallel, Perpendicular/Orthogonal or Neither?m = < 1 , 5 > n = < 3 , 15 >
1) To find out we need to calculate the dot product of those two vectors
[tex]\begin{gathered} m\cdot n=\mleft\langle1,5\mright\rangle\cdot\mleft\langle3,15\mright\rangle=1\cdot3+5\cdot15=3+75=78 \\ \end{gathered}[/tex]Since these vectors have a dot product different than zero, then they are not Orthogonal.
2) Let's now check if they are perpendicular, calculating the norm of each one and the angle between them:
[tex]\begin{gathered} \mleft\|m\mright\|=\sqrt[]{1^2+5^2}=\sqrt[]{26} \\ \|n\|=\sqrt[]{3^2+15^2}=\sqrt[]{9+225}=\sqrt[]{234} \end{gathered}[/tex]And finally the angle theta between them:
[tex]\begin{gathered} \theta=\cos ^{-1}(\frac{u\cdot v}{\|m\|\cdot\|n\|}) \\ \theta=\cos ^{-1}(\frac{78}{\sqrt[]{26}\cdot\sqrt[]{234}}) \\ \theta=0 \end{gathered}[/tex]3) Since the angle is 0, these vectors are parallel since parallel vectors for 0º or 180º
please help with this problem I have a test in a few minutes which be about this kind of topic but I don't Understand
We want to find f(2) for the following function
f(x) = 2x² + 3x
This means that we want to find the value of f(x) when x = 2. So, we replace all the x by 2:
f(x) = 2x² + 3x
f(2) = 2(2)² + 3 · 2
Since 2² = 4 and 3 · 2 = 6 then
f(2) = 2(2)² + 3 · 2
f(2) = 2 · 4 + 6
f(2) = 8 + 6
f(2) = 14
Answer: f(2) = 14Consider the system of linear equalities:2+>26+3<12This is the graph of the solution.
Given inequality is 2x+y>2 and 6x+3y<12.
Solve 2x+y>2 for y:
[tex]undefined[/tex]find the slope of the line. (-3,0), (2,2), (7,4), (12,6)
find the slope of the line.
(-3,0), (2,2), (7,4), (12,6)
To find the slope we need two points
so
we take
(-3,0), (2,2)
so
m=(2-0)/(2+3)
m=2/5
Verify with the other two points
(7,4), (12,6)
m=(6-4)/(12-7)
m=2/5 ----> is ok
therefore
the slope is 2/5Suppose that the functions p and q are defined as follows.p(x) = -2x-1q(x)=x²+1Find the following.(q*p)(-2)=(p*q)(-2)=
Answer:
(a)10
(b)-11
Explanation:
Given the function p(x) and q(x) defined as follows:
[tex]\begin{gathered} p(x)=-2x-1 \\ q(x)=x^2+1 \end{gathered}[/tex]Part A
[tex]\begin{gathered} (q\circ p)(x)=q(p(x)) \\ =(-2x-1)^2+1 \\ (q\circ p)(-2)=(-2(-2)-1)^2+1 \\ =(4-1)^2+1 \\ =3^2+1 \\ (q\circ p)(-2)=10 \end{gathered}[/tex]Part B
[tex]\begin{gathered} (p\circ q)(x)=p(q(x)) \\ =-2(x^2+1)-1 \\ (p\circ q)(-2)=-2((-2)^2+1)-1 \\ =-2(4+1)-1 \\ =-2(5)-1=-10-1 \\ (p\circ q)(-2)=-11 \end{gathered}[/tex]Raina got a prepaid debit card with $30 on it .For her first purchase with the card ,she bought some bulk ribbon at a craft store. The price of the ribbon was 6 cents per yard.If after that purchase there was $27.90 left on the card ,how many yards of ribbon did Raina buy?
Data:
• Total: $30
,• Price of ribbon: 6 cents per yard
,• Left after purchase: $27.90
Procedure:
[tex]30-27.90=2.1[/tex]Raina spent a total of $2.1 on ribbons.
To know how many yards she bought, we have to homogenize the units. For that, a conversion to dollars must be made.
[tex]6\frac{cents}{\text{yard}}\cdot\frac{0.01dollar}{1cent}=0.06\frac{\text{dollars}}{\text{yard}}[/tex]The yard is $0.06 per yard.
Finally, we can get the yards bought:
[tex]2.1dollars\cdot\frac{1\text{yard}}{0.06\text{dollars}}=35[/tex]Answer: 35 yards
In ΔKLM, the measure of ∠M =90°, the measure of ∠K=,86°, and MK = 86 feet. Find the length of KL to the nearest tenth of a foot.
ΔKLM is a right triangle, the following measures are known:
∠M= 90º
∠K=86º
MK= 86 feet
The side KL is the hypotenuse of the right triangle. To determine its measure you have to apply the trigonometric ratios. In this case, we know one of the angles of the triangle and the length of the side next to the angle, the trigonometric ratio that relates the adjacent side and the hypotenuse is the cosine:
[tex]\cos \theta=\frac{adjacent}{hypotenuse}[/tex]From this expression, you can calculate the measure of the hypotenuse.
[tex]\cos 86=\frac{86}{x}[/tex]-Multiply both sides of the expression by x to take the term out of the denominators place:
[tex]\begin{gathered} x\cos 86=x\cdot\frac{86}{x} \\ x\cos 86=86 \end{gathered}[/tex]-Divide both sides of the expression by cos 86 to reach the value of x:
[tex]\begin{gathered} x\cdot\frac{\cos86}{\cos86}=\frac{86}{\cos 86} \\ x=\frac{86}{\cos 86} \\ x=1232.86 \\ x\approx1232.9ft \end{gathered}[/tex]The length of KL is equal to 1232.9ft
???can anyone please help???
Because he wants to mantain a constant rate over the first hours, we can see that on the first hour he should travel 26 miles.
How many miles should he travel in the first hour?The total race is 150 miles, if we discount the last 20 miles (that Jackson will bike as fast as he can) we get:
150mi - 20mi = 130mi
We know that he travels these 130 miles at a constant rate over 5 hours, so the distance that he moves each hour (an particularly the first hour) is given by the quotient between the distance and the time.
R = 130mi/5h = 26mi/h
So he should travel 26 miles in the first hour.
Learn more about constant rates:
https://brainly.com/question/25598021
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Simplify the following polynomials. All final answers must be in standard form!-3x(4x + 12)
Given the expression:
[tex]-3x(4x\text{ + 12)}[/tex]Let's simplify the expression by multiplying -3x to each of the terms inside the parenthesis.
We get,
[tex]-3x(4x\text{ + 12)}[/tex][tex](-3x)(4x)\text{ + (12)}(-3x)[/tex][tex](-12x^2)\text{ + (-36x)}[/tex][tex]-12x^2\text{ - 36x}[/tex]Therefore, the simplified form of the given expression is -12x^2 - 36x.
Options for the first box: inverse and direct Options: 275, 50, 5,000, 13,750 Options for the third box: $275.00, $137.50, $550.00
• The proportional situation represents an Inverse Variation
,• The constant of variation, k = 13750
,• If all 100 students participate in the fundraiser, each will contribute $137.5
Explanation:From the given information, we notice that the more students involved in the fundraiser, the less the amount each student needs to contribute.
This is an INVERSE VARIATION
Let s represent the number of students who participated in the fundraiser, and t represents the amount needed to be contributed by each student, we have:
[tex]\begin{gathered} s\propto\frac{1}{t} \\ \\ \Rightarrow s=\frac{k}{t} \\ \\ OR \\ st=k \end{gathered}[/tex]To find k, we use the information that s = 50 when t = 275
So,
[tex]\begin{gathered} k=50\times275 \\ =13750 \end{gathered}[/tex]From the above, we have the formula:
[tex]st=13750[/tex]If 100 students participate in the fundraiser, we have:
[tex]\begin{gathered} 100t=13750 \\ t=\frac{13750}{100}=137.5 \end{gathered}[/tex]Each student needs to contribute $137.5
Bob had three 10' lengths of conduit. If he used a total of 13.75' of conduit to install a motor, how much total conduit does he have left?
Given that Bob had three 10' lengths of conduit and he used a total of 13.75' of conduit to install a motor.
To find how much total conduit he has left we would subtract the initial from what was used.
The initial length = three 10' lengths = 30'
Length used = 13.75'
[tex]\begin{gathered} \text{Conduit left = Initial length - length used} \\ \text{Conduit left }=\text{ 30 - 13.75} \\ =16.25 \\ \end{gathered}[/tex]
The total conduit he has left is
Answer: 16.25 feet
Find f(5) for f(x)= 2(2)*O A. 32B. 2O C. 4D. 8
Given function is
[tex]2^x[/tex]Fo
Please help me to classify each of the numbers below as an integer or not
The integers are the number that are written without a factional component.
The fraction -6/19 can be expressed with fraction. So it is not a integer.
The fraction -10/5 can be expressed as -2, so this is a integer.
The number -40 is expressed wth fraction component. So it is a integer.
The number -33 is expressed with fraction component. So it is a integer.
The decimal number 8.98 can be expressed as fraction and not expressed without fraction. So it is not a interger.
Answer:
-6/19 No
-10/5 Yes
-40 Yes
PLS HELP ME I BEG OF YPU PLS PLS
Answer: Y= 2x + 4
Step-by-step explanation:
The line is going up which means it is positive, giving it a positive slope of 2x. It intersects with the Y-axis at (0, 4), giving it a positive Y-int of 4. You can use rise over run to find the slope; pick two easy points and count how many units up and over it is to the next point!
In a canoe race, Team A is traveling 6 miles per hour and is 2 miles ahead of Team B. Team B is also traveling 6 miles per hour. The teams continue traveling at their current rates for the remainder of the race. Using d for distance (in miles) and t for time (in hours), write a system of linear equations that represent this situation.Equation for Team A:Equation for Team B:Will team B catch up to Team Ao Yeso No
First of all, we need to remember the equation of the speed and time:
distance = speed x time
Now, for the first team (team A):
[tex]\begin{gathered} x=6\frac{miles}{hour}\cdot t+2; \\ x\text{ - distance} \\ t\text{ - time} \end{gathered}[/tex]After that, for the second team (team B):
[tex]x=6\frac{miles}{hour}\cdot t[/tex]Finally, the team B never catch up to team A!
Jackson has a points card for a movie theater.He receives 55 rewards points just for signing upHe earned 12.5 points for each visit to the movie theaterHe needs at least 210 points for a free movie ticketWrite and solve an inequality which can be used to determine x, the number of visits Jackson can make to earn his first free movie ticket
Given data:
The given reward is 55.
The point earned by visit is 12.5.
The number of point for the movie ticket 210.
The given expression for the inequality is,
55+12.5x ≥ 210
12.5x ≥ 155
x ≥12.4
Thus, the minimum number of visit is 13.
round each number to the nearest ten, hundred, and thousand5,999
SOLUTIONS
Round each number to the nearest ten, hundred, and thousand
5,999
[tex]5999=6000\text{ \lparen nearest thousand\rparen}[/tex][tex]5999=6000\text{ \lparen nearest ten\rparen}[/tex][tex]5999=6000\text{ \lparen nearest hundred\rparen}[/tex]