The first terms is
[tex]undefined[/tex]Substitute 2, 3, 4, and 5 for n in the equation to find first four next terms.
Find (3/5x+3/4)−(1/3x−1/8)
The answer is [tex]\frac{32x+105}{120}[/tex].
The area of mathematics known as algebra is used to represent situations or problems using mathematical expressions. In algebra, we combine integers with variables like x, y, and z.
A fraction is a number that is a component of a whole. In algebra, fractions can be added, subtracted, multiplied, and divided just like in basic arithmetic.
The given equation is an algebraic fraction having variables in the numerator. This equation is written as,
[tex]\left(\frac{3}{5}x+\frac{3}{4}\right)-\left(\frac{1}{3}x-\frac{1}{8}\right)[/tex]
First combine, x with the nearby fraction,
[tex]\begin{aligned}\left(\frac{3}{5}x+\frac{3}{4}\right)-\left(\frac{1}{3}x-\frac{1}{8}\right)&=\left(\frac{3x}{5}+\frac{3}{4}\right)-\left(\frac{x}{3}-\frac{1}{8}\right)\\&=\frac{3x}{5}+\frac{3}{4}-\frac{x}{3}+\frac{1}{8}\end{aligned}[/tex]
Now, group the fraction with a common denominator,
[tex]\begin{aligned}\left(\frac{3}{5}x+\frac{3}{4}\right)-\left(\frac{1}{3}x-\frac{1}{8}\right)&=\left(\frac{3x}{5}-\frac{x}{3}\right)+\left(\frac{3}{4}+\frac{1}{8}\right)\\&=\left(\frac{3x\cdot3}{15}-\frac{x\cdot5}{15}\right)+\left(\frac{3\cdot2}{8}+\frac{1}{8}\right)\\&=\left(\frac{9x-5x}{15}\right)+\left(\frac{6+1}{8}\right)\\&=\frac{4x}{15}+\frac{7}{8}\\&=\frac{32x+105}{120}\end{aligned}[/tex]
Therefore, the final answer is [tex]\frac{32x+105}{120}[/tex].
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Tony used a photocopier to dilate the design for a monorail track system. The figure below shows the design and its photocopy:The ratio of CD:GH is 2:3. What is the length, in meters, of side EH on the photocopied image?
Solution
The length of EH is;
[tex]\begin{gathered} \frac{2}{3}=\frac{8}{EH} \\ \\ \Rightarrow EH=\frac{3}{2}\times8=12 \\ \\ \Rightarrow EH=12 \end{gathered}[/tex]Determine the domain and range of the quadratic function. f(x)=−2(x+8)^2−4
Since the given is a polynomial with a degree of 2, there are no restrictions to its domain. The domain therefore is
[tex]\text{Domain: }(-\infty,\infty)[/tex]The given function is in the vertex form
[tex]\begin{gathered} f(x)=a(x-h)^2+k \\ \text{where} \\ (h,k)\text{ is the vertex} \end{gathered}[/tex]By inspection, we determine that the vertex of the function is at (-8,-4), and since a = -2, then the parabola opens up downwards. This implied that its output peaks at y = -4, and the graph continues towards negative infinity.
We can conclude therefore that the range is
[tex]\text{Range: }(-\infty,-4\rbrack[/tex]Find the sum of the first 46 terms of the following series, to the nearestinteger.12, 15, 18, ...
We can see that this is an arithmetic sequence. The first term is 12 and the common difference is 3.
Using the formula to calculate the sum of the first 46 terms, we have:
[tex]undefined[/tex]The rabbit population in a certain area is 200% of last year's population. There are 1100 rabbits this year. How many were there last year?
As per the given percentage, the population of rabbits in the area is 2200 in last year.
Percentage:
Percentage refers the ratio or the fraction that is multiplied and divided by 100. And it will be represented by the symbol "%".
Given,
The rabbit population in a certain area is 200% of last year's population. There are 1100 rabbits this year.
Now, we need to find the population of rabbit in last year.
Let us consider x be the total number of rabbit in last year.
We know that the rabbit in the current year is 110.
And we also know that, there are 200% of rabbits in last year.
So, we have to write it in the following expression,
200% of 1100 = x
so, the value of x is,
x = 200/100 x 1100
x = 2200
Therefore, there are 2200 rabbits in last year.
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Meri invests 15000 into an account the interest is compounded monthly for 17 years. The account balance will be 87,219.93 at the end of 17 years. What is the annual interest rate?
Annual interest rate will be 11.95% or 12% approx.
What is compound interest?
The interest earned on savings that is computed using both the original principal and the interest accrued over time is known as compound interest.
It is thought that Italy in the 17th century is where the concept of "interest on interest" or compound interest first appeared. It will accelerate the growth of a total more quickly than simple interest, which is solely calculated on the principal sum.
Money multiplies more quickly thanks to compounding, and the more compounding periods there are, the higher the compound interest will be.
P = principal
i = nominal annual interest rate in percentage terms
n = number of compounding periods
formula for compound interest is [tex]P [(1 + i)^n - 1][/tex]
According to the question
P=15000
i = ?
compound interest = 87,219.93
n=17 years
Therefore
[tex]87219.93=15000 [(1 + i)^1^7 - 1][/tex]
[tex]\frac{87219.93}{15000} = [(1 + i)^1^7 - 1][/tex]
[tex]6.814662 = (1+i)^1^7[/tex]
[tex](6.814662)^\frac{1}{17} = (1+i)[/tex]
i =1.1195-1
i =0.1195
i.e. 11.95%
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reduce the radical of 200
Bryce is cutting tree trunks into circular pieces of wood 1 inch thick to make wall art for log cabins. Match the circumferences of each wood circle to its diameter or radius.
The circumference of a circle of radius r can be calculated as:
[tex]C=2\pi r[/tex]If the diameter d is given, then the formula is:
[tex]C=\pi d[/tex]Calculate the circumference of the following circles:
1 d = 8 inches.
[tex]\begin{gathered} C=\pi(8\text{ inches}) \\ C=3.14\cdot8\text{ inches} \\ C=25.12\text{ inches} \end{gathered}[/tex]1 matches with b
2 d = 7 inches
[tex]\begin{gathered} C=\pi(7\text{ inches}) \\ C=3.14\cdot7\text{ inches} \\ C=21.98\text{ inches} \end{gathered}[/tex]2 matches with d
3 r = 2 inches
[tex]\begin{gathered} C=2\pi(2\text{ inches\rparen=4}\cdot3.14\text{ inches} \\ C=12.56\text{ inches} \end{gathered}[/tex]3 matches with a
4 r = 3 inches
[tex]\begin{gathered} C=2\pi(3\text{ inches\rparen=6}\cdot3.14\text{ inches} \\ C=18.84\text{ inches} \end{gathered}[/tex]4 matches with c
Solve for a.2(a+4)+6a=48 Enter your answer in the box.a =
Step 1
Given
[tex]2(a+4)+6a=48[/tex]Required : To find the value of a
Step 2
Expand the bracket
[tex]2a+8+6a=48[/tex]Step 3
Bring like terms together
[tex]\begin{gathered} 2a+6a=48-8 \\ \end{gathered}[/tex]Step 4
Find the value of a
[tex]\begin{gathered} 8a\text{ = 40} \\ \frac{8a}{8}=\frac{40}{8} \\ a=\text{ 5} \end{gathered}[/tex]Hence, a = 5
Find the area of the figure. (Sides meet at right angles.) Check 7 yd 5 yd 3 yd 3 yd 13 yd 3 yd 5 yd 7 yd yd²
Given:
The figure with sides measurements.
Required:
Find area of the figure.
Explanation:
First we will draw figure
In figure, we can see that all figure ABIJ, CDHI and EFGH are rectangles.
So, we need area of rectangle formula. That is
[tex]A=length\times width[/tex]So, area of given figure
[tex]\begin{gathered} A=\text{ Area of ABIJ +Are of CDHI + Area of EFGH} \\ ABIJ=EFGH \\ So, \\ A=2\times(ABIJ)+CDHI \\ A=2\times(7\times5)+(4\times3) \\ A=70+12 \\ A=82yd^2 \end{gathered}[/tex]The volume V of a cone varies jointly as the square of the radius of the base,r, and the height, h. Find the equation of the joint variation if v =285, r=4, and h = 17.
Answer:
V = 1.05r²h
Explanation:
The expression ''the volume V of a cone varies jointly as the square of the radius of the base,r, and the height, h'' can be represented as:
[tex]V=k\cdot r^2\cdot h[/tex]Where the k is a constant.
So, replacing V = 285, r = 4, and h = 17, we get:
[tex]285=k\cdot4^2\cdot17[/tex]Solving for k, we get:
[tex]\begin{gathered} 285=k\cdot16\cdot17 \\ 285=k\cdot272 \\ \frac{285}{272}=\frac{k\cdot272}{272} \\ 1.05=k \end{gathered}[/tex]So, the equation of the joint variation is:
[tex]V=1.05r^2h[/tex]Donovan took a math test and got 20 correct questions and 5 incorrect answers. What was the percentage of correct answers?
ANSWER
80%
EXPLANATION
Donovan got 20 questions correct and 5 questions incorrect.
This means that the total number of questions he attempted in the test is the sum of correct and incorrect questions:
Total = 20 + 5
Total = 25
To find the percentage of correct answers, we have to divide the number of correct answers by the total number of questions attempted and multiply by 100 (per cent).
That is:
[tex]\begin{gathered} \frac{20}{25}\cdot\text{ 100 = }\frac{4}{5}\cdot\text{ 100} \\ =\text{ 80\%} \end{gathered}[/tex]That is the percentage of correct answers.
Write a two column proofGiven: q is parallel to r Prove: angle 1 is supplementary to angle 3
Answer:
Proved.
Explanation:
Given: q is parallel to r
Statement: m∠1 = m∠2
Reason: Vertically Opposite Angles
Statement: m∠2+m∠3=180°
Reason: Same-side Interior Angles
Recall that m∠1 = m∠2
Statement: m∠1+m∠3=180°
Reason: Congruent Angles (m∠1 = m∠2)
Therefore, angle 1 is supplementary to angle 3.
Proved.
*Express the end behavior of the followingFunction in limit notation.G(x)=-x(x^2 + 3) (x - 2)^3 (x + 5)^2
we have the function
[tex]g(x)=-x(x^2+3)(x-2)^3(x+5)^2[/tex]In this problem, we have that
the leading coefficient is negative (-1)
The degree of the function is 8 (even)
therefore
the end behavior of the function is
f(x)→−∞, as x→−∞
f(x)→−∞, as x→+∞
Find FG.FL x + 113x + 1EН.FG=
Answer:
FG = 16
Explanation:
The triangles EFG and EHG are congruent because they share the side EG and they have the same interior angles.
If they are congruent the lengths of the corresponding sides are equal, so we can write the following equation:
FG = GH
Then, subtitute FG = x + 11 and GH = 3x + 1 to get:
x + 11 = 3x + 1
So, solving for x, we get:
x + 11 = 3x + 1
x + 11 - 1 = 3x + 1 - 1
x + 10 = 3x
x + 10 - x = 3x - x
10 = 2x
10/2 = 2x/2
5 = x
Then, replacing x by 5, we get that FG is equal to:
FG = x + 11
FG = 5 + 11
FG = 16
So, the answer is FG = 16
4312345L2-3445To find the rate of change of the function, Kevin did the following:
Please help I was sick and missed out on class.Thank you
The slope of the line passing through given points is 5/7
We know that the slope is the ratio of the change in y-values to the change in x-values.
We use slope formula,
m = (y2 - y1)/(x2 - x1)
For pair of points (1, 3) and (8, 8),
m1 = (8 - 3)/(8 - 1)
m1 = 5/7
For pair of points (8, 8) and (15, 13),
m2 = (13 - 8)/(15 - 8)
m2 = 5/7
For pair of points (22,18) and (29, 23),
m3 = (23 - 18)/(29 - 22)
m3 = 5/7
For pair of points (15, 13) and (22, 18),
m4 = (18 - 13)/(22 - 15)
m4 = 5/7
Since the rate of change of output values to the input values is constant i.e., 5/7, the slope of the line is 5/7
Therefore, the slope of the line passing through given points is 5/7
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Consider the following expression and determine which statements are true. z? + 5y: -8 Choose 2 answers: There are 3 terms. The variables are z, y. and . The coefficient of zis 2 The term Syz is made up of 2 factors.
we have:
the expression has 3 terms and 3 variables, they are x, y and z. Therefore
answer:
A and B
A rectangular garden is 15 feet wide. If its area is 1050ft², what is the length of the garden?
The width is w=15 ft.
The area is A=1050 sq ft.
The length of the garden is,
[tex]\begin{gathered} L=\frac{A}{w} \\ =\frac{1050ft^2}{15ft} \\ =70ft \end{gathered}[/tex]Thus, the length of the garden is 70 ft.
Write a linear function rule for the data in the table.x01234y31–1–3–5A.f(x) = 2x + 3B.f(x) = –2x + 3C.f(x) = 2x – 6D.f(x) = –2x – 6
From the given table
Choose two-point for x and y from the table
So,
The point will be:
(0, 3) and (1, 1)
Now,
From the standard for of the linear function
[tex]y=mx+b[/tex]Then,
First, find the value of the slope (m) from the given point
So,
From the formula of the slope:
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]Then,
[tex]\begin{gathered} m=\frac{y_2-y_1}{x_2-x_1} \\ m=\frac{1_{}-3_{}}{1_{}-0_{}} \\ m=-2 \end{gathered}[/tex]Now,
To find the value of b, put the value of m = -2, x = 0 and y = 3 into the standard form of the equation
[tex]\begin{gathered} y=mx+b \\ 3=-2(0)+b \\ 3=b \end{gathered}[/tex]Then,
Put the value of b into the standard form of the equation
[tex]\begin{gathered} y=mx+b \\ y=-2x+3 \end{gathered}[/tex]Hence, the correct option is B.
What is the slope of a linear function that passes between (2, 7) and (5, 12)?
The formula of slope,
→ (y2 - y1)/(x2 - x1)
Then the slope will be,
→ m = (12 - 7)/(5 - 2)
→ [ m = 5/3 ]
Hence, thy slope is 5/3.
arina runs up 4 flights and runs down 4 flights of stairs does this situation repreasent additive inverses explain . A. Yes; The numbers combine to eight B. Yes; The numbers are combine to zero C. No;The numbers are both represented by the same integer. D. No; The numbers cannot be added together.
B. Yes; The numbers are combine to zero
is additive inverses because the sum is 0
[tex]\begin{gathered} 4+(-4) \\ =0 \end{gathered}[/tex]For the polynomial below, -3 and 1 are zeros. Express f (x) as a product of linear factors.
Explanation
Since -3 and 1 are zeros of the functions, it implies that
[tex](x+3)\text{ }and\text{ }(x-1)[/tex]are factors of the equation.
Therefore we can find the remaining factors below
[tex](x+3)(x-1)=x^2+2x-3[/tex]By long division
[tex]remaining\text{ expression =}\frac{x^4+6x^3+7x^2-8x-6}{x^2+2x-3}=x^2+4x+2[/tex]By quadratic formula
[tex]\begin{gathered} x_{1,2}=\frac{-4\pm\sqrt{4^2-4\times1\times2}}{2\times1} \\ x_1=\frac{-4+2\sqrt{2}}{2},x_2=\frac{-4-2\sqrt{2}}{2} \\ x=-2+\sqrt{2},x=-2-\sqrt{2} \\ therefore \\ (x+2-\sqrt{2})(x+2+\sqrt{2}) \end{gathered}[/tex]The linear factor are
Answer:
[tex]f(x)=(x+3)(x-1)(x+2-\sqrt{2})(x+2+\sqrt{2})[/tex]
In 10 seconds, Jake travels 550 feet on his bike. At this speed. How many fert can he travel in 1 minute.
We can find out how many feet can Jake travel using a rule of three:
[tex]\begin{gathered} 10s\rightarrow550ft \\ 60s\rightarrow xft \\ \Rightarrow x=\frac{60\cdot550}{10}=3300 \\ x=3300ft \end{gathered}[/tex]therefore, Jake can travel 3300ft in 1 minute
I need help with this, I need it step by step please.
In this problem we have a translation
the rule is
(x,y) ------> (x-6,y)
that means
the translation of the point is 6 units at left
so
we have
E(2,4) ------> E'(2-6,4)
E'(-4,4)
you must to subtract 6 units from the x coordinate
F(4,4) -----> F'(4-6,4)
F'(-2,4)
G(2,1) -----> G'(2-6,1)
G'(-4,1)
The graph below shows the cost for going roller skating at 2 roller rinks . Bianca is going roller skating with a group of friends . Roller Rink A charges $3.00 per person and a $60 group fee . Roller Rink B charges $7.00 per person and an $8.00 group fee . When comparing costs ,which statement is true ? • Roller Rink B always cost less • Roller Rink A always cost less • Roller Rink B costs less if Bianca's group has fewer then 13 people• Roller Rink A costs less if Bianca's group has fewer then 13 people
In this case we can see that the cost of each company is increasing but the slopes are diferent. also we can see that the cost of company B is is cheaper at the begining but after some peaple is more expensive so the correct statement will be:
Roller Rink B costs less if Bianca's group has fewer than 13 people
Debra is going to rent a truck for one day. There are two companies she can choose from, and they have the following prices: company A has no initial fee but charges 80 cents for every mile driven. Company B charges an initial fee of $75 and an additional 70 cents for every mile driven. For what mileages will company A charge at least as much as company B? Use m for the number of miles driven, and solve your inequality for m.
Given that m is the number of miles driven by the truck.
From the information given,
The charge by Company A would be:
[tex]0+80\times m=80m[/tex]The charge by Company B would be:
[tex]75+70\times m=75+70m[/tex]If company A charges at least as much as company B, then
[tex]A\ge B[/tex]Therefore,
[tex]\begin{gathered} 80m\ge75+70m \\ 80m-70m\ge75 \\ 10m\ge75 \\ m\ge\frac{75}{10}=7.5 \\ \text{The mileage is 7.5}\frac{miles}{cent} \end{gathered}[/tex](8.5-2x)(11-2x)(x) what is the approximate value of x that would allow you to construct an
open-top box with the largest volume possible from one piece of paper
The largest volume possible from one piece of paper for open-top box is 64.296 cubic unit.
What is meant by the term maxima?The maxima point on the curve will be the highest point within the given range, and the minima point will be the lowest point just on curve. Extrema is the product of maxima and minima.For the given question dimensions of open-top box;
The volume is given by the equation;
V = (8.5-2x)(11-2x)(x)
Simplifying the equation;
V = x(4x² - 39x + 93.5)
Differentiate the equation with respect to x using the product rule.
dV/dx = x(8x -39) + (4x² - 39x + 93.5)
dV/dx = 8x² - 39x + 4x² - 39x + 93.5
dV/dx = 12x² - 72x + 93.5
Put the Derivative equals zero to get the critical point.
12x² - 72x + 93.5 = 0.
Solve using quadratic formula to get the values.
x = 4.1 and x = 1.9
Put each value of x in the volume to get the maximum volume;
V(4.1) = 4.1(4(4.1)² - 39(4.1) + 93.5)
V(4.1) = 3.44 cubic unit.
V(1.9) = 1.9(4(1.9)² - 39(1.9) + 93.5)
V(1.9) = 64.296 cubic unit. (largest volume)
Thus, the largest/maximum volume possible from one piece of paper for open-top box is 64.296 cubic unit.
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21 The number of students in each of 2 exercise classes was the same. The box and whiskerplots below represent the average amount of time the students in each class spent exercisingdaily outside class.First class九术也Second class A+153012010545 6075 90Time Spent Exercising(minutes)Based on the information in the box and whisker plots, which statement about the time spentexercising outside class appears to be true?A The median amount of time the first class spent exercising was greater than the medianamount of time the second class spent exercising.B The range for the second class was less than the range for the first class.C The interquartile range for the first class was less than the interquartile range for thesecond class.D The minimum amount of time the second class spent exercising was greater than theminimum amount of time the first class spent exercising.
We are given a box and whiskers plot and we are asked the following questions:
A. The median amount of time the first class spent exercising was greater than the median amount of time the second class spent exercising.
The median of the first class is 60 and the median of the second class is 75, therefore, the median of the second class is greater than the median of the first class.
B. The range for the second class was less than the range for the first class.
The range of the first class is:
[tex]r_1=90-30=60[/tex]The range of the second class is:
[tex]r_2=105-30=75[/tex]Therefore, the rage of the second class is greater than the range of the first class.
C. The interquartile range for the first class was less than the interquartile range for the
second class.
The Interquartile range of the first class is:
[tex]IQ_1=75-45=30[/tex]The interquartile range of the second class is:
[tex]IQ_2=90-45=45[/tex]Therefore, the interquartile range of the first class is less than the interquartile range of the second class.
D The minimum amount of time the second class spent exercising was greater than the minimum amount of time the first class spent exercising.
The minimum time for the first class is 30 and the minimum time for the second class is 30, therefore, the minimum times are equal.
A) Find the simple interest amount earned for $5500 at 6.5% for 5 months. b)What is the total value of the investment?
The simple interest I on an amount P invested at an interest rate R %, for a period of time T per annum is evaluated as
[tex]I\text{ = }P\times R\times T[/tex]A) The interest earned at $5500 at 6.5% for 5 months is thus evaluated as
[tex]\begin{gathered} P\text{ = 5500} \\ R\text{ = 6.5\% = }\frac{6.5}{100} \\ T\text{ = 5 months = }\frac{5}{12}\text{ year} \\ thus, \\ I\text{ = }5500\times\frac{6.5}{100}\times\frac{5}{12} \\ \Rightarrow I\text{ = \$ 148.958} \end{gathered}[/tex]thus, the interest earned for $5500 at 6.5% for 5 months is $ 148.985.
B) Total value of the investment.
The total value of the investment is the sum of the interest earned and the initial amount invested.
Thus,
[tex]\begin{gathered} Total\text{ value of investment = interest earned + amount invested} \\ A\text{ = I + P} \\ A\text{ = 148.985 + 5500} \\ \Rightarrow A\text{ = \$ 5648.985} \end{gathered}[/tex]Hence, the total value of the investment is $ 5648.985.