The table of values to represent the relationship between the number of days and the number of puzzles she has completed will be:
Days Puzzle completed
1. 4
2. 8
3. 12
4. 16
5. 20
How to illustrate the information?From the information, Maddie is determind to finish some extra credit math homework by completeing 4 math puzzles each day.
Therefore, the first day, she will do 4 puzzles. The second day will be:
= 2 × 4 = 8 puzzles
Third day = 3 × 4 = 12 puzzles.
In this case, the method that was used is simply to multiply the number of days by 4 since he does 4 puzzles everyday.
It should be noted that this illustrates a proportional relationship
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Apollo Enterprises has been awarded an insurance settlement of $6,000 at the end of each 6 month period for the next 12 years. calculate how much (in $) the insurance company must set aside now at 6% interest compounded semiannually to pay this obligation to Apollo
$12180 the insurance company must set aside now at 6% interest compounded semiannually to pay this obligation to Apollo.
This is a problem from the compound interest system. We can solve this problem by following a few steps.
Apollo Enterprises has been awarded an insurance settlement of $6,000 at the end of each 6-month period for the next 12 years with a 6% interest rate. We have to calculate the total amount after 12 years.
To solve this problem we should know the formula for the compound interest method.
Formula:-
A = P {(1 + r/n)^(n.t)}
Here,
A denotes the final amount, we have to find this.P denotes the initial principal balance which is $6,000r denotes the interest rate which is 6%n denotes the number of times interest is applied per time period which is 12/6 = 2. t denotes the number of time periods elapsed which is 12 years.Now, we can calculate the value of A.
A = 6000 {( 1 + 6/200 )^2.12} = 6000 ( 1 + 6/200 )^24 = 6000 × 2.03 = 12180
Therefore, the total amount after 12 years is $12180
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A line has a slope of 2 and passes through the point ( - 8 , 1 ). Which of the following represents the equation for this line?
A) y = 2x - 7
B) y = 2x + 8
C) y = 2x + 9
D) y = 2x + 17
Answer:
D
Step-by-step explanation:
Equation of line is given as y = mx + c, where m is the slope and c is the y-intercept.
Since line has slope of 2, equation is y = 2x + c.
Substitute (-8, 1) into the equation to find c
1 = 2(-8) + c
c = 17
Hence, equation of line is y = 2x + 17.
The correct answer is [D]
Hence, the equation of line is y = 2x + 17.
What is slope?Finding the ratio of "vertical change" to "horizontal change" between any two unique locations on a line yields the slope. Occasionally, the ratio is written as a quotient (also known as a "rise over run"), which produces the same number for every two unique points on the same line. Negative "rise" refers to a diminishing line. The line could be functional, established by a road surveyor, or depicted in a graphic that represents a road or a roof as a description or a design.
The absolute value of the slope is used to determine how steep, incline, or grade a line is. The steeper the line, the larger the absolute magnitude of the slope. A line's direction might be either horizontal, ascending, decreasing, or vertical.
If a line rises from left to right, it is said to be growing. The slope is upward, or m>0.If a line slopes downward from left to right, it is diminishing. The slope, m0, is negative.The slope of a line is 0 if it is horizontal. This function is constant.The slope is unknown if a line is vertical.Y = mx + c, where m is the slope and c is the y-intercept, is the equation for a line.
Since the line has slope of 2, equation is y = 2x + c.
Substitute (-8, 1) into equation to find c
1 = 2(-8) + c
c = 17
Hence, the equation of line is y = 2x + 17.
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What is the area of triangle ADC is ? Square units?
Given:
The triangle ADC is formed by reflecting the triangle ABC across the line segment AC
So, the triangles ABC and ADC are congruent
The area of the triangle ADC = Area of the triangle ABC
The area of the triangle = 1/2 * base * height
Base = AC = 4 units
Height = BE = 3 units
So, the area will be as follows:
[tex]Area=\frac{1}{2}*AC*BE=\frac{1}{2}*4*3=6[/tex]So, the answer will be:
Area of the triangle ADC is 6 square units
You roll a 6-sided die two times.What is the probability of rolling a 6 and then rolling a number less than 2?Simplify your answer and write it as a fraction or whole numb
We are asked to determine the probability of rolling a 6 and then rolling a number less than 2. To do that we will use the product rule probabilities since we want to find the probability of two independent events happening:
[tex]P(AandB)=P(A)P(B)[/tex]Where:
[tex]\begin{gathered} A=\text{ rolling a 6} \\ B=\text{ rolling a number less than 2} \end{gathered}[/tex]To determine the probability of rolling a 6 we need to have into account that there are 6 possible outcomes out of which only one is a 6. Therefore, the probability is:
[tex]P(A)=\frac{1}{6}[/tex]To determine the probability of B we need to have into account that in a 6-sided die the numbers that are less than 2 are (1), this means that there is only one number less than 2 out of 6 possible numbers. Therefore, the probability is:
[tex]P(B)=\frac{1}{6}[/tex]Now, we substitute in the product rule:
[tex]P(AandB)=(\frac{1}{6})(\frac{1}{6})[/tex]Solving the product:
[tex]P(AandB)=\frac{1}{36}[/tex]Therefore, the probability is 1/36.
Solve |3x + 7 = 4x for x.A) Infinitely many solutionsB) 7 and -7C) No solutionOD) -7 or 1
The given equation is
[tex]\lvert{3x+7}\rvert=4x[/tex]We will solve it in this way
[tex]3x+7=4x,3x+7=-4x[/tex]For the 1st equation
[tex]3x+7=4x[/tex]Subtract 7 from each side
[tex]\begin{gathered} 3x+7-7=4x-7 \\ 3x=4x-7 \end{gathered}[/tex]Subtract 4x from both sides
[tex]\begin{gathered} 3x-4x=4x-4x-7 \\ -x=-7 \\ x=7 \end{gathered}[/tex]For the 2nd equation
[tex]3x+7=-4x[/tex]Subtract 3x from both sides
[tex]\begin{gathered} 3x-3x+7=-4x-3x \\ 7=-7x \end{gathered}[/tex]Divide both sides by -7
[tex]\begin{gathered} \frac{7}{-7}=\frac{-7x}{-7} \\ -1=x \end{gathered}[/tex]The solutions are
7, -1
The answer should be B
the graph shows the mass of the bucket containing liquid depends on the volume of liquid in the bucket. Use the graph to find the range of the function.
The graph's range is 0 to M plus or minus 5.5.
Where can I find the function's range?The set of graph output values that make up a function's range.
This means that the set of y values in the graph is the range of a function.
How do you figure out the domain and range?The domain
We can observe the following on the function's graph:
The x values range from zero to seven and a half.
This indicates that the domain is 0=x=7.5.
The range
We can observe the following on the function's graph:
Beginning at 0, the x values go all the way up to 5.5.
In other words, the range is 0 = M = 5.5.
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two students determined the value of this expression
these are the steps each student used
analyze the steps and describe any eras made by student p and student q
Answer:
Student P made a mistake at step 1
Student Q made a mistake at step 3
Explanation:
First, let's analyze what are the correct steps to find the value of the expression.
So, the first step is to apply the distributive property:
[tex]\begin{gathered} -2.5(1.4+3.1)+6.9(-4.3) \\ -2.5(1.4)-2.5(3.1)+6.9(-4.3) \\ -3.5-7.75+6.9(-4.3) \end{gathered}[/tex]Then, we need to solve the multiplication of the last term:
[tex]-3.5-7.75-29.67[/tex]Now, we can factorize the sign minus, but we will need to change the signs of all terms:
[tex]-(3.5+7.75+29.67)[/tex]Finally, we can add the term to get:
[tex]\begin{gathered} -(40.92) \\ -40.92 \end{gathered}[/tex]Therefore, the errors made by student P were at step 1, when this student applies the distributive property, he or she made a mistake with the sign of 7.75. This number should be negative. The next steps are correct by taking into account that the error was in the first step, the result will be wrong.
For student Q, we have that everything was correct until step 3, where he or she factorize the minus sign but he or she doesn't change the signs of all signs. After that, the procedure is correct but the answer will be wrong due to the mistake made at step 3.
3) Find the equation of the line:
a) with a gradient of 2 and cutting the y-axis at 7
b) with a gradient of -2 and passing through the point (2;4)
c) passing through the points (2; 3) and (-1; 2)
d) parallel to the x-axis cutting the y-axis at 5
Step-by-step explanation:
this is very much doing the exact same things as the previous question, just with a little bit different numbers.
remember, gradient = slope.
the slope is always the factor of x in the slope-intercept form
y = ax + b
our in the point-slope form
y - y1 = a(x - x1)
"a" is the slope, b is the y-intercept (the y- value when x = 0).
(x1, y1) is a point on the line.
the slope is the ratio (y coordinate change / x coordinate change) when going from one point on the line to another.
a)
y = 2x + 7
b)
y - 4 = -2(x - 2) = -2x + 4
y = -2x + 8
c)
going from (2, 3) to (-1, 2)
x changes by -3 (from 2 to -1)
y charges by -1 (from 3 to 2)
the slope is -1/-3 = 1/3
we use one of the points, e.g. (2, 3)
y - 3 = (1/3)×(x - 2) = x/3 - 2/3
y = x/3 - 2/3 + 3 = x/3 - 2/3 + 9/3 = x/3 + 7/3
d)
y = 5
this is a horizontal line (parallel to the x-axis) and represents every point on the grid, for which y = 5.
the slope is 0/x = 0, as y never changes at all.
the y- intercept is 5, of course.
Answer:
[tex]\textsf{a) \quad $y=2x+7$}[/tex]
[tex]\textsf{b) \quad $y=-2x+8$}[/tex]
[tex]\textsf{c) \quad $y=\dfrac{1}{3}x+\dfrac{7}{3}$}[/tex]
[tex]\textsf{d) \quad $y=5$}[/tex]
Step-by-step explanation:
Part (a)Slope-intercept form of a linear equation:
[tex]y=mx+b[/tex]
where:
m is the slope.b is the y-intercept.Given values:
Slope = 2y-intercept = 7Substitute the given values into the formula to create the equation of the line:
[tex]\implies y=2x+7[/tex]
---------------------------------------------------------------------------
Part (b)Point-slope form of a linear equation:
[tex]y-y_1=m(x-x_1)[/tex]
where:
m is the slope.(x₁, y₁) is a point on the line.Given:
Slope = -2(x₁, y₁) = (2, 4)Substitute the given values into the formula to create the equation of the line:
[tex]\implies y-4=-2(x-2)[/tex]
[tex]\implies y-4=-2x+4[/tex]
[tex]\implies y=-2x+8[/tex]
---------------------------------------------------------------------------
Part (c)Slope formula:
[tex]\textsf{slope}\:(m)=\dfrac{y_2-y_1}{x_2-x_1}[/tex]
where (x₁, y₁) and (x₂, y₂) are points on the line.
Given points:
(x₁, y₁) = (2, 3)(x₂, y₂) = (-1, 2)Substitute the points into the slope formula to calculate the slope of the line:
[tex]\implies m=\dfrac{2-3}{-1-2}=\dfrac{-1}{-3}=\dfrac{1}{3}[/tex]
Substitute the found slope and one of the points into the point-slope formula to create the equation of the line:
[tex]\implies y-y_1=m(x-x_1)[/tex]
[tex]\implies y-3=\dfrac{1}{3}(x-2)[/tex]
[tex]\implies y-3=\dfrac{1}{3}x-\dfrac{2}{3}[/tex]
[tex]\implies y=\dfrac{1}{3}x+\dfrac{7}{3}[/tex]
---------------------------------------------------------------------------
Part (d)Slope-intercept form of a linear equation:
[tex]y=mx+b[/tex]
where:
m is the slope.b is the y-intercept.If the line is parallel to the x-axis, its slope is zero.
If the line intersects the y-axis at y = 5, then its y-intercept is 5.
Therefore:
m = 0b = 5Substitute the given values into the formula to create the equation of the line:
[tex]\implies y=0x + 5[/tex]
[tex]\implies y=5[/tex]
2. A ladder rises 20 feet for every horizontal change of 4 feet. What is the slope of the ladder?A. O 5C. O 20D.OChoose
The slope is defined as rise/run - in other words, for every step you take horizontally, how many steps you should take vertically to get to a point on the same line.
In our case, the ladder rises 20 feet for every horizontal change of 4 feet; therefore, the slope is
[tex]\frac{\text{rise}}{\text{run}}=\frac{20ft}{4ft}[/tex]dividing 30 ft by 4 ft gives 5; therefore,
[tex]\frac{\text{rise}}{\text{run}}=5.[/tex]Hence, the correct answer to choose is A. 5.
The diameter of Jupiter is about 1.43•10^5km. The diameter of the Earth is about 12,700km. About how many times greater is the diameter of Jupiter that the diameter of Earth
The diameter of the Earth is 11.3 times less than the diameter of the Jupiter
Ratio and proportionsFractions are written as a ratio of two integers. Given the following parameters;
Diameter of Jupiter = 1.43•10^5km
Diameter of Earth = 1.27 * 10^4km
Find the ratio
Ratio = Jupiter/Earth
Ratio = 1.43•10^5/1.27*10^4
Ratio = 1.13 * 10^1
Ratio = 11.3
Jupiter = 11.3 of Earth
This shows that the diameter of Jupiter if 11.3 times greater than Earth.
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A new heating and aip constitioner will cost the Benguin fomily $4122,theymake a down payment of 20 percent and finance the remaining amount theyObtain an instaliment loan for 36 months at an APR of 9%A What is the down payment?B What is the amount of the loan?
The cost of the new heating and air conditioner equipment is:
A = $4122
They make a down payment of 20%
A. The down payment is:
[tex]\begin{gathered} DP=\$4122\times\frac{20}{100} \\ \\ DP=\$824.40 \end{gathered}[/tex]B The amount of the loan is the remaining amount after paying the down payment:
L = $4122 - $824.40
L = $3297.60
Name two rays that contain the following line segments:• BC• GH
Two rays that contain the given line segment BC is [tex]\overrightarrow {EC}[/tex] and line segment GH is [tex]\overrightarrow {EH}[/tex] .
The length of a line segment is its measurement. Unlike a line that extends continuously, a line segment has a set length and is easy to measure.
The next link in the chain is Ray. It is made up of a line and even a mix of line segments with one terminating end and an eternally extending end.
Due to one of its ends not terminating, its length cannot be determined. Line segments are parts of a line that have two endpoints.
The construction of various shapes, such as triangles, polygons, hexagons, and squares, involves the use of a number of line segments.
From the diagram we can see that the rays EC and EH contains the given line segments. From the Rays the other line segments are BD, CD , GH.
AH is another ray.
Two rays that contain the given line segment BC is [tex]\overrightarrow {EC}[/tex] and line segment GH is [tex]\overrightarrow {EH}[/tex] .
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What is the x-intercept?
Answer:2.5
Step-by-step explanation:
l
I
------------------ x intersept
I
I
y intersept
Money set aside to pay for small, unforeseen expenses is called a(n) _____.
A.
down payment
B.
emergency fund
C.
mutual fund
D.
equity fund
Answer:
B. Emergency fund.
Step-by-step explanation:
Analyze the diagram. Which quadrilateral is a kite?
Quadrilateral N M O P is shown. Sides P N and N M are congruent.
Quadrilateral A B C D is shown. Sides A D and D C are congruent. Sides A B and B C are congruent.
Quadrilateral N M O P is shown. All sides are different lengths.
Answer: in the picture
Answer:
Quadrilateral ABCD
Step-by-step explanation:
I need help with unit rate fractions pls try to explain very very easily and well and answer quickly i gave an example
To find out the unit rate
Divide cups of sugar by the teaspoon of vanilla
so
[tex]\frac{2}{3}\colon2=\frac{2}{3*2}=\frac{1}{3}[/tex]The answer is 1/3
Option A
Yep I am watching the last one on my birthday
Given the functions
[tex]\begin{gathered} f(x)=2x-10 \\ g(x)=2x^2+12x+18 \\ h(x)=2x^3-8x^2-10x \end{gathered}[/tex]The graph of the given functions
From the graph of the functions above, it can be seen that each of the functions have the same domain
[tex]-\inftyThus, the domain is the same for each function.The answer is the first option.
7 1/5 - 6 2/5= ?
A. 1 4/5
B. 4/5
C. 1 1/5
D. 13 3/5
Hello!
So, we are given the following to solve:
[tex]7\frac{1}{5} -6\frac{2}{5}[/tex]
Convert the mixed numbers to improper fractions, then find the LCD and combine.
Exact Form of Solution:
[tex]\frac{4}{5}[/tex]
Hence, the correct choice is B. Hope this helps!
Answer:
B. 4/5
Step-by-step explanation:
7 1/5= to 36/5
6 2/5= to 32/5
subtract both numbers on top and keep the bottom, which equals to 4/5
Equation 1: A [-4x+7y=4]
Equation 2: B [3x-3y=6]
Think about using the elimination method to solve this system.
Question 1: If you wanted to eliminate the x variables with the lowest common value by multiplying each equation by numbers A and B, the value of A must be ______ and the value of B must be ______.
Question 2: If you wanted to eliminate the y variables with the lowest common value by multiplying each equation by numbers A and B, the value of A must be ______ and the value of B must be _____.
Using the elimination method, the value of A and B must 3 and 4 respectively to eliminate x in the equation. The value of A and B must be 3 and 7 respectively to eliminate y.
How to solve system of equation?System of equation can be solved using elimination method, substitution method of graphical method.
But we are asked to use elimination method.
Therefore, the system of equation are as follows:
Equation 1: A [-4x + 7y = 4]
Equation 2: B [3x - 3y = 6]
Therefore,
If you wanted to eliminate the x variables with the lowest common value by multiplying each equation by numbers A and B, the value of A must be 3 and the value of B must be 4.
If you wanted to eliminate the y variables with the lowest common value by multiplying each equation by numbers A and B, the value of A must be 3 and the value of B must be 7.
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Please help me, I am happy to contribute and learn .
Explanation:
We are told that the rate at which the pump dumps the pollutant per day to be
[tex]\frac{\sqrt{t}}{15}[/tex]To solve the question, let us assume that t is the number of days
So, to find the amount dumped after 3 days, we will put t =3 into the equation
[tex]\frac{\sqrt{3}}{15}=\frac{1.732}{15}=0.11547[/tex]Therefore, the answer is 0.115
complex vector question.A bolt is tightened by applying a force to one end of a wrench.
The Scalar and Cross Product of Vectors
Given two vectors:
[tex]\begin{gathered} \underline{r_1}=(a,b,c) \\ \underline{r_2}=(d,e,f) \end{gathered}[/tex]The scalar product is defined as:
[tex]\underline{r_1}\cdot\underline{r_2}=ad+be+cf[/tex]The cross product is the result of computing the following determinant:
[tex]\underline{r_1}\times\underline{r_2}=\begin{bmatrix}i & j & {k} \\ {a} & {b} & {c} \\ {d} & {e} & {f}\end{bmatrix}[/tex]Where i, j, and k are the unit vectors in each of the directions x, y, and z, respectively.
This concept will be applied to the following physics problem.
Given a force F= (2, 3, 0) and the distance vector d = (4, 0, 0), the torque is defined by:
[tex]\tau=r\times F[/tex]Calculating:
[tex]\tau=(4,0,0)\times(2,3,0)[/tex][tex]\tau=\begin{bmatrix}{i} & {j} & {k} \\ {4} & {0} & {0} \\ {2} & {3} & {0}\end{bmatrix}[/tex]Calculating the determinant:
[tex]\begin{gathered} \tau=0i+12k+0j-(0k+0j+0i) \\ \tau=0i+0j+12k \end{gathered}[/tex]Expressing in vector form τ = (0, 0, 12) <= should use angle brackets
The magnitude of the torque is:
[tex]\begin{gathered} |\tau|=\sqrt[]{0^2+0^2+12^2} \\ |\tau|=\sqrt[]{144} \\ |\tau|=12 \end{gathered}[/tex]The power P is equal to the scalar product of the torque by the angular velocity w. We are given the angular velocity w = (3, 3, 2), thus:
[tex]\begin{gathered} P=(0,0,12)\cdot(3,3,2) \\ P=0\cdot3+0\cdot3+12\cdot2 \\ P=24 \end{gathered}[/tex]P = 24
You flip a coin twice what is the probability of getting tails and then getting tails
The probability a single event A occurs is:
[tex]P(A)=\frac{\text{ number of favorable outcomes to A}}{\text{ number of total outcomes}}[/tex]The probability P two consecutive events A and B occurs is:
[tex]P=P(A)*P(B)[/tex]So, let's consider the probability of the first event: getting tails.
Favorable outcomes: Tail
Number of favorable outcomes: 1
Total outcomes: Tail, Head
Number of total outcomes: 2
So, the probability of getting a tail is:
[tex]P(A)=\frac{1}{2}[/tex]So, let's consider the probability of the second event: getting tails in the second time.
Favorable outcomes: Tail
Number of favorable outcomes: 1
Total outcomes: Tail, Head
Number of total outcomes: 2
So, the probability of getting a tail is:
[tex]P(B)=\frac{1}{2}[/tex]Finally, let's calculate the probability of getting a tail twice.
[tex]\begin{gathered} P=P(A)*P(B) \\ P=\frac{1}{2}*\frac{1}{2} \\ P=\frac{1}{4} \end{gathered}[/tex]Answer: The probability is 1/4.
choose the equation that could be used to find two consecutive integers whose sum is 67.
d) n + (n +1) = 67
1) Let's call the first number as n, and its consecutive as n +1
Then we can write:
n + (n +1) = 67
n +n +1 = 67
2n +1 = 67
2n = 66
n= 33
and n+1 = 34
2) Hence, the answer is n + (n +1) = 67
What is the arc length of CD in the circle below?
Solution
[tex]\begin{gathered} \theta=35^0 \\ r=8ft \end{gathered}[/tex]The formula for arc length is;
[tex]\begin{gathered} A=\frac{\theta}{360}\times2\pi r \\ \\ \Rightarrow A=\frac{35}{360}\times2\pi\times8=4.88\text{ feet} \end{gathered}[/tex]the formula for the volume of a cylinder is V=πr²h A cylinder has a volume of 300p feet³ and a radius of 5 feet (A) Solve the formula V= πr²h for h (B) Find the height of the cylinder
A) To solve the formula for h:
1. Divide both sides of the equation into π*r²:
[tex]\begin{gathered} \frac{V}{\pi\cdot r^2}=\frac{\pi\cdot r^2\cdot h}{\pi\cdot r^2} \\ \\ \frac{V}{\pi\cdot r^2}=h \end{gathered}[/tex]B) You have the next data:
V=300πfeet³
r=5feet
Substitute those values in the formula you get in A) and calculate the h:
[tex]\begin{gathered} h=\frac{300\pi\cdot ft^3}{\pi\cdot(5ft)^2} \\ \\ h=\frac{300ft^3}{25ft^2}=12ft \end{gathered}[/tex]Then, the height of the cylinder is 12 feetIf f(x) = 1 x - 2 v
x-2, what is f¹(x)?
Answer:
f¯¹(x) =9x+18, for the pictorial(image) question
what is the send question is it asking derivative or inverse
If it is inverse f¯¹(x)=-x-2 as it is
Or if it is derivative f'(x)=-1
Step-by-step explanation:
For the image question f(x)=1/9x-2,f¯¹(x)=?
f(x)=1/9x-2............given
y=1/9x-2................swapping f(x) by y to Easily write
x=1/9y-2................interchanging x and y
1/9y-2=x................changeling side of equation
9(1/9y-2)=(x)9.......multiplying both sides by 9 to override the fraction on the right side
y-18=9x
y=9x-18.................Return to where it were
f¯¹(x)=9x+18..........swap back f¯¹(x) in the y
For the question f(x)=-x-2,f¯¹(x)=?
following the ☝️ arrangement
y=-x-2
x=-y-2
-y-2=x
-y=x+2
y=-x-2
f¯¹(x)=-x-2
1. Last year the price of a college textbook(b) was $197. This year the price will be 13% higher. Which expression shows the difference in price from last year to this year? 1. b * 0.13 B. b.1.13 C.b-0.13 D.b - 13
B) b*1.13
1) Since the College textbook's price has been raised up 13%, then we can write:
b(1 +0.13)
b(1.13)
Plugging b = $197 we have:
197(1 +0.13)
197 (1.13)
$222.61
2) So the factor that expresses that difference (13% up) is 1.13
For 1 is equivalent to 100% and 0.13 to 13%
3) Hence, the answer is b* 1.13 calling b that $197.
Make three problem about finding DOMAIN X-intercept Y-intercept Vertical Asymptote Horizontal asymptote
A graph's domain, which is defined as the entire set of input values visible on the x-axis, refers to the set of possible input values. The possible output values are displayed on the y-axis and make up the range.
What is Vertical and Horizontal asymptote?Asymptotes are a distinctive feature of the graphs of rational functions. When a curve is nearing the edges of a coordinate plane, it is said to be asymptote. A rational function's vertical asymptotes happen as its denominator gets closer to zero.
In order to cross a vertical asymptote, a rational function must divide by one, which is impossible. When the x-values increase significantly in size, either positively or negatively, horizontal asymptotes develop. You can pass through horizontal asymptotes.
A vertical asymptote of a graph is a vertical line with the equation x = a, where the graph tends toward positive or negative infinity as the inputs get closer to a.
A graph's horizontal asymptote is a horizontal line, y = b, where the graph moves toward the line as the inputs move toward ∞+ or ∞-.
Three problem about finding DOMAIN, X-intercept, Y-intercept, Vertical Asymptote, Horizontal asymptote
1) Determine the vertical asymptote(s), horizontal or slant asymptote, x-intercept(s), y-intercept, and domain. Then, sketch a graph of the function on the given set of axes. Label all asymptotes and intercepts.
[tex]m(x) = \frac{3x^2 -12}{x^2 -7x + 6}[/tex]
2) Determine the Domain, Y-intercept, x-intercept(s), Vertical Asymptote(s), and Horizontal Asymptote, if the exist: Include the multiplicity of the x-intercepts if the multiplicity is greater than 1. Then graph the ratio function.
[tex]v(x) = \frac{3x - 1}{x^2+5x +6}[/tex]
3) What are the Domain, x-intercept, y-intercept, vertical asymptote and horizontal asymptote of the rational function [tex](x^3-x+12/x^2-3x-4)[/tex]?
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A person has a rectangular board 14 inches by 18 inches around which she wants to put a uniform border of shells. If she has enough shells for a border whose area is 320 square inches, determine the width of the border.
The width of the border will be 8 inches.
In the given question, it is stated that a rectangular board has dimensions of 18*14 around which a person has to put a border of shells. If there are enough shells for the border of an area of 320 square inches, we need to find out the Width of the border.
Firstly, we know the Area of Rectangle is l*b => Length*Breadth
So, now we calculate the area of the board with dimensions l = 18 and b = 14
=> A₁ = l*b
=> A₁ = 18*14
=> A₁ = 252 square inches
We get the Area of the Board as 252 Square inches.
Now, there are shells enough to cover a border area of 320 square inches.
So, the Board and Border area will be A₁ + A, where A₁ and A are the areas of the Board and border respectively.
So, Total Area = 252 + 320
Total = 572 Square inches.
Now, let 'w' be the width of the border. Then the dimensions for the board will be
l = (18 + w)
b = (14 + w)
Using the formula for the Area of the rectangle, we get
=> Total = l * b
=> Total = (18 + w)*(14 + w)
=> 572 = w² + 32m + 252
=> w² + 32m - 320 = 0
=> w² + 40w -8w -320
=> w(w + 40) -8(w + 40)
=> (w+40) (w-8)
We get factors as w = -40 and 8
Since Width can not be negative quantity, width w will be 8 inches.
Hence, the width will be 8 inches for the border.
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The office manager orders computer paper every 6 weeks and printer ink cartridges every 8 weeks.
QUESTION: If she places an order for both paper and ink this week, how many weeks will it be until she orders them both in the same week again?
The number of weeks that she orders them both in the same week again is 24 weeks.
How to illustrate the information?From the information, the office manager orders computer paper every 6 weeks and printer ink cartridges every 8 weeks.
The weeks when they will order same will be the least common multiple for 6 and 8. This will be:
6 = 6, 12, 18, 24
8 = 8, 16, 24
The multiple is 24 weeks. This illustrates the information.
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