The answer is either Option (a) or option(b) or both the options i.e, 120 or √144
What is Integer ?
An integer is a whole number from the set of -ve , non-negative , +ve and 0 numbers.
Let's see how both the option is correct
we know, integers are 0,1, 2, 3, .... or -1, -2 , -3 ......
Means the number which is not in fraction or decimal value or root.
so, by seeing option(a) we can say it's an integer
but in option(b) it's the root so how it is integer
The answer is simple it's perfect square of 12 so √144 = 12 so that means it have the value which is not in fraction. By that we can say that option(b) is also an integer
Now, in option(c) and (d) they are not the perfect square and their root results in decimal values. so we cannot tell them as an integer value.
Hence, the answer is either Option (a) or option(b) or both the options.
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f(x)=[tex]\sqrt{x}[/tex], g(x)=x+9
A: (fg)(x)= ??, Domaine of fg=?
B(gf)(x)=??, Domaine of gf=?
The values are as:
a) f(g(x)) = √(x+ 9)
b) (gf)(x)= √x +9
What is function?The core concept of mathematics' calculus is functions. The unique varieties of relations are the functions. In mathematics, a function is represented as a rule that produces a distinct result for each input x. In mathematics, a function is indicated by a mapping or transformation. Typically, these functions are identified by letters like f, g, and h. The collection of all the values that the function may input while it is defined is known as the domain. The whole set of values that the function's output can produce is referred to as the range. The set of values that might be a function's outputs is known as the co-domain.
Given:
f(x)=√x, g(x)=x+9
a) (fg)(x)=
f(g(x)) = f( x+9)
= √(x+ 9)
Now, domain is all the input values
i.e., x=2, 4, 7
f(g(2)) = √(2+ 9)
= √11
and, f(g(4)) = √(4+ 9)
= √13
and, f(g(7)) = √(7 + 9)
= √16
= 4
b) (gf)(x)= g(f(x))
= g(√x)
= √x +9
Now, domain is all the input values
i.e., x=2, 4, 7
f(g(2)) = √2+ 9
and, f(g(4)) = √(4+ 9)
= 2+9
= 11
and, f(g(7)) = √7 + 9
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Elena is making an open top box by cutting squares out of the corners of a piece of paper that is 11 inches wide and 17 in long and then folding up the sides if the side length of a square cut outs RX in in the volume of the box is given by [tex]v(x) = x(11 - 2x)(17 - 2x)[/tex]what is a reasonable domain for V of x?approximately which value of x will give her a box with the greatest volume round to the nearest whole numberfor approximately which values of X is the volume of the Box increasing round to the nearest whole number
The expression for the volume of the box is:
[tex]v(x)=x(11-2x)(17-2x)[/tex]Mathematically, there is no restriction for the values of x, but phisically we know that x is a length and has a positive value, so x>0.
Also, we know that x can not be largest than half of the width, that is the smallest dimension of the piece of paper.
As the width is 11, we then know that x is smaller than 11/2=5.5.
In conclusion, the domain for x is:
[tex]0To calculate the maximum volume for the box we have to derive the volume function and equal to zero:[tex]\begin{gathered} v(x)=x(11-2x)(17-2x) \\ v(x)=x(11\cdot17-11\cdot2x-2x\cdot17+4x^2) \\ v(x)=x(4x^2-56x+187) \\ v(x)=4x^3-56x^2+187x \end{gathered}[/tex][tex]\begin{gathered} \frac{dv}{dx}=4(3x^2)-56(2x)+187(1)=0 \\ 12x^2-112x+187=0 \\ x=\frac{-(-112)\pm\sqrt[]{(-112)^2-4\cdot12\cdot187}}{2\cdot12} \\ x=\frac{112\pm\sqrt[]{12544-8976}}{24} \\ x=\frac{112\pm\sqrt[]{3568}}{24} \\ x=\frac{112}{24}\pm\frac{59.73}{24} \\ x=4.67\pm2.49 \\ x_1=4.67-2.49=2.18\approx2 \\ x_2=4.67+2.49=7.16\approx7 \end{gathered}[/tex]The solutions are x=2 and x=7 approximately.
Because of our domain definition, we know that x=7 is not a valid solution, so the value of x that maximizes the volume is x=2.
The volume for x=0 is 0. Then, it will increase its value until x=2, where it reaches the maximum volume. From x=2 to x=5.5, the volume decrease until reaching v=0 at x=6.5.
Answer:
Domain: 0
Value of x that maximizes the volume: x=2.
From x=0 to x=2 the volume of the box increases.
URGENT!! ILL GIVE
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Angles are given below.
Define angles.When two straight lines or rays intersect at a single endpoint, an angle is created. The vertex of an angle is the location where two points come together. The Latin word "angulus ," which means "corner," is where the term "angle" originates. When a transversal connects two coplanar lines, alternate interior angles are created. They are located on the transverse sides of the parallel lines, but on the inner side of the parallel lines. At two different locations, the transversal passes through the two lines that are coplanar.
Given,
∠6 and ∠7 = Vertical angles
∠2 and ∠8 = Same side exterior angles
∠1 and ∠5 = Corresponding angles
∠3 and ∠6 = Adjacent angles
∠2 and ∠7 = Alternate exterior angles
∠4 and ∠6 = Same side interior angles
∠1 and ∠2 = Linear pair
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Bonus: Write the equation of a line in slope intercept form that is parallelto y=4/3x-7 and contains the point (5,-8)
The given equation is
[tex]y=\frac{4}{3}x-7[/tex]The slope of the given line is 4/3 because it's the coefficient of x.
Now, the new line has a slope of 4/3 too because parallel lines have equal slopes.
We know that the new line passes through the point (5, -8). Let's use the point-slope formula to find the equation.
[tex]y-y_1=m(x-x_1)[/tex]Replacing the points and the slope, we have.
[tex]\begin{gathered} y-(-8)=\frac{4}{3}(x-5) \\ y+8=\frac{4}{3}x-\frac{20}{3} \\ y=\frac{4}{3}x-\frac{20}{3}-8 \\ y=\frac{4}{3}x+\frac{-20-24}{3} \\ y=\frac{4}{3}x-\frac{44}{3} \end{gathered}[/tex]Therefore, the equation of the new line is y = (4/3)x - (44/3).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 shows that the range of the function is 0.9 ≤ M ≤ ∞.
Linear FunctionA linear function can be represented by a line. The standard form for this equation is: y=mx+b , for example, y=5x+8.
All functions present their domain and range. The domain of a function is the set of input values for which the function is real and defined. In the other words, when you define the domain, you are indicating for which values x the function is real and defined. While the domain is related to the values of x, the range is related to the possible values of y that the function can have.
From the graph it is possible to see that: the function is a linear function, the values of the coordinate x are represented by the volume (liters) while the values of the coordinate y are represented by the mass (kg).
The question asks for the range of the function. Therefore, you should indicate the possible values of y that the function can have.
For this, you should analyze the axis-y. See that for x=0, the graph shows y =0.9. Therefore, the function starts for values of y >= 0.9 kg. It is possible to verify that when the volume increases, the mass also increases. With this information, you can find that the range is 0.9 ≤ M ≤ ∞.
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A single fair die is tossed. Find the probability of rolling a number greater than 5
Using the probability concept, the odds of rolling a number greater than 5 is 1 :5
What is probability ?
probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur or how likely it is that a proposition is true . the probability of an event is a number between 0 and 1 , where ,roughly speaking ,0 indicates impossibility of the event and 1 indicates certainty.
The odd of a particular experiment is defined thus :
Number of possible outcomes greater than 5 : number of possible below or equal to 5
Sample space = {1, 2, 3, 4, 5, 6}
Outcomes greater than 5 = {6} = 1
Outcomes below or equal to 5 = {1, 2, 3, 4, 5} = 5
The odds equals to 1 : 5
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Rick Takei has a 4-wheel drive vehicle whose average retail value is $15,857. A used vehicle guide adds $60 for heated outside mirrors, $250 for rear and side air bags. $175 for cruise control, and $100 for remote keyless entry. It suggests deducting $750 for excessive mileage. What is the average retail price?
We are asked to find the final average retail price of the vehicle after the given additions and deductions.
The average retail value is $15,857
Add $60 for heated outside mirrors.
[tex]$\$15,857+\$60=\$15,917$[/tex]Add $250 for rear and side airbags.
[tex]\$15,917+\$250=\$16,167[/tex]Add $175 for cruise control.
[tex]\$16,167+\$175=\$16,342[/tex]Add $100 for remote keyless entry.
[tex]\$16,342+\$100=\$16,442[/tex]Deduct $750 for excessive mileage.
[tex]\$16,442-\$750=\$15,692[/tex]Therefore, the average retail price is $15,692
Find the area of ABC with vertices A(3,-6), B(5,-6), and C(7,–9).
Area of a triangle ABC with the given vertices is 3 square units.
Given that, the vertices of a triangle ABC, A(3,-6), B(5,-6), and C(7,–9).
What is the area of triangle formula in coordinate geometry?In Geometry, a triangle is a three-sided polygon that has three edges and three vertices. The area of the triangle is the space covered by the triangle in a two-dimensional plane.
Area of a triangle = [tex]\frac{1}{2} ( |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|)[/tex]
Here, (x1, y1) = A(3,-6), (x2, y2) = B(5,-6), and (x3, y3) = C(7,–9)
Now, the area of a triangle = 1/2 (|3(-6+9)+5(-9+6)+7(-6+6)|)
= 1/2 (|3(3)+5(-3)+7(0)|)
= 1/2 (|(9-15)|)
= 1/2 × 6
= 3 square units
Therefore, area of a triangle ABC with the given vertices is 3 square units.
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Ricky grows plants for a science project at the start of his project the plants average a height of 3 inches after three weeks the average player height was 7 inches what was the percent of the change in the average height of the plants?
To answer this question, we will use the following formula for the percent of change:
[tex]P=\frac{F-I}{I}\times100,[/tex]where F is the final height and I is the initial height.
Substituting F=7, and I=3 we get:
[tex]P=\frac{7-3}{3}\times100.[/tex]Simplifying the above result we get:
[tex]F=\frac{400}{3}\text{.}[/tex]Answer: 400/3 %.
The mathematics section of a standardized college entrance exam had a mean of and an SD of for a recent year. Assume these are well modeled by a Normal distribution.
Normal distributions are crucial to statistics because they are widely used in the natural and social sciences to represent real-valued random variables whose distributions are unknown.
Normal distribution is a type of continuous probability distribution for a random variable with a real value.
They are important in part because of the central limit theorem. This claim asserts that, in some cases, the mean of many samples (observations) of a random variable with finite mean and variance is itself a random variable, whose distribution tends to converge to a normal distribution as the number of samples increases. Because of this, the distributions of physical quantities, such as measurement errors, which are assumed to represent the sum of multiple distinct processes, are usually close to normal. It may also be referred to as a Laplace-Gauss, Gaussian, or Gaussian distribution.[tex]{\displaystyle f(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {1}{2}}\left({\frac {x-\mu }{\sigma }}\right)^{2}}}[/tex]Another name for a normal distribution is a bell curve.The Cauchy, Student's t, and logistic distributions, among many others, have a bell-shaped structure.The univariate probability distribution is generalized for vectors in the multivariate normal distribution and for matrices in the matrix normal distribution.Hence we can use the normal distribution to find the mean and standard deviation of a data set.
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Point m represents the opposite of negative 1/2 and point n represents the opposite of positive 5/2 which number line correctly shows points m and n great
If Point m represents the opposite of negative 1/2 and point n represents the opposite of positive 5/2. Then M is 1/2 and N is -5/2.
What is Number system?A number system is defined as the representation of numbers by using digits or other symbols in a consistent manner.
A number line is a picture of a graduated straight line that serves as visual representation of the real numbers.
Given that point M represents the opposite of negative 1/2. Which means opposite of -1/2. The opposite of -1/2 means positive of 1/2. The sign changes.
Point N represents the opposite of positive 5/2. Which means opposite of 5/2. The opposite of 5/2 means negative of 5/2. The sign changes.
opposite of positive 5/2 is -5/2.
Now let us plot this values on a number line. 1/2 is 0.5 and -5/2 means -2.5.
The graph is attached below.
Hence M is 1/2 and N is -5/2
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If f (x) = x
2 − 2 x , g (x) = x − 2
1) prove that : f(2) = g(2)
2) If g (K) = 7 , find : the value of k
The value of k is 9 for the function g.
To solve this problem we should have a brief concept of algebraic functions.
To solve this problem we have to follow a few steps.
Here f is a function of x and the relation with the function denotes as x²-2x. Also, g is a function of x and the relation with the function denotes as x− 2.
If we put, x = 2 on f(x) = x²-2x. We can write, f(2) = 2²-2.2 = 4 - 4 = 0.
If we put, x = 2 on g (x) = x − 2. We can write, g(2) = x− 2= 2- 2 = 0.
Hence, we can conclude that f(2) = g(2) = 0. ( proved)
Here, g(k) = 7. So, x = k in this relation.
We have to put x= k on g(x) = x− 2 ; now we can write, g (k) = k− 2.
g (k) = k− 2 = 7 as per the question. Therefore k = 7 + 2 = 9
The value of k is 9.
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The correct question is,
If f (x) = x²-2x
g (x) = x − 2
1) prove that : f(2) = g(2)
2) If g (K) = 7, find the value of k
I need the answers as soon as you can! I’m on the clock
1) The first thing to do is to find the vertex. So let's do it, considering that we already have the vertex and then we can find the coefficients. Note that all functions have a=1:
[tex]\begin{gathered} h=-\frac{b}{2a} \\ 9=\frac{-b}{2(-1)} \\ 9\cdot-2=-b \\ -b=-18 \\ b=18 \\ ----- \\ \end{gathered}[/tex]2) Now, let's find the other coefficient "c", since we've got a,b.
[tex]\begin{gathered} k=\frac{-\Delta}{4a} \\ 7=\frac{-(18^2-4(-1)(c))}{4(1)} \\ 28=-(324+4c) \\ 28=-324-4c \\ 28+324=-4c \\ 352=-4c \\ c=\frac{352}{-4} \\ c=-88 \end{gathered}[/tex]3) Thus the answer is:
[tex]D[/tex]please helpjjjjjjjjjjjjjjjj
Answer:
Diverge i think.
Step-by-step explanation:
See the photo
Find in the exact simplified form of an exact expression for the sum of the first n terms of the following series 1+11+111+1111+11111+.... Binary notation is used to represent numbers on a computer. For example, the number 1111 in base two represents 1(2)^3 + 1(2)^2 +1(2)^1+1, or 15 in base ten. (i) Why is the sum above an example of a geometric series? (ii) Which number in base ten is represented by 11 111 111 111 111 111 111 in base two? Explain your reasoning.
Step-by-step explanation:
so, I understand, the given series is written in binary form.
a1 = 1 = 1×2⁰ = 1
a2 = 11 = 1×2¹ + 1× 2⁰ = 3
a3 = 111 = 1×2² + 1×2¹ + 1×2⁰ = 7
a4 = 1111 = 1×2³ + 1×2² + 1×2¹ + 1×2⁰ = 15
a5 = 11111 = 1×2⁴ + 1×2³ + 1×2² + 1×2¹ + 1×2⁰ = 31
...
we see, that
an = 2×(an-1) + 1
a1 = 1
a2 = 2×a1 + 1
a3 = 2×a2 + 1 = 2×(2×a1 + 1) + 1 = 4×a1 + 2 + 1
a4 = 2×a3 + 1 = 2×(2×a2 + 1) = 2×(2×(2×a1 + 1) + 1) + 1 =
= 8×a1 + 2×2 + 2 + 1 = 8×a1 + 7
...
an = (2^(n-1))×a1 + an-1
because
an = 2×(an-1) + 1,
an-1 = (2^(n-1))×a1 - 1
therefore,
an = 2×(2^(n-1))×a1 - 1 = (2^n)×a1 - 1
the sequence of the sums of the first n elements
s1 = a1 = 1
s2 = a1 + a2 = 1 + 3 = 4
s3 = a1 + a2 + a3 = 7 + 3 + 1 = 11
s4 = a1 + a2 + a3 + a4 = 15 + 7 + 3 + 1 = 26
...
(i)
it is NOT a geometric sequence.
for a geometric sequence
an/an-1 = r, and r must be a constant ratio for any n.
but
7/3 = 2.333333...
15/7 = 2.142857143...
these are different, so, the sequence itself is not geometric.
neither is the sequence of the sums of the series. because
11/4 = 2.75
26/11 = 2.363636363...
are different.
1, 2, 4, 8, 16, 32, ... is a geometric sequence (constant r = 2).
but not
1, 3, 7, 15, 31, ...
(ii)
11 111 111 111 111 111 111 in base 2.
the utmost right position is the 2⁰ position. every position further to the left multiples the position value by 2. it is the same process as for numbers in base 10 (just there every position value is multiplied by 10).
we have 6×3 + 2×1 positions = 20 positions.
so, the position values go from 2⁰ to 2¹⁹.
as per the formula for "an" up there, we get
a20 = (2²⁰)×a1 - 1 = 1,048,576 - 1 = 1,048,575
Lesson 2: Exit Ticket
Flipping Ferraris
Find the inverse of each function:
b. g(x)= √x
The inverse of the functions given as y = 5(x-2) and f^-1(x) = x²
Inverse of a functionA function's inverse function reverses the action of a function, or f.
a) From the given table, we can use the coordinate points (-5, 1) and (0, 2)
The standard linear equation is y = mx + b
m = 2-1/0-(-5)
m = 1/5
Since the y-intercept is (0, 2), hence the required function is y = 1/5 x + 2.
y = 1/5x + 2
x = 1/5 y + 2
1/5 y = x - 2
y = 5(x-2)
b) For the function g(x) = √x
y = √x
x = √y
y = x²
f^-1(x) = x²
This gives the inverse of the function.
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Line a is parallel to line b and line c is parallel to line d, using the diagram what can be said about angle 7 and 12
Answer:
C
Step-by-step explanation:
Angle 7 is congruent to angle 5 by the corresponding angles theorem, and angles 5 and 12 are supplementary because they are consecutive interior angles.
Thus, angles 7 and 12 are supplementary.
For each expression, combine like terms and write an equivalent expression with fewer terms.a. 4x+3xb. 3x+5x-1c. 5+2x+7+4xd. 4-2x+5xe. 10x-5+3x-2
To simplify the expressions you have to combine the like terms.
This means that you'll solve the operations between the terms that have the same variables, for example x + 2x=3x
Or the terms that have no variables and are only numbers, for example 4+5=9
a. The expression is
[tex]4x+3x[/tex]Both terms have the same variable "x", so you can add them together. To do so, add the coefficients, i.e. the numbers that are being multiplied by x
[tex]4x+3x=(4+3)x=7x[/tex]And you get that the simplified expression is 7x
b. The expression is
[tex]3x+5x-1[/tex]In this expression you have two types of terms, the x-related terms and one constant. In this case you have to solve the operation for the x-related terms together and leave the constant as it is
[tex](3x+5x)-1=(3+5)x-1=8x-1[/tex]The simplified expression is 8x-1
Question 4 2 pts A fireman leaned a 36-foot ladder against a building. If he placed the ladder 12 feet from the base of the building, what angle is formed between the ladder and the ground? 0 78.8 Degrees 70.5 Degrees O 77.2 Degrees O 80.4 Degrees O 75.5 Degrees « Previous
x is the ladder ,x =36
y is the distance between ladder and wall y=12
z is the wall
we have in this triangle , only the hypotenuse(x) and the adjacent side
so,'
[tex]\cos \theta=\frac{y}{x}=\frac{12}{36}=0.33[/tex][tex]\theta=\cos ^{-1}(0.33)[/tex][tex]\theta=70\circ(approximately)[/tex]Graph the linear function f(x)=4x+1 by plotting points.
To plot points, click on a point on the graph and drag it to the desired location.
A graph of the linear function (f(x) = 4x + 1) is shown in the image attached below.
What is a graph?A graph is a type of chart that's commonly used for the graphical representation of data on both the horizontal and vertical lines of a cartesian coordinate, which are the x-axis and y-axis.
Generally speaking, the graph of any proportional relationship such as a linear function is characterized by a straight line with the points passing through the origin (0, 0) because as the values on the x-axis increases or decreases, the values on the y-axis increases or decreases simultaneously as shown in the image attached below.
In conclusion, a graph which represents the given linear function shows a proportional relationship between the value of x and y.
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Consider that AABC is similar to AXYZ and the measure of ZB is 68º. What is the measure of ZY? A) 70° B) 68° C) 41° D) 22°
Answer
Option B is correct.
Angle Y = 68º
Explanation
Similar triangles have the same set of angles in them.
All the corresponding angles are equal to each other.
So, if triangle ABC is similar to triangle XYZ
Angle A = Angle X
Angle B = Angle Y
Angle C = Angle Z
The order in which they are named determines the angles that are corresponding to each other.
So, if Angle B = 68º
Angle Y = Angle B = 68º
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what is the answer to 389+_=2,897
Answer:
the answer is 2508
Evaluate the expression for the given variable.9 - k ÷ 3/4 k=2/3
We are given the following expression:
[tex]9-k\div\frac{3}{4}[/tex]We are also given that k is equal to 2/3. So, we can substitute that into the expression:
[tex]9-\frac{2}{3}\div\frac{3}{4}[/tex]Due to order of operations, we have to do the division first, and then do the subtraction. To do division with fractions, we keep the first fraction the same and take the reciprocal of the second fraction. Then, we can multiply the two fractions. Let's do that:
[tex]9-(\frac{2}{3}\div\frac{3}{4})=9-(\frac{2}{3}*\frac{4}{3})=9-\frac{8}{9}[/tex]Now, we can do the subtraction:
[tex]9-\frac{8}{9}=\frac{81}{9}-\frac{8}{9}=\frac{73}{9}[/tex]Therefore, our answer is 73/9
Given f(x) = 2x² + 2x + k, and the remainder when f(x) is divided by x - 1 is
13, then what is the value of k?
Answer:
Step-by-step explanation:
Set up either long division or synthetic div. I'd do the latter.
Your divisor for synth. div. should be -7.
__________
Then -7 / 2 9 k
-14 +35
--------------------
2 -5 k+35 Rem is 32;
Let 32 = k + 5 and solve for k:
k = -3.
Let's check that. Is k correct?
Then -7 / 2 9 -3
-14 +35
--------------------
2 -5 -3+35 = 32
Since the rem is 32, we are correct; k = -3.
Give a negation of each inequality.
p < 9
Answer: P can be anything from 8 to below
Example: 8, 7, 6, 5, 4, 3, 2, 1, 0, -1 .....
Give three value to x such that |xl= -x.
We have the following equation:
[tex]\left|x\right|=-x[/tex]And we want to identify values who satisfy the equation. And the possible answers for this case are:
x=0
Since :
[tex]\left|0\right|=-0=0[/tex]Other two possible answers are:
[tex]\left|\frac{0}{10}\right|=-\frac{0}{10}=0[/tex]28282838383833883+182827272727277227
Answer:
[tex]211,110,111,111,111,110[/tex]Explanation:
We want to find the sum of the numbers;
[tex]28282838383833883+182827272727277227[/tex]solving;
Therefore;
[tex]28,282,838,383,833,883+182,827,272,727,277,227=211,110,111,111,111,110[/tex]Jackson puts 600.00 into an account to use for school expenses the account earns 2% interest compounded quarterly monthly how much will be in the account after 10 years round your answer to the nearest cent
By the compound interest formula, you know that
[tex]undefined[/tex]Input x Output y3. -56. -49. -3What is a equation
We are to determine the equation of line by interpreting tabulated results between an independent variable ( x ) and a dependent variable ( y ).
A function is usually expressed as follows:
[tex]y\text{ = f ( x )}[/tex]The above notation gives us the output ( y ) which is a function of input variable ( x ). This means that whatever relationship these two variables have the value of output ( y ) is related to the imput variable ( x ).
We are given a table/list of values of output ( y ) corresponding to each value of input variable ( x ) as follows:
Input ( x ) Output ( y )
3 -5
6 -4
9 -3
There are a series of steps that we must take to arrive at the equation that relates two variables.
Step 1: Determine the type of relationship between two variables by intuition
The first step in the process is the hardest of all. We have to critically analyze each input value ( x ) and its corresponding output value ( y ) with successive pair of values.
There are many types of relationships possible ( polynomial order, exponential, logarithmic, trigonometric, radical, etc .. ).
We can conjure up a way by comparing outputs of successive values to determine the type of relationship possible.
So looking at the first value:
[tex]y\text{ = f ( 3 ) = -5}[/tex]The successive value:
[tex]y\text{ = f ( 6 ) = -4}[/tex]The next successive value:
[tex]y\text{ = f ( 9 ) = -3}[/tex]Here if scrutinize between each successive value of input variable ( x ) we see that there is a "3 unit step-up" in each pair of values i.e ( 3 -> 6 -> 9 ).
Next we compare each output values ( y ) for successive pairs. We see that with every step increase of 3 units in ( x ) value there is an increase of ( 1 ) unit in output value i.e ( -5 -> -4 -> -3 ).
Conclusion: Combing the result of above analysis we see that with each 3 step increase in input value ( x ) there is an increase in output value ( y ) by 1 unit.
This gives us the idea that the two variables are linearly related to one another.
Therefore, the type of relationship is:
[tex]\text{straight line }\text{ }[/tex]Step 2: Recall the equation for the type of relationship between two vairbales x and y
Once we have determined the type of relationship between two variables. We will have to resort to our equation bank and pluck out the corresponding equation that expresses a LINEAR relationship i.e equation of a straight line.
The slope-intercept form of a straight line is:
[tex]y\text{ = m}\cdot x\text{ + c}[/tex]Step 3: Determine the complete equation of function by defining the arbitrary constants.
The above equation is valid for all straight lines that express a linear relationship. However, we seek to find a unique straight line for the given set of points.
Every unique straight line equation would have either of the constants different. The constants defined in a striaght line equation are:
[tex]\begin{gathered} m\colon\text{ The slope( gradient ) of the line} \\ c\colon\text{ The y-intercept} \end{gathered}[/tex]To determine these constants we will use the given pairs of coordinates of input and output variables, x and y respectively.
To determine the slope (m) of an equation:
[tex]m\text{ = }\frac{y_2-y_1}{x_2-x_1}[/tex]The above expression relates the change in output value ( y ) with respect to change in input variable ( x ).
To determine the constant ( m ) we will use the conclusion from Step 1:
"3 step increase in input of ( x ) value there is an increase in output value ( y ) by 1 unit."
Therefore,
[tex]m\text{ = }\frac{+1}{+3}\text{ = }\frac{1}{3}[/tex]To determine the value of y-intercept ( c ). We will plug in the value of ( m ) into the general equation of a straight line written in step 2:
[tex]y\text{ = }\frac{1}{3}x\text{ + c}[/tex]Now, we will use any pair of input and output value.
[tex]x\text{ = 3 , y = -5}[/tex]Substitute the pair of values into the derived equation expressed above and solve for constant ( c ):
[tex]\begin{gathered} -5\text{ = }\frac{1}{3}\cdot(3)\text{ + c} \\ -5\text{ = 1 + c} \\ c\text{ = -6} \end{gathered}[/tex]Note: The above step implies that following equation must satisfy each and every data pair of point given to us ( table ). Or each and every value must lie on the line. For that each value must satisfy the equation of line.
Step 4: Write the complete equation of the relationship
Once we have evaluated the values of equation defining constants ( m and c ). We can simply plug in the values into the general equation relationship ( Linear - slope intercept form ) as follows:
[tex]m\text{ = }\frac{1}{3}\text{ , c = -6}[/tex]Therefore, the equation for the set of values given to us is:
[tex]\begin{gathered} \textcolor{#FF7968}{y}\text{\textcolor{#FF7968}{ = }}\textcolor{#FF7968}{\frac{1}{3}\cdot x}\text{\textcolor{#FF7968}{ - 6}} \\ OR \\ y\text{ = }\frac{x\text{ - 18}}{3} \\ \textcolor{#FF7968}{3y}\text{\textcolor{#FF7968}{ = x - 18}} \end{gathered}[/tex]Find the area of the shaded region show or explain your reasoning
The area of the shaded region will be 28 cm². The shaded region is a combination of two rectangles.
What is the area of the shaded region?The area of the shaded area is the difference between the total area of the polygon and the area of the portion of the polygon that is not shaded. In polygons, the area of the shaded component might appear in two different ways. A polygon's sides or its center are both potential locations for the shaded area.
From the triangle, it is obtained that the shaded region is the combination of a 4×6 rectangle and a 2×2 square.
Area of shaded region = Area of rectangle + Area of square
Area of shaded region = (6-2)×6 + 2×2
Area of shaded region = 4×6 + 4
Area of shaded region = 24 + 4
Area of shaded region = = 28 cm²
Thus, the area of the shaded region will be 28 cm². The shaded region is a combination of two rectangles.
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