Recall the formula for the Future value of a principal amount, compounded continuously.
[tex]\begin{gathered} A=Pe^{rt} \\ \text{where} \\ A\text{ is the Future Value} \\ P\text{ is the Principal amount} \\ r\text{ is the rate in decimals} \\ t\text{ is time in years} \end{gathered}[/tex]Rearrange the equation so that we can solve for time t.
[tex]\begin{gathered} A=Pe^{rt}^{} \\ \frac{A}{P}=\frac{Pe^{rt}}{P} \\ \frac{A}{P}=e^{rt} \\ e^{rt}=\frac{A}{P} \\ \ln e^{rt}=\ln \mleft(\frac{A}{P}\mright) \\ rt\ln e^{}=\ln \mleft(\frac{A}{P}\mright) \\ rt^{}=\ln \mleft(\frac{A}{P}\mright) \\ \frac{rt}{r}=\frac{\ln (\frac{A}{P})}{r} \\ t=\frac{\ln(\frac{A}{P})}{r} \end{gathered}[/tex]Now substitute the following given and we have
[tex]\begin{gathered} t=\frac{\ln(\frac{A}{P})}{r} \\ t=\frac{\ln (\frac{15000}{6750})}{0.0525} \\ t=\frac{\ln (\frac{20}{9})}{0.0525} \\ t=15.2096704\text{ years} \end{gathered}[/tex]Rounding the answer to the nearest tenth, the investment will reach $15000, in approximately 15.2 years.
Solve the following system of equations using an augmented matrix and Gauss-Jordan Elimination. Be sure to show your work and explain what you are doing. Then, interpret your answer in terms of the original system.
Okay, here we have this:
Considering the provided equation, we are going to solve the system using an augmented matrix and Gauss-Jordan Elimination. So we obtain the following:
[tex]\begin{gathered} \begin{bmatrix}3x+2y-4z=4 \\ x-3y-10z=8 \\ -5x-4y+12z=-2\end{bmatrix} \\ \begin{bmatrix}\frac{4-2y+4z}{3}-3y-10z=8 \\ -5\cdot\frac{4-2y+4z}{3}-4y+12z=-2\end{bmatrix} \\ \begin{bmatrix}\frac{-11y-26z+4}{3}=8 \\ \frac{-2y+16z-20}{3}=-2\end{bmatrix} \\ \begin{bmatrix}\frac{-2\left(-\frac{26z+20}{11}\right)+16z-20}{3}=-2\end{bmatrix} \\ \begin{bmatrix}\frac{4\left(19z-15\right)}{11}=-2\end{bmatrix} \\ y=-\frac{26\cdot\frac{1}{2}+20}{11} \\ y=-3 \\ x=\frac{4-2\left(-3\right)+4\cdot\frac{1}{2}}{3} \\ x=4 \\ \end{gathered}[/tex]Finally we obtain that the solution to the system is:
[tex]x=4,\: z=\frac{1}{2},\: y=-3[/tex]A scientist mixes water (containing no salt) with a solution that contains 15% salt. She wants to obtain 105 ounces of a mixture that is 5% salt. How many ounces of water and how many ounces of the 15% salt solution should she use?
Let's call X the ounces of water and Y the ounces of the salt solution.
Of she
Using the point (-5, 4) has one endpoint, State a possible location of the other endpoint given the line segment is 7 units long. Apply the distance formula to create a possible endpoint(s) from a given location.
EXPLANATION
Since the line segment is 7 units long, we can apply the following relationship:
(x_1+ 7 , y_1) = (x_2 , y_2)
[tex](-5+7)=2[/tex]The coordinate of the endpoint is as follows:
[tex](x_{endpoint},y_{endpoint})=(2,4)[/tex]We can get to this point by applying the distance formula as follows:
[tex]distance=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]Applying the square power to both sides:
[tex]7^2=(x_2-(-5))^2+(y_2-4)^2[/tex]Subtracting numbers:
[tex]49=(x_2+5)^2+(y_2-4)^2[/tex]Now, if the x_2 coordinate is -3, the value of y_2 will be as follows:
[tex]49=(-3+5)^2+(y_2-4)^2[/tex][tex]49=4+(y_2-4)^2[/tex]Subtracting -4 to both sides:
[tex]45=(y_2-4)^2[/tex]Applying the square root to both sides:
[tex]\sqrt{45}=y_2-4[/tex]Adding +4 to both sides:
[tex]4+\sqrt{45}=y_2[/tex]In conclusion, the equation to get the coordinate from a given point is,
[tex]49=(x_{2}+5)^{2}+(y_{2}-4)^{2}[/tex]X+87°+2x degrees i need to solve for angle 2x
Answer:
62
Step-by-step explanation:
x + 87 and 2x are linear pair angles.
Sum of linear pair angles is 180,
x + 87 + 2x = 180
x + 2x + 87 = 180
3x + 87 = 180
3x = 180 - 87
3x = 93
x = 93 / 3
x = 31
2x
= 2 * 31
= 62
Please Help!!!Use division to determine if the binomial is a factor. Show all work (x^3+7x^2+16x+12) ; (x+2)
Clearly, we can see that x+2 is a factor of the given polynomial.
Find the point of intersection of the pair of straight lines.
y = −10x − 3
−y = 11x + 5
(x, y) =
The point of intersection is a point where the value of both functions will be the same thus the point of intersection of the lines y = −10x − 3 and −y = 11x + 5 is at (-2,17).
What is a linear function?A straight line on the coordinate plane is represented by a linear function.
A linear function always has the same and constant slope.
The formula for a linear function is f(x) = ax + b, where a and b are real values.
As per the given lines,
y = −10x − 3
−y = 11x + 5 → y = -11x - 5
The value of the function at the point of intersection is always the same.
So,
−10x − 3 = -11x - 5
-10x + 11x = -5 + 3
x = -2
So,y = -10(-2) - 3 = 17
Hence "The point of intersection is a point where the value of both functions will be the same thus the point of intersection of the lines y = −10x − 3 and −y = 11x + 5 is at (-2,17)".
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In an all boys school, the heights of the student body are normally distributed with a
mean of 67 inches and a standard deviation of 5 inches. Using the empirical rule,
determine the interval of heights that represents the middle 95% of male heights
from this school.
The interval of heights calculated by using empirical rule which represents the middle 95% of male heights from this school is between 57 inches and 77 inches.
What is Empirical rule?
Empirical rule has three rule of statement based on the normal distribution and they as as follows:
1) About 68% of the x values lie between 1 standard deviation below and above the mean.
2) About 95% of the x values lie between 2 standard deviations below and above the mean.
3) About 99.7% of the x values lie between 3 standard deviations below and above the mean.
According to the question, let us assume 'x' be a random variable representing the heights of males from this school. With the mean and standard deviation given, then the Empirical Rule number two states that the:
From the information given,
mean = 67 inches
Standard deviation = 5 inches
We want to determine where 95% of x values lies. This is 2 standard deviations from the mean Therefore,
2 standard deviations = 2 × 5 = 10 inches
The heights are
67 - 10 = 57 inches and
67 + 10 = 77 inches
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in the figure below, what is the value of x?
Answer:
the value of x is 32 degree
Step-by-step explanation:
100 + 68 +x = 180
168 +x=180
x= 180- 168
x= 32 degree
Write an equation for the function graphed below?
Replace the variable in an equation with to write it in function notation. In function notation, the equation would be stated as follows: f (x) = x + 30000, where is the mileage displayed on the odometer.
How can you determine a graph's curve's equation?The values of the parameters m and c, and hence the equation for the curve, can be obtained by taking the coordinates of the two points as (x1,y1) and (x2,y2) and inserting them into the equation y=mx+c. Similarly, by swapping the coordinates, we may determine the equation for any other curve. Students can pick out specific points on the graph and enter them into the equation y = mx+b, where m is the slope, to determine the equation for a non-parabolic, non-quadratic line.To learn more about Function notation refer to:
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what is the value that will correctly replace the missing part? 6(2x + 4) = ? + 24
Answer:?=12x
Step-by-step explanation: 6(2x+4)=?+24, 6 x 2x= 12x. 6 x 4= 24, 24 is already in the equation, so 12x is our missing number.
1. Find the amount of tax owed: vacant land valued at $24,000.00; assessed at 18%; taxed at $11.00 per hundred dollars worth of property.
The amount of tax owed is $ 6,960.
It is given in the question that:-
Vacant land value = $ 24,000
Tax percentage = 18 %
Tax per hundred dollars = $ 11
We have to find the tax owed.
Tax = (18/100)*24000 = $ 4,320
Tax per hundred for $ 24000 land = 11*(24000/100) = $ 2,640
Hence, total tax owed = $ 4,320 + $ 2,640 = $ 6,960
Percentage
In mathematics, a percentage is a number or ratio that can be expressed as a fraction of 100. If we have to calculate percent of a number, divide the number by the whole and multiply by 100.
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answers asap
im failing this class
Answer:
y=2/3x+4
Step-by-step explanation:
Write an equation in point-slope form for the line that passes through the point with the given slope.
(2, 1); m=−32
[tex](\stackrel{x_1}{2}~,~\stackrel{y_1}{1})\hspace{10em} \stackrel{slope}{m} ~=~ - 32 \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies {\Large \begin{array}{llll} y-\stackrel{y_1}{1}=\stackrel{m}{- 32}(x-\stackrel{x_1}{2}) \end{array}}[/tex]
3x^3-15x-13/x+2
Using synthetic division
The quotient when [tex]3x^{3} -15x-13[/tex] is divided by (x+2) using the synthetic division is ([tex]3x^{2} -6x-3[/tex]) and remainder is -7.
According to the question,
We have the following expressions:
[tex]3x^{3} -15x-13[/tex] is divided by (x+2)
First, we will look at some of the rules of the synthetic division:
We have to make the quotient in such a way that it is same as the term with the highest power in the dividend.
We change signs after solving using one complete term of the quotient.
We have to solve this using the synthetic division method:
x+2 )[tex]3x^{3} -15x-13[/tex]( [tex]3x^{2} -6x-3[/tex]
[tex]3x^{3} +6x^{2}[/tex]
_-___-_____
[tex]-6x^{2} -15x-13[/tex]
[tex]-6x^{2} -12x[/tex]
+ +
_________
-3x-13
-3x-6
+ +
_______
-7
Hence, the quotient when [tex]3x^{3} -15x-13[/tex] is divided by (x+2) using the synthetic division is ([tex]3x^{2} -6x-3[/tex]) and remainder is -7.
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Find the product
-7÷5 ×4÷5
Answer:
-1.12 or -28?/25 or -1 3/25
Step-by-step explanation:
Liz does screen-printing. When she screen-prints a batch of T-shirts, there is an initial set-up time of 15 minutes. After that, it takes 3 minutes to print each shirt. How long does it take to screen-print a batch of 14 shirts?
The time taken to screen-print a batch of 14 shirts are 42 minutes
In the above question, it is given that,
Lizz does screen-printing and she screen-prints a batch of T-shirts
The time taken in initial set-up to screen print the first t-shirt is = 15 minutes
After that,
The time taken in screen printing further t-shirts of the batch is = 3 minutes
We need to find the time taken to screen-print a batch of 14 shirts = 3 x 14 = 42 minutes
Hence, the time taken to screen-print a batch of 14 shirts are 42 minutes
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decide if you think the method described would result in a good random sample, and explain your answer. random phone numbers are dialed in a given area code to survey people as to whether or not they've needed the services of a food pantry to feed their families.
The given situation represents a random sample because, in statistics, a simple random sampling refers to the subset of individuals chosen from a larger set called population, where the sample is chosen randomly.
In this case, the dialed number are a random event, so it can be called a simple random sample.
If the probability that a vaccine you took will protect you from getting the flu is 0.977, what is the probability that you will get the flu?
Probability that you will get the flu =
Answer:
Step-by-step explanation:
0.033
Write each number
1. 1,000 more than 3,872
The value for a number that is 1000 more than 3,872 will be 4,872
In the given question, it is stated that we have to find out the value of the expression given. The expression states that we have to find out a number that is 1000 more than 3872.
This can easily be done. To find out the value for a number that is 1000 greater than 3872, we just need to add the value to the number i.e. we need to add 1000 to 3872. Let the new number be 'x'.
So, by solving this condition, we get
=> x = 3872 + 1000
=> x = 4872
Here we get x = 4872.
Hence, 1000 more than 3,872 will be 4,872.
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Use the slope-intercept form to graph the equation 2x - 5y = - 15.
To do this, this equation must be taken to the form
[tex]\begin{gathered} y=mx+b \\ \text{Where} \\ m\colon\text{Slope oth the line} \\ b\colon\text{ intercept with the y-axis} \end{gathered}[/tex]So,
[tex]\begin{gathered} 2x-5y=-15 \\ \text{ Substract 2x from both sides of the equation} \\ 2x-5y-2x=-15-2x \\ -5y=-15-2x \\ \text{Divide by -5 from both sides of the equation} \\ \frac{-5y}{-5}=\frac{-15}{-5}-\frac{2}{-5}x \\ y=3+\frac{2}{5}x \\ y=\frac{2}{5}x+3 \end{gathered}[/tex]Then a positive slope of 2/5 tells you that for every 2 units on the y-axis there are 5 units on the x-axis. And 3 tells you that the line intercepts the y-axis at 3.
a pancakes recipe asks for 3 and 1 half times as much milk as flour. if 4 and two thirds cups of milk is used, what quantity of flour would then be needed, according to the recipe?
A pancakes require 3.5 cups of milk and 1 cup of flour, or a ratio of 3.5:1 or a fraction of
(3.5) / (1)
If you have 4.6 cups of milk you need x cups of flour
Set up a ratio of
[tex]\frac{3.5}{1}=\frac{4.6}{x}[/tex][tex]x=\frac{4.6}{3.5}[/tex][tex]undefined[/tex]
What is a(b(x)) if a(x) = 3x 2 and g(x) = -3x + 1?
The equivalent composite function of the given one is expressed as; -9x + 5
What is a function?Function is a type of relation, or rule, that maps one input to specific single output. Composite functions are also known as function of a function.
Given the following functions as;
a(x) = 3x+2 and
b(x) = -3x + 1
To determine the value of a(b(x));
a(b(x)) = a(-3x+1)
a(b(x)) = 3(-3x+1) + 2
a(b(x)) = -9x + 3 + 2
a(b(x)) = -9x + 5
Therefore, This gives the equivalent composite function of the given one.
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2
80. Air Conditioning The yearly profit p of Arnold's Air Con-
ditioning is given by p = x² + 15x - 100, where x is the
number of air conditioners produced and sold. How many
air conditioners must be produced and sold to have a
yearly profit of $45,000?
205 air conditioners must be produced and sold to have a yearly profit of $45,000.
As per the relation representing p and x, x is the number of air conditioners so p must be the profit. Keep the value of profit to find the value of x.
45000 = x² + 15x - 100
Shifting 100 to Left Hand Side of the equation
45000 + 100 = x² + 15x
Performing addition on Left Hand Side of the equation
45100 = x² + 15x
Rearranging the equation
x² + 15x - 45100 = 0
Factorising the equation to find the value of x
x² + 220x - 205x - 45100 = 0
x(x + 220) -205 (x + 220) = 0
(x - 205) (x + 220) = 0
x = 205, - 220
The number of air conditioners produced and sold can not be negative. So, the value of x and hence the number of air conditioners is 205.
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Temperature (°C) Number of People Day х у 1 20 280 2 24 360 3 36 450 4 32 420 5 2 28 400 ON 38 500 7 34 475 8 26 520 Question: The table shows data for the number of people using a swimming pool over 8 days in summer, and the corresponding maximum temperature (in degrees Celsius) on each day. A Find the equation of the line of best fit for the data. Round decimals to the nearest tenth. B. Can a set of data have more than one line of best fit? Why or why not?
First Step: Calculate the mean of the values in X and Y
[tex]\begin{gathered} \bar{x}=\frac{20+24+36+32+28+38+34+26}{8}=29.75 \\ \bar{y}=400.625 \end{gathered}[/tex]Second step: we have to find the sum of Squares and sum of products
Third step: find the linear equation
[tex]\begin{gathered} Y=aX+b, \\ a=\frac{SP}{SS}=\frac{3201.25}{275.5}=11.61978 \\ b=\bar{y}-a\cdot\bar{x}=54.93648 \\ Y=11.61978X+54.93648 \end{gathered}[/tex]please help me out thanks
The value for the A³ matrix will be [tex]A^{3} = \left[\begin{array}{ccc}75&0 \\0&1\end{array}\right][/tex]
In the given question, it is stated that A is a given matrix. We have to find out the values of A³. This can be done by product of A*A*A. The product should follow the properties of the Matrix. First, we will find out the value of A². So calculating, we get:
=> [tex]A^{2} = \left[\begin{array}{ccc}-5&-1\\0&1\end{array}\right] * \left[\begin{array}{ccc}-5&-1\\0&1\end{array}\right][/tex]
=> [tex]A^{2} = \left[\begin{array}{ccc}-25+0 &1 -1\\0+0&0 + 1\end{array}\right][/tex]
=> [tex]A^{2} = \left[\begin{array}{ccc}-25 &0\\0&1\end{array}\right][/tex]
Now we will calculate the value of A³. We can calculate the value of A³ by using the values of A² because we know that A³ = A².A. So, after calculating we get:
=> [tex]A^{3} = \left[\begin{array}{ccc}-25&0\\0&1\end{array}\right] * \left[\begin{array}{ccc}-5&-1\\0&1\end{array}\right][/tex]
=> [tex]A^{3} = \left[\begin{array}{ccc}75+0&0 + 0\\0+0&0+1\end{array}\right][/tex]
=> [tex]A^{3} = \left[\begin{array}{ccc}75&0 \\0&1\end{array}\right][/tex]
Hence we get value for [tex]A^{3} = \left[\begin{array}{ccc}75&0 \\0&1\end{array}\right][/tex]
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consider the three circles, where R represents the radius of each circle which statement is true?
If two shapes are similar to each other, they have the same shape, but not necessarily the same size.
All circles are similar to each other. All points on the circumference of any circle are equidistant from its center.
The ratio of the radii of the pair of circles will give the scale factor.
The correct option is the SECOND OPTION.
Need help on this question thanks
The linear regression equation is y = 15430.5034 + 301.2586x .
What is the substitution method?The algebraic technique for resolving multiple linear equations at once is called the substitution method. As the name implies, this method involves substituting a variable's value from one equation into another. The first stage in the substitution approach is to determine any variable's value in terms of the other variable from one equation. If there are two equations, x + y=7 and x - y=8, for instance, we can deduce from the first equation that x=7-y. The substitution approach is applied in this manner as the first stage. There are three steps in the substitution technique: For each variable, solve a single equation. Solve the other equation by substituting (plugging in) this expression. Find the corresponding variable by substituting the value back into the original equation.
∑x = 56
∑ y = 109453.5
∑ x² = 760
∑ x × y = 1093064.7
Substitute the upper values
6a + 56b = 109453.5
56a + 760b = 1093064.7
Solve the above 2 equations
a = 15430.5034
b = 301.2586
Now substitute the values in y = a + bx
y = 15430.5034 + 301.2586x
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A farmer is building a fence to enclose a rectangular area against an existing wall, shown in the figure below.
A rectangle labeled, Fenced in Region, is adjacent to a rectangle representing a wall.
A rectangle labeled, Fenced in Region, is adjacent to a rectangle representing a wall.
Three of the sides will require fencing and the fourth wall already exists.
If the farmer has 144 feet of fencing, what are the dimensions of the region with the largest area?
The most appropriate choice for maxima and minima of a function will be given by
Rectangle of length 72 feet and breadth 36 feet has largest area
What is maxima and minima?
Maxima of function f(x) is the maximum value of the function and minima of function f(x) is the minimum value of the function.
Here,
Let the length be x feet and breadth be y feet
The farmer has 144 feet of fencing
Three of the sides will require fencing and the fourth wall already exists.
So,
x + y + y =1 44
x + 2y = 144
Area of rectangle(A) = xy [tex]ft^2[/tex]
= (144 - 2y)y
= [tex]144y - 2y^2[/tex]
[tex]\frac{dA}{dy} = \frac{d}{dy}(144y - 2y^2)[/tex]
= [tex]144 - 2\times 2y^{2-1}\\144 - 4y[/tex]
For largest area,
[tex]\frac{dA}{dy} = 0[/tex]
[tex]144 - 4y = 0 \\4y = 144\\y = \frac{144}{4}\\y = 36[/tex]
[tex]\frac{d^2A}{dy^2} = \frac{d}{dy}(144 - 4y)\\=0-4\\=-4 < 0[/tex]
Hence area is maximum
For largest area, y = 36 feet
[tex]x = 144 - 2\times 36\\x = 144-72\\[/tex]
[tex]x = 72[/tex] feet
So length of rectangle is 72 feet, breadth of rectangle is 36 feet
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Find :
1) (-50+1)÷49
Answer: The answer is -1
Ta Question
Unit Activity: Polynomials and Factors
Question 1
The gray-banded kingsnake requires an enclosure in which the length is at least 20 inches greater than the width and the
height is 33 inches.
w+20
음) 11 of
√6
Vo 0₁
Aw² + Bw + C
What expression models the volume of this enclosure?
Replace the values of A, B, and C to write the expression.
+
4
X
< > S
33
2
W
It
ap CO
A P PO
sin cos tan sin cos tan
csc sec cot log log, In
0
11
1-4
-
0
ZAN
IN
CO
.
U
888
Answer:
33w² +660w
Step-by-step explanation:
You want the expression for the volume of a cuboid with dimensions in inches of 33, (w+20), and (w).
VolumeThe volume of a cuboid is the product of its dimensions:
V = HLW
V = (33)(w +20)(w) = 33(w² +20w)
V = 33w² +660w . . . . expression for volume
__
Additional comment
Comparing coefficients, you see ...
A = 33B = 660C = 0