Answer:I think $1064
Step-by-step explanation:
First you multiply 1/5 by 420 and then add, so 1/5 x 420 = 84 & 420 + 84 = 504. Then you multiply 1/3 by 420 and also add, so 1/3 x 420 = 140 & 420 + 140 = 560. After you add both totals to get your final answer, so 504 + 560 = 1064. Her income each month is $1064.
a process that fills packages is stopped whenever a package is detected whose weight falls outside the specification. assume that each package has probability 0.04 of falling outside the specification and that the weights of the packages are independent. find the variance of the number of packages that will be filled before the process is stopped.
The variance of the number of packages that will be filled before the process is stopped will be 600 when the probability of falling outside the specification is 0.04.
What is variance?Variability is measured by the variance. The average of the squared deviations from the mean is used to calculate it. The degree of spread in your data set is indicated by variance. The variance is greater in relation to the mean the more dispersed the data. In statistics, the variance is used to assess how accurately the mean summarizes all of the data. For instance, more range exists within the set the higher the variance.
P(fail)=0.04
P(pass)=0.96
variance=(1-P)/P²
=(1-0.04)/0.0016
=0.96/0.0016
variance=600
When the likelihood of falling outside the specification is 0.04, the range of packages that will be filled before the process is stopped is 600.
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Two lines. C and D. are represented by the equations given below:
Line C. y = x + 12
Line D. y = 3X + 2
Which of the following shows the solution to the system of equations and explains why?
The correct result would be (x, y) = (5, 17); because both lines pass through this point.
Given,
In the question:
Two lines. C and D are represented by the equations.
We have :
Line C. y = x + 12
Line D. y = 3X + 2
Now, According to the question:
y = x + 12
y = 3X + 2
Substitute into one of the equations
x + 12 = 3x + 2
Solve the equation:
Combine like terms:
x - 3x = 2 - 12
-2x = -10
x = -10/-2
x = 5
Put the value of x in equation of Line C
y = x + 12
y = 5 + 12
y = 17
Hence, The correct result would be (x, y) = (5, 17); because both lines pass through this point.
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What are the advantages of bankruptcy? a. Exempted items are allowed. b. Certain incomes are unaffected. c. Debts are erased. d. "Exempted items are allowed" and "An automatic stay is enacted". e. All of these choices are correct. f. An automatic stay is enacted.
The advantages of bankruptcy include the following: e. All of these choices are correct.
What is bankruptcy?In Financial accounting, bankruptcy can be defined as a legal process that is initiated when an individual or business organization (debtors) is relieved of the responsibility to pay their debts, honor its financial obligations to creditors, or protect them while they are trying to repay their debt.
The advantages of bankruptcy.Generally speaking, there are several advantages of bankruptcy and these include the following:
Some certain incomes are unaffected by bankruptcy.All of the debts are erased.An automatic stay would be enacted.Exempted assets such as unemployment benefits and retirement accounts are retained or allowed.In conclusion, we can reasonably infer and logically deduce that bankruptcy is generally considered as both a friend and foe (enemy).
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What are the workings of this question? How to arrive at the answer?
The variation equation that connects x and y is y³ = 4.5x(x - 1) and the value of y is 4.48 when x equals 5
The equation that connects x and y?From the question, the variation statement is given as
The cube of y is directly proportional to x(x - 1)
Also, we have the initial values of x and y to be
x = 3 and y= 3
When the variation statement is represented as an equation, we have:
y³ = kx(x - 1)
Where k is the variation constant
Substitute the known values in the above equation So, we have the following equation
3³ = k x 3 x (3 - 1)
Evaluate
27 = 6k
Divide both sides by 6
k = 4.5
Substitute k = 4.5 in y³ = kx(x - 1)
y³ = 4.5x(x - 1)
Hence, the equation is y³ = 4.5x(x - 1)
The value of yHere, we are given that
x = 5
Substitute 5 for x in the equation y³ = 4.5x(x - 1)
So, we have
y³ = 4.5 x 5 x (5 - 1)
Evaluate
y³ = 90
Take the cube root of both sides
y = 4.48
Hence, the value of y is 4.48
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The area of the rectangle ABCD is 28[tex]cm^{2}[/tex]
a) What is the length of side AB in terms of x?
length of AB = (Answer here) (1)
b) If the area of rectangle ABCD is 28[tex]cm^{2}[/tex], we can show that [tex]x^{2}[/tex] + ax = b where a and b are integer values.
Work out the values of a and b.
a = (Answer here) (1) b = (Answer here) (1)
Answer:
see explanation
Step-by-step explanation:
(a)
AB = 4 + x
(b)
the area (A) of rectangle ABCD is calculated as
A = length × breadth
= AB × AD
= (4 + x)(6 + x) ← expand using FOIL
= 24 + 10x + x²
given A = 28 , then
x² + 10x + 24 = 28 ( subtract 24 from both sides )
x² + 10x = 4 ← in the form x² + ax = b
with a = 10 and b = 4
A city doubles its size every 5 years. If the population is currently 469,900, what will the
population be in 20 years?
the population would be 7,518,400
469,900 • 2^20/5
* 20/5 is part of the exponent
Answer:
7518400
Step-by-step explanation:
Doubling time model:
[tex]P= P_o 2^{\frac{t}{D}}[/tex]
Here, P is the initial value, D is the time taken to double the quantity, t is the given period of time.
[tex]P_o = 469,900\\\\\\D = 5 \ years\\t = 20 \ years[/tex]
[tex]P = 469,900 * 2^\frac{20}{5}\\\\P = 469,900 * 2^4[/tex]
= 469,900 * 2 * 2 * 2 * 2
= 7518400
a solid triangular prism has a height of 12 inches. the triangular base of this prism is shown below. what is the surface area, in square inches, of this prism?
The surface area of the triangular prsim is 290.4 in² and the right option is B. 290.4 in².
What is surface area?Surface area is the space occupied by a two-dimensional flat surface is called the area. It is measured in square units.
To calculate the surface area of the triangular prism, we use the formula below.
Fromula:
S.A = (a+b+c)L+bh.................... Equation 1Where:
S.A = Surface area of the prisma , b, c = Three sides of the base triangle respectively.h = Height of the triangular baseL = Lenght of the prismFrom the question,
Given:
L = 12 ina = 5 inb = 9 inc = 7.2 inh = 4 inSubstitute these values into equation 1
S.A = (5+9+7.2)12+(4×9)S.A = 254.4+36S.A = 290.4 in²Hence, the right option is B. 290.4 in².
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Michael is having party. He'll have 12 tables for every 32 guests. Complete the table below to find the number of guests he can seat with 3
tables.
Number of guests 32
Number of tables 3
Which statement is true? All decimals are rational. All rational numbers can be written as the quotient of Integers
The statement that is true is All rational numbers can be written as the quotient of Integers
What are rational numbers ?Any number that can be written as a fraction and has an integer as both the numerator and the denominator is referred to as rational number.
In a rational number, the denominator cannot be zero.
Integer refers to whole numbers. They are generally of two types the positive integers and the negative integers
Quotients refers to the end product of division example 2 ÷ 1 = 2/1
2/1 is the quotient, both 2 and 1 are integers
All decimals are rational - A counter example to this is √2
It has decimal and not a rational number because it cannot be represented as a fraction
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The statement that is true is All rational numbers can be written as the quotient of Integers
What are rational numbers ?
Any number that can be written as a fraction and has an integer as both the numerator and the denominator is referred to as rational number.
In a rational number, the denominator cannot be zero.
Integer refers to whole numbers. They are generally of two types the positive integers and the negative integers
Quotients refers to the end product of division example 2 ÷ 1 = 2/1
2/1 is the quotient, both 2 and 1 are integers
All decimals are rational - A counter example to this is √2
It has decimal and not a rational number because it cannot be represented as a fraction
Janet attends state university and lives in an on campus dorm suite with 5 friends. They share cost of the monthly upgraded cable bill for their suite. Below is a listing of the bills for their freshman year. 9. Round the following value
Σ
619
X₁
to the nearest dollar.
Interpret the answer in the context of the problem.
Answer:
Step-by-step explanation: Step 1
1 of 2
step 1.2659 ;[26 Discovery Group
October 20
Last
Trade Time
Chg
Open
52-week High
52-week Low
Sales in 100s
High
Low
$38.50
4:00 P.M. ET
$1.56
$37.22
$76.19
$22.78
19,700
$40.10
$36.77
\begin{matrix} \text{Discovery Group} & \text{}\\ \text{October 20} & \text{}\\ \text{Last} & \text{\$42.00}\\ \text{Trade Time} & \text{4:00 P.M. ET}\\ \text{Chg} & \text{\$1.50}\\ \text{Open} & \text{\$42.50}\\ \text{52-week High} & \text{\$76.19}\\ \text{52-week Low} & \text{\$22.78}\\ \text{Sales in 100s} & \text{23,600}\\ \text{High} & \text{\$42.50}\\ \text{Low} & \text{\$42.00}\\ \end{matrix}
Discovery Group
October 20
Last
Trade Time
Chg
Open
52-week High
52-week Low
Sales in 100s
High
Low
$42.00
4:00 P.M. ET
$1.50
$42.50
$76.19
$22.78
23,600
$42.50
$42.00
A triangle is shown with its exterior angles. The interior angles of the triangle are angles 2, 3, 5. The exterior angle at angle 2 is angle 1. The exterior angle at angle 3 is angle 4. The exterior angle at angle 5 is angle 6. Which statements are always true regarding the diagram? Select three options. m∠5 + m∠3 = m∠4 m∠3 + m∠4 + m∠5 = 180° m∠5 + m∠6 =180° m∠2 + m∠3 = m∠6 m∠2 + m∠3 + m∠5 = 180°
Considering the triangle with exterior angles and interior angles as described. The statements that are true include
m∠5 + m∠6 =180° m∠2 + m∠3 = m∠6 m∠2 + m∠3 + m∠5 = 180°How to determine the statements that are correctThe problem is solved considering this 3 theorems about triangles
exterior angle of a triangle theoremlinear pair theoremsum of angles in a triangleexterior angles theorem: m∠2 + m∠3 = m∠6
If a triangle side is created, the outside angle that results is equal to the sum of the two opposite internal angles.
Linear pair theorem: m∠5 + m∠6 =180°
Linear pairs are particular instances of supplementary angles that only involve two neighboring angles.
Sum of angles in a triangle is equal to 180 degrees:
m∠2 + m∠3 + m∠5 = 180°
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Hannah swims in a swimming pool of 25 metres long.
How many laps does she need to complete for a total distance of 0.5 kilometre?
She needs to complete 20 laps for a total distance of 0.5 kilometer.
1 kilometers = 1000 meters
0.5 kilometers = 500 meters
Length of swimming pool = 25 meters
That means, she covers 25 meters in 1 lap.
Laps to cover 500 meters = 500/25
= 20
Hence, she needs to do 20 laps.
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Activity 8. Counting the roots of polynomial equation
by inspection determine the number of real roots of each polynomial equation. Roots multiply n are counted n times
1. (x-4)(x+3)^2(x-1)^3=0
2. X^2(x^3-1)=0
3. X(x+3)(x-6)^2=0
4. 3x(x^3-1)^2=0
5. (x^3-8)(x^4+1)=0
patulong po plss.
The number of real roots found are-
1. (x-4)(x+3)^2(x-1)^3=0 ; 6 real roots.
2. X^2(x^3-1)=0 ; 3 real roots.
3. X(x+3)(x-6)^2=0 ; 4 real roots.
4. 3x(x^3-1)^2=0 ; 3 real roots.
5. (x^3-8)(x^4+1)=0 ; 1 real roots.
Define the term roots of polynomial?If a root is not imaginary (defined as a number as in form a + bi, whereby I is a number which is not on any real number line, which is why the name imaginary), it is said to be real.Consider the given cases-
1. (x-4)(x+3)^2(x-1)^3=0,
Put each equals zero.
x – 4 = 0 or (x + 3)^2 = 0 or (x – 1)^3 = 0,
Thus,
x = 4
x = –3 twice
x = 1 thrice
All are real numbers, thus given polynomial has 6 real roots.
2. x^2(x^3-1)=0
Put each equals zero.
x² = 0
x³ – 1 = x³ – 1³ = (x – 1)(x² + x + 1) = 0
The quadratic expressions has 2 imaginary roots.
Use the discriminant formula to find-
D = b² – 4ac = 1² – 4(1)(1) = 1 – 4 = –3 < 0.
Thus, roots that are not real.
So, the real roots of this polynomial : 0(twice) and 1.
Hence, thus given polynomial has 3 real roots.
3. x(x+3)(x-6)^2=0
Put each equals zero.
x = 0
x = 3
x = 6 twice
Hence, thus given polynomial has 4 real roots.
4. 3x(x^3-1)^2=0
Put each equals zero.
3x = 0 and x = 0
(x^3 – 1)² = [(x – 1)(x² + x + 1)]² = (x – 1)²(x² + x + 1)² = 0,
Here, only two roots are real
The quadratic expression do have unreal roots, thus 1 twice are the real roots.
Hence, thus given polynomial has 3 real roots..
5. (x^3-8)(x^4+1)=0
Put each equals zero.
x³ – 8 = x³ – 2³ = (x – 2)(x² + 2x + 4) = 0
There is only one real root i.e, 2, the quadratic expression do have imaginary roots
Check using the discriminant formula
D = b² – 4ac = 2² – 4(1)(4) = 4 – 16 = – 12 < 0.
x⁴ + 1 = 0 has imaginary roots.
Hence, thus given polynomial has 1 real roots..
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Alan travels eight and two thirds kilometers to get to school. He ran one and ten elevenths kilometers and walked the rest. How many kilometers were left to walk? (4 points)
a
six and two thirty thirds kilometers
b
five and twenty five thirty thirds kilometers
c
six and twenty five thirty thirds kilometers
d
six and twenty thirty thirds kilometers
The Kilometer that Alan was left to walk is given as c six and twenty five thirty thirds kilometers
How to solve for the kilometerIn order to solve for the solution, we would have to write out the expressions in mathematical forms first
eight and two thirds kilometers = [tex]8\frac{2}{3}[/tex]
This is is the time that Alan is said to travel for him to get to school
The expression one and ten elevenths is written mathematically as [tex]1\frac{10}{11}[/tex]
This is the kilometer that he spent running.
We are to find the time that was used to walk to school.
To get this we have to subtract the kilometer that he ran from the total kilometer that he traveled.
[tex]8\frac{2}{3} - 1\frac{10}{11}[/tex]
we have to express as a mixed fraction then we would subtract
26/3 - 21/11
= 286 - 63 / 33
= 223/33
= 6.75
= 6 25/33
Hence the solution is C.
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Fill in the blanks below in order to justify whether or not the mapping shown
represents a function.
Set A
Submit Answer
+
2
Ń
9.
The mapping diagram above
where there
Set B
5
C
a function since
C
in
attempt 1 out of 2
Dev’ car currently ha 18 gallon of ga in the tank. He ue about 0. 04 gallon for every mile he drive. Dev want to go on a car trip over the weekend and have at leat 2 gallon of ga left in hi tank to get to work on Monday. What i the olution of the inequality that determine how far Dev can travel over the weekend? (Example 2) Let x repreent the number of mile Dev travel
The solution of the inequality that determines how far Dev can travel over the weekend is -
x ≤ 400 miles.
What is a expression? What is a mathematical equation? What is Equation Modelling?A mathematical expression is made up of terms (constants and variables) separated by mathematical operators. A mathematical equation is used to equate two expressions. Equation modelling is the process of writing a mathematical verbal expression in the form of a mathematical expression for correct analysis, observations and results of the given problem.
We have Dave who is planning a car trip for weekend.
Assume that [x] represents the number of miles Dev can travel.
We can write the inequality as -
18 - 0.04x ≤ 2
0.04x ≤ 16
x ≤ 400 miles
Therefore, the solution of the inequality that determines how far Dev can travel over the weekend is -
x ≤ 400 miles.
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air is pumped into a spherical balloon at the constant rate of 200 cubic centimeters per second. how fast is the surface area of the balloon changing when the radius is 5 centimeters?
The rate of change of the surface area of the balloon is 32cm²/sec
We know that volume of the sphere is,
V = 4/3πr³
Given,
dV / dt = +200
r = 5cm
d(SA) / dt = ?
dV / dt = 4πr² dr/dt
80 = 4π (25) dr/dt
dr / dt = 0.8 / π
SA = 4πr²
d(SA) / dt = 8πr dr / dt
d(SA) / dt = 8π × 5 × 0.8 / π
d(SA) / dt = 32cm²/sec
Hence the rate of change of the surface area is 32cm²/sec
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Select the correct symbol to make 11⎯⎯⎯⎯√ , 1 of 1.
Select Choice
323 a true statement.
The correct inequality symbol that makes the statement true is:
1000 < (6²)³.
How to choose the inequality symbol?The inequality symbols, and their meaning, are presented as follows:
> x: greater than x.< x: less than x.≥ x: at least x.≤ at most x.In the context of this problem, the numbers being compared are presented as follows:
1000 and (6²)³.
The power of power property of a number with multiple exponents means that the base is kept while the exponents are multiplied.
The number in which this property is applied in this problem is (6²)³, in which:
The base is of 6.The exponents are of 2 and 3.Hence the equivalent number is given as follows:
(6²)³ = 6^6 = 6 x 6 x 6 x 6 x 6 x 6 = 46,656
Which is a greater number than 1,000, meaning that 1,000 is less than the number, and the inequality is:
1000 < (6²)³.
Missing InformationThe problem is given by the image at the end of the answer.
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please help me with this
Answer: final answer = 3/8
Step-by-step explanation:
it says 1/8 + 1/4. before you can add them you have to make sure that both fractions have the same denominator first you have to make both of the bottom denominators the same so its 1/8+2/8.
1/8+2/8=3/8
if he uses upper and lower control limits of 520 and 480 hours, what is the probability that a sample mean is out of control?( assuming the process standard deviation is known to be 20 hours)
0.0456 is the probability that a sample mean is out of control.
What is probability in math?
Probability refers to potential. The subject of this area of mathematics is the occurrence of random events.The range of the value is 0 to 1. To forecast how likely events are to occur, probability has been introduced in mathematics.UCL = 520
LCL = 480
Mean (X ) = 500
Standard Deviation of sample (Sn) = 11.55
Z (for UCL) = (UCL - Mean)/Sn
Z (for UCL) = (520-500)/10
Z (for UCL) = 2
Z (for LCL) = (LCL - Mean)/Sn
Z (for LCL) = (480 - 500)/10
Z (for LCL) = -2
*Use Z table to find confidence level between Z value of -2 and 2.
Confidence level = 0.4772 + 0.4772
Confidence level = 0.9544
Risk (α ) = 1 - confidence level
Risk (α) = 1 - 0.9544
Risk (α) = 0.0456
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Write 10x - 5y = -30 in slopeintercept form.
y =
X +
Answer:
[tex]y=2x+6[/tex]
Step-by-step explanation:
Slope-intercept form is [tex]y=mx+b[/tex].
Therefore, rearranging the given formula:
[tex]10x-5y=-30[/tex]
[tex]-5y=-10x-30[/tex]
[tex]\frac{-5y}{-5}=\frac{-10x}{-5}-\frac{30}{-5}[/tex]
[tex]y=2x-(-6)[/tex]
[tex]y=2x+6[/tex]
Answer: y=2x+6
Step-by-step explanation:
A slope-intercept form is y=mx+b
Hence,
[tex]10x-5y=-30\\\\10x-5y+5y=-30+5y\\\\10x=-30+5y\\\\10x+30=-30+5y+30\\\\10x+30=5y[/tex]
Divide both parts of the equation by 5:
2x+6=y
Thus, y=2x+6
A catering company loses $110 as a result of a delay.
The 5 owners of the company must share the loss equally.
Which quotient represents each owner's share of the loss?
I know the answer but, is 110 negative or positive??
Answer: Positive
Step-by-step explanation: It would be positive because the question is asking how much money they lost. The lost a positive amount of money.
an educational psychologist wanted to know the effects of four different methods of teaching arithmetic. using 12 classrooms, each method was used in the classes. at the end of the school year, all the pupils took the same achievement test, and the means for groups receiving different methods were compared. answer the following questions about the study. a. what was the dependent variable? b. what was the independent variable? c. what was a controlled variable? d. give an example of a parameter.
The dependent variable depends on a particular situation. The independent variable does depends on the result of an experiment. The controlled variable is a variable whose value is constant.
What is dependent variable?
The variable being measured or tested in an experiment is known as the dependent variable. The results of the participants' tests, for instance, since that is what is being measured, would be the dependent variable in a study looking at how tutoring affects test scores.
What is independent variable?
This isn't the same as an experiment's independent variable, which is a variable that can stand on its own. Tutoring would be the independent variable in the aforementioned case. The dependent variable (test results), however, may alter depending on the independent variable (tutoring).
What is controlled variable?
Anything kept constant or constrained in a research study is referred to as a control variable. Despite not being relevant to the study's objectives, this variable is controlled because it might have an impact on the results.
Numbers called parameters are used to define the characteristics of entire populations. Statistics are numerical representations of sample characteristics. One population parameter is the average income in the United States.
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if sec2a is 3 find the value of sina.
By using trigonometry, it can be calculated that
The value of sina is [tex]\pm \frac{1}{\sqrt{3}}[/tex]
What is Trigonometry?
Trigonometry shows the relationship between sides and angles of a right angled triangle.
There are six trigonometrical functions. They are
[tex]sin \theta, \ cos\theta, \ tan \theta, \ cot\theta, \ sec\theta, \ cosec\theta[/tex]
[tex]sin\theta = \frac{perpendicular}{hypotenuse}\\\\cos\theta = \frac{base}{hypotenuse}\\\\tan\theta = \frac{perpendicular}{base}\\\\cot\theta = \frac{base}{perpendicular}\\\\sec\theta = \frac{hypotenuse}{base}\\\\cosec\theta = \frac{hypotenuse}{perpendicular}[/tex]
Trigonometry is a very important tool in mathematics.
Here, the trigonometric equation is
sec 2a = 3
cos 2a = [tex]\frac{1}{3}[/tex]
we know,
[tex]1 - cos 2a = 2 sin^2a\\\\1 - \frac{1}{3} = 2sin^2a\\\\2 sin^2a = \frac{2}{3}\\\\sin^2a = \frac{2}{3} \times \frac{1}{2}\\\\sin^2a = \frac{1}{3}\\\\ sina = \pm \frac{1}{\sqrt{3}}[/tex]
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30% of the tickets sold at a school carnival were early-admission tickets. If the school sold 100 tickets in all, how many early-admission tickets did it sell?
The number of early-admission tickets that it sells is 30.
How to calculate the number of early admission?From the information, 30% of the tickets sold at a school carnival were early-admission tickets and the school sold 100 tickets in all.
Therefore, the number of early admission will be:
= Percentage for early admission × Total number of tickets
= 30% × 100
= 0.3 × 100
= 30
Therefore, there are 30 early admission.
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I need help with number 7.
With the help of the given trendline equation, the length of the flight will be 3,613.34 miles.
What are equations?A mathematical statement known as an equation is made up of two expressions joined together by the equal sign. A mathematical statement that has an "equal to" symbol between two expressions with equal values is called an equation. As in 3x + 5 Equals 15, for instance. Equations come in a variety of forms, including linear, quadratic, cubic, and others.So, the length of the flight will be:
The trendline equation is: T = 458 + 0.15DT is the cost which is given as $1000 and D is the distance.Now, calculate for D as follows:
T = 458 + 0.15D1000 = 458 + 0.15D1000 - 458 = 0.15D0.15D = 542D = 542/0.15D = 3,613.33We can write as D = 3,613.34
Therefore, with the help of the given trendline equation, the length of the flight will be 3,613.34 miles.
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Please help me, I need the answer asap
Find m<2
28
62
56
142
The given equation m<2 = 28⁰ so here in this case option A is correct .
What three characteristics do angles have?A transversal's corresponding angles are equal when it crosses two parallel lines.Angles that are vertically opposite are equal.All of the interior angles alternately are equal.Every other external angle is equivalent.On the same side of the transversal, the pair of internal angles is additional.Simplification:
<1 = <3
<1 = 5x + 8 ………………equation 1.
<3 = 12x – 20 ………………equation 2.
By angle property <1 = <3 [ given ]
and
<1 = <2 [ vertically opposite angles ]
We have some options so we can use it to get answer more quickly:
Put <1=28 and <3=28
And solve it for X.
28 = 5x + 8 Eq1
20=5x => x=4
28 = 12x - 20 Eq2
28 + 20 = 12x
48 = 12x
4 = x
X = 4
As <1 = <2 ( vertically opposite angles )
From the first equation <1 = <2 = 5x + 8 ......( equation 3)
Putting the value of x = 4 in (equation 3)
<2 = 5* 4+ 8
<2=28
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Partition a line segment in the given ratio.
Point V lies on TU such that TV:UV is 5:2. Graph V.
Check the picture below.
[tex]\textit{internal division of a line segment using ratios} \\\\\\ T(-5,9)\qquad U(9,2)\qquad \qquad \stackrel{\textit{ratio from T to U}}{5:2} \\\\\\ \cfrac{T\underline{V}}{\underline{V} U} = \cfrac{5}{2}\implies \cfrac{T}{U} = \cfrac{5}{2}\implies 2T=5U\implies 2(-5,9)=5(9,2)[/tex]
[tex](\stackrel{x}{-10}~~,~~ \stackrel{y}{18})=(\stackrel{x}{45}~~,~~ \stackrel{y}{10}) \implies V=\underset{\textit{sum of the ratios}}{\left( \cfrac{\stackrel{\textit{sum of x's}}{-10 +45}}{5+2}~~,~~\cfrac{\stackrel{\textit{sum of y's}}{18 +10}}{5+2} \right)} \\\\\\ V=\left( \cfrac{ 35 }{ 7 }~~,~~\cfrac{ 28}{ 7 } \right)\implies {\Large \begin{array}{llll} V=(5~~,~~4) \end{array}}[/tex]
20-foot extension ladder set base six feet from house How far up will ladder reach?
The top of the 20-foot ladder used for painting will be 19 feet up from the base of the house
How to determine how high up the house the ladder will goinformation given in the question
Brian borrowed a 20-foot extension ladder: hypotenuse = 20-foot ladder
he sets the base of the ladder six feet from the house
adjacent = 6 feet from the house
opposite = how far up will the top of the ladder reach = ?
The problem is solved using the Pythagoras theorem is applicable to right angle triangle. the formula of the theorem is
hypotenuse² = adjacent² + opposite²
substituting the values into the equation
let x be thow far up will the top of the ladder reach
20² = 6² + x²
400 = 36 + x²
400 - 36 = x²
x² = 364
x = √364
x = 19.08
The ladder Brain borrowed will go 19 feet high
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complete question
Brian borrowed a 20-foot extension ladder to use to paint his house. If he sets the base of the ladder six feet from the house how far up will the top of the ladder reach?
a tank contains 1360l of pure water. a solution that contains 0.02kg of sugar per liter enters a tank at the rate 2 l/min. the solution is mixed and drains from the tank at the same rate. how much sugar is in the tank initially? find the amount of sugar in the tank after t minutes. find the concentration of sugar in the solution in the tank after 60 minutes
a) Amount of sugar in the tank initially is 0.
b) The amount of sugar in the tank after t minutes is 13.6(1 - [tex]e^{\frac{4t}{1360} }[/tex]).
c) The concentration of sugar in the solution in the tank after 60 minutes is 3.672.
a) Assign the sugar content of the tank for time t to the symbol A(t). Starting with only clean water, the tank
A(0) = 0
b) Sugar flows in at a rate of
(0.02 kg/L) × (2 L/min) = 0.04 kg/min
and flows out at a rate of
(A(t) ÷ 1360 kg/L) × (2 L/min) = (2A(t) ÷ 1360) kg/min
so that the net rate of change of A(t) is governed by the ODE,
dA(t) ÷ dt = (4 ÷ 100) - (2A(t) ÷ 1360)
A'(t) + (2A(t) ÷ 1360) = 2 ÷ 100
Multiply both sides by the integrating factor [tex]e^{\frac{4t}{1360} }[/tex] to condense the left side into the derivative of a product:
[tex]e^{\frac{4t}{1360} }[/tex] A'(t) + (2 [tex]e^{\frac{4t}{1360} }[/tex] A(t) ÷ 1360) = 4 [tex]e^{\frac{4t}{1360} }[/tex] ÷ 100
( [tex]e^{\frac{4t}{1360} }[/tex] A(t))' = 4 [tex]e^{\frac{4t}{1360} }[/tex] ÷ 100
Integrate both sides:
[tex]e^{\frac{4t}{1360} }[/tex] A(t) = ∫ 4 [tex]e^{\frac{4t}{1360} }[/tex] ÷ 100
[tex]e^{\frac{4t}{1360} }[/tex] A(t) = 13.6 [tex]e^{\frac{4t}{1360} }[/tex] + C
Solve A(t);
A(t) = 13.6 + C [tex]e^{\frac{4t}{1360} }[/tex]
A(0) = 0, substitute the value
0 = 13.6 + C [tex]e^{\frac{4t}{1360} }[/tex]
C = -13.6
so that the amount of sugar at any time t is,
A(t) = 13.6 - 13.6[tex]e^{\frac{4t}{1360} }[/tex]
A(t) = 13.6(1 - [tex]e^{\frac{4t}{1360} }[/tex])
c) As t → 60, the exponential term converges to 0 and is left with
[tex]\lim_{t \to \(60}[/tex] A(t) = 13.6 × (1 - 0.73)
[tex]\lim_{t \to \(60}[/tex] A(t) = 3.672 kgs of sugar.
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