Answer: There are several statements that could be false if f(x) is continuous on [a, b] and differentiable on (a, b):
1 The function f(x) has a maximum or minimum value at some point in the interval [a, b].
2 The function f(x) is monotonic (either increasing or decreasing) on the interval [a, b].
3 The function f(x) has an inflection point (a point where the concavity of the function changes) in the interval [a, b].
4 The function f(x) has an asymptote (a line that the function approaches but never touches) in the interval [a, b].
5 Note that these statements are not necessarily false, but they could be false depending on the specific function f(x).
A librarian measured the number of young adult books in a library. The number of books in the library as a function of time (in years since 2008) is shown in the scatterplot.
scatterplot with points at 0 comma 10, 4 comma 42, 8 comma 74 and 12 comma 106
Choose the linear function that best describes the scatterplot relating the number of books in the library, y, to time, x. (1 point)
Group of answer choices
y = 8 − 10x
y = 8 + 10x
y = 10 − 8x
y = 10 + 8x
Therefore ,the linear function that best describes the scatterplot relating the number of books in the library is y = 10 + 8x
What is linear function?Two separate but related concepts are referred to as linear functions in mathematics: A polynomial function of degree 0 or 1 is referred to as a linear function in calculus and related fields if its graph is a straight line.
Here,
The points given on the scatterplot are (0,10),(4,42),(8,74) & (12,106)
By putting in the points in the given linear functions
y = 8 − 10x
y = 8 + 10x
y = 10 − 8x
y = 10 + 8x
Thus ,the linear function which best satisfy the points given on the scatterplot is y = 10 + 8x
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Answer:
y = 10 + 8x
Step-by-step explanation:
we can find the y-intercept first, by observing the point at which x=0, (0,y)
when x = 0, y = 10. The y-intercept is 10.
Let's find the slope. Given the 2 points (0, 10) and (4, 42), we can use the slope calculating formula:
[tex]m(slope)=\frac{y_{2}-y_{1} }{x_{2} -x_{1} }[/tex]
we have:
m = (42-10)/(4-0)
m = 32/4
m=8.
we have y = 8x + 10 (slope-intercept form), which is equal to y = 10 + 8x (unofficial confusing form.)
Hope this helped!
In a certain city the number of hours of daylight on day t (where t is the number of days after January 1) is modeled by the function
n(2π (t-80)).
365
(a) Which days of the year have about 9 h of daylight? (Enter your answers as a comma-separated list.)
L(t) = 11+ 2.81 sin
(b) How many days of the year have more than 9 h of daylight?
(a) February 4 and November 3 of the year have about 9 h of daylight.
(b) 271 days with more than 9 hours of daylight.
What is a function?
A function, according to a technical definition, is a relationship between a set of inputs and a set of potential outputs, where each input is connected to precisely one output.
Here, we have
Given
(a) L(t) = 11+ 2.86 sin(2π/365 (t-80))
L(t) = 9
9 = 11+ 2.86 sin(2π/365 (t-80))
-2 = 2.86 sin(2π/365 (t-80))
-2/2.86 = sin(2π/365 (t-80))
-2/2.86 = -0.6993
Sin function is negative on [π, 3π/2] and [3π/2, 2π]
sin⁻¹(-0.6993) = 2π/365 (t-80)
2π/365 (t-80) = 3.9160 or 5.5087
t = 80 + 365/2π(3.9160 or 5.5087)
t = 35 or 307
After January 31 days and February 4 days
For t = 35 days = February 4
For t =307days = November 3
Hence, February 4 and November 3 of the year have about 9 h of daylight.
(b) There are 35 + (365-307) = 35 + 58 = 94 days with 9 hours or less
Hence, 365 - 94 = 271 days with more than 9 hours of daylight.
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Points A, B, and C are collinear. If BC= 8.5 and AC= 13.2, select all possible values of AB
A. 4.7
B. 3.8
C. 21.7
D. 8.5
E. 13.2
I know that A is for sure an answer but I think there’s supposed to be another one
Points A, B, and C are collinear then the possibility of AB will be equal to 4.7. Hence, option A is correct.
What are Collinear Points?Points that are parallel to or within a single line are referred to as collinear points. In Euclidean space, two or more points have been shown to be collinear if they are situated on a line that is either close to or far from another.
As per the given information in the question,
The case is that when B is in between A and C, which means A, B, and C are collinear.
So,
AB + BC = AC
AB + 8.5 = 13.2
AB = 13.2 - 8.5
AB = 4.7
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Find the dimensions of the box with volume 4096 cm3 that has minimal surface area. (Let x, y, and z be the dimensions of the box.)
The dimensions of the box with a volume of 4096 cm³ that has a minimal surface area are 16cm×16cm×16 cm.
A solid object's surface area is a measurement of the overall space that the object's surface takes up. It is also known as the overall area of a three-dimensional shape's surface. The formula to calculate surface area is S = 2(xy+yz+zx) where x is length, y is width and z is the height of the object.
Given the volume is 4096 cm³.
Then, V = xyz = 4096. Deriving the value of z from this, we get z = 4096/xy. Substitute value of z in S = 2(xy+yz+zx), we get,
[tex]\begin{aligned}S&=2\left(xy+y\times\frac{4096}{xy}+x\times\frac{4096}{xy}\right)\\&=2\left(xy+\frac{4096}{x}+\frac{4096}{y}\right)\\&=2xy+\frac{8192}{x}+\frac{8192}{y}\\&=2xy+8192\left(\frac{1}{x}+\frac{1}{y}\right)\end{aligned}[/tex]
Differentiating S with respect to x, we get,
[tex]\frac{dS}{dx}=2y-\frac{8192}{x^2}[/tex]
Equating this to zero and multiplying with x² we get,
[tex]\begin{aligned}2y-\frac{8192}{x^2}&=0\\2x^2y-8192&=0\\x^2y&=\frac{8192}{2}\\x^2y&=4096\end{aligned}[/tex]
Differentiating S with respect to y, we get,
[tex]\frac{dS}{dy}=2x-\frac{8192}{y^2}[/tex]
Equating this to zero and multiplying with y² we get,
[tex]\begin{aligned}2x-\frac{8192}{y^2}&=0\\2xy^2-8192&=0\\xy^2&=\frac{8192}{2}\\xy^2&=4096\end{aligned}[/tex]
Dividing x²y by xy², we get,
[tex]\begin{aligned}\frac{x^2y}{xy^2}&=\frac{4096}{4096}\\\frac{x}{y}&=1\\x&=y\end{aligned}[/tex]
Substitute x = y in x²y = 4096, we get,
[tex]\begin{aligned}y^3&=4096\\y&=16=x\end{aligned}[/tex]
Then, the z value is
[tex]\begin{aligned}z &= \frac{4096}{16\times 16}\\&=16\end{aligned}[/tex]
The required answer is 16cm×16cm×16 cm.
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Let $ABC$ be a triangle. Points $A_1, B_1, C_1$ are, respectively, on sides $BC, AC, AB$ and:
$\frac{AC_1}{C_1B} = \frac{BA_1}{A_1C} = \frac{CB_1}{B_1A} \neq 1$
If $\frac{AB}{A_1B_1} = \frac{BC}{B_1C_1} = \frac{CA}{C_1A_1}$ prove that triangle $ABC$ is equilateral.Let ABC be a triangle. Points A1, B1, C1 are, respectively, on sides BC, AC, AB and:
AC1/C1B} = BA1/A1C = CB1/B1A which is not equal to 1
If AB/A1B1 = BC/B1C1} = CA/C1A1 prove that triangle ABC is equilateral.
Triangle ABC is equilateral and hence proved.
According to the question,
Let ABC be a triangle
And the given points A1, B1, and C1 are on its sides BC, AC, and AB respectively
Since according to the first equality says that A1, B1, and C1 are not the midpoints and just normal points on the lines.
According to what equality says that ΔABC = ΔA1B1C1 because the sides are in proportion.
Let us consider 3 more triangles = ΔAB1C1, ΔBC1A1, and ΔCA1B1
Now, these above-mentioned triangles will also be similar due to the proportionality insides.
Which results in ⇒ ∠A = ∠B = ∠C due to the phenomenon of corresponding angles of triangles.
Therefore, Triangle ABC is equilateral and hence proved.
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A line passes through point (9, -2) and has a slope of - 2/3
Write an equation in Ax+By = C form for this line.
Use integers for A, B, and C.
The equation -2/3x + y = -2 is in the slope-intercept form, which is a common form for linear equations.
What is the slope of the equation?
The slope of a line is a measure of its steepness. Mathematically, the slope is calculated as "rise over run" (change in y divided by change in x).
we can use the two-point form to find the equation of the line:
y - y1 = m(x - x1)
Substituting the coordinates of the given point and the slope, we get:
y - (-2) = (-2/3)(x - 9)
Combining like terms and multiplying through the parentheses, we get:
y + 2 = -2/3x + 6
This equation is already in the form Ax + By = C, so we can simply use it as is:
-2/3x + y + 2 = 0
This equation can also be written as -2x + 3y = -6 by multiplying all the terms by 3 to get rid of the fraction.
The equation -2/3x + y = -2 is in the slope-intercept form, which is a common form for linear equations. In this form, the coefficient of x (-2/3) is the slope of the line, and the constant term (-2) is the y-intercept, which is the point where the line intersects the y-axis.
Hence, The equation -2/3x + y = -2 is in the slope-intercept form, which is a common form for linear equations.
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Closside is 14 km north and 20 km east of Farlisle.
Use Pythagoras' theorem to calculate the direct distance between Closside and
Farlisle.
Give your answer in kilometres (km) to 1 d.p.
Answer:
24.4 km
Step-by-step explanation:
[tex]\sqrt{14^2 + 20^2} = \sqrt{196 + 400} = \sqrt{596}[/tex] = 24.413111 = 24.4 km
Using the concept of Pythagorean theorem, the distance between Closside and Farlisle is 24.4km
What is Pythagorean theorem?The Pythagorean theorem is a fundamental principle in geometry that relates the lengths of the sides of a right triangle. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
Mathematically, the Pythagorean theorem can be expressed as:
c² = a² + b²
In this problem, we can substitute the values and solve the distance;
c² = 14² + 20²
c² = 596
c = 2√149
c = 24.4km
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Kendra opened an account with a deposit of $3,500.
.
• The account earns 2.5% interest compounded annually.
.
• Kendra makes no additional deposits or withdrawals.
Which amount is closest to the balance of the account at the end of 3 years?
A $3,762.50
B) $3,587.50
c) $3,507.50
D) $3,769.12
Answer:
its A
Step-by-step explanation:
cause 2.5 x 3 is 7.5 and 3500+7.5% is 3762.50
Willa Schaefer deposited $3,000 in a savings plan with her credit
union. The credit union savings plan pays 3% interest compounded
semiannually. If she makes no other deposits or withdrawals:
a. What is the amount in the account after 2½ years?
b. How much interest did her money earn?
c. What is the amount in the account after 4 years?
d. How much interest did her money earn?
The amount in the account after 2½ years is $3231.85 and the interest is $231.85
The amount in the account after 4 years is $3379.47and the interest is $379.47
How to determine the amount and the interest?The amount after 2½ years
The given variables can be represented as:
Deposit amount P = $3,000
Rate of interest, R = 3%
Time, t = 2½ years
The amount from compound interest formula can be calculated as:
A = P(1 + r/n)^nt
Also from the question, we have
n = 2 i.e. compounded semi-annually.
Solving further, we replace the variables with their values in the above equation
A = 3000 * (1 + (3%/2))^(2*2.5)
Evaluate
A = 3231.85
The interest is
I = 3231.85 - 3000
I = 231.85
The amount after 4 years
Here, we have
Time, t = 4 years
The amount from compound interest formula can be calculated as:
A = P(1 + r/n)^nt
Solving further, we replace the variables with their values in the above equation
A = 3000 * (1 + (3%/2))^(2*4)
Evaluate
A = 3379.47
The interest is
I = 3379.47 - 3000
I = 379.47
So, the interest is 379.47
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¿Consideras usted que las empresas de servicios pueden incluir todas las cuentas de las empresas comerciales?
Answer:
Step-by-step explanation:
es posible que desee cambiar a la sección de negocios del sitio web porque es posible que puedan responderle mejor
A family of 4 goes out to eat at a restaurant. They buy a family dinner platter that costs $29. Each of the 4 family members also gets a dessert for $3 apiece.
Let the variable x represent the total cost of the meal. Which of the following equations is correct?
Select ALL that apply.
A. x = 3 * 4 + 29
B. x = 3 + 4 + 29
C. x = 41
D. x = 36
The equations that model the total cost of the meal are:
A. x = 3 * 4 + 29
C. x = 41
Which of the following equations is correct?
We know that the family has 4 members and they buy:
A family dinner platter, which costs $29Each member also gets a dessert, which costs $3 per piece.Then the total cost is the $29 for the dinner plate plus four times the $3 for the dessert, this is:
x = $29 + $4*3
x = $29 + $12
x = $41
Then the correct options are A and C.
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3. provide an equation for converting gain g in db to a linear scale. for example, a 6db gain translates to a factor of 2, so y[n]
The equation for converting gain in decibels to a linear scale is dB → gain-multiplier: [tex]g=2^{\frac{d}6} }[/tex]
gain-multiplier → dB: [tex]6*log_{2} g[/tex]
The decibel is used in a wide range of applications. Decibels are especially used when referring to power or a derived measure, which values can vary in a wide range. The most prominent usage of decibels is in sound volume. So, for example, a sound of 0dB is barely hearable, whereas a vacuum cleaner on average has 75dB and a rock concert reaches about 110 dB.
Practical version-
dB → gain-multiplier: [tex]g=2^{\frac{d}6} }[/tex]
gain-multiplier → dB: [tex]6*log_{2} g[/tex]
So if we set that 0 dB gain as 1.0 factor, and -∞ dB gain is 0.0 factor, it means that (if we are considering voltage gain as in a mixing desk fader) :
gain = 20.0*log10(factor)
therefore :
factor = 10^(gain/20.0)
If, as described in a comment, the 0 dB gain is at 0.65 factor, it means the ref is 0.65.
gain in dB = 20*log10(factor/0.65)
factor = 0.65*10^(gain/20.0)
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Find the length of side x in simplest radical form with a rational denominator.
Answer: x =
45°
4
X
45°
Submit Answer
Answer: [tex]x=4\sqrt{2}[/tex]
Step-by-step explanation:
Using the properties of a 45-45-90 triangle, the hypotenuse is [tex]\sqrt{2}[/tex] times the length of a leg.
Therefore, [tex]x=4\sqrt{2}[/tex].
the number t if 1/4 of t is 15
Answer:
60
Step-by-step explanation:
1/4=15 then multiply 15 by 4 and 1 by 1then the answer will be 60How much would you need to deposit in an account now in order to have $5000 in the account in 5 years? Assume the account earns 5% interest compounded daily.
Answer:
$3893.97
Step-by-step explanation:
5000 = x (1 + 0.05/365)^365 * 5
5000 = x ( 1 + 0.000137)^1825
5000 = x (1.000137)^1825
5000 = 1.284036 x
1.284036 / 1.284036 x = 5000 / 1.284036
x = 3 893.97182
x = 3 893.97
Answer:
$3,894.07
Step-by-step explanation:
[tex]\boxed{\begin{minipage}{8.5 cm}\underline{Compound Interest Formula}\\\\$ A=P\left(1+\frac{r}{n}\right)^{nt}$\\\\where:\\\\ \phantom{ww}$\bullet$ $A =$ final amount \\ \phantom{ww}$\bullet$ $P =$ principal amount \\ \phantom{ww}$\bullet$ $r =$ interest rate (in decimal form) \\ \phantom{ww}$\bullet$ $n =$ number of times interest is applied per year \\ \phantom{ww}$\bullet$ $t =$ time (in years) \\ \end{minipage}}[/tex]
Given:
A = $5,000r = 5% = 0.05n = 365 (daily)t = 5 yearsSubstitute the given values into the formula and solve for P:
[tex]\implies 5000=P\left(1+\dfrac{0.05}{365}\right)^{365 \times 5}[/tex]
[tex]\implies 5000=P\left(1.000136986...\right)^{1825}[/tex]
[tex]\implies P=\dfrac{5000}{\left(1.000136986...\right)^{1825}}[/tex]
[tex]\implies P=3894.070588...[/tex]
Therefore, the amount you would need to deposit in an account now in order to have $5,000 in the account in 5 years time is $3,894.07.
Ruby decides to research the relationship between the length in inches and the weight of a certain species of catfish. She measures the length and weight of a number of specimens she catches then throws back into the water. After plotting all her data, she draws a line of best fit. What is the meaning of the x-value on the line when y=25 ?
The required value of x for y = 25 is 32.4
What is a linear function?
A polynomial function of degree 0 or 1 that has a straight line as its graph is said to be linear.
A linear function is given by:
y = mx + b
where:
The slope, or m, indicates how much y changes when x changes by one, or the rate of change.
b is the y-intercept, which is also known as the function's starting point and is the value of y at x = 0.
From the given graph, the line passes through points (32,25) and (48,50), then the slope of the line is given by:
m = (50-25)/(48-32) = 25/16 = 1.6
Then, y = 1.6x + b
For points (48,50) then,
or, 50 = 1.6*48 + b
or, 50 = 76.8 + b
or, b = 50 - 76.8 = -26.8
Then, the equation will be
y = 1.6x - 26.8
So, for y = 25 then x becomes
25 = 1.6x - 26.8
or, 1.6x = 25 + 26.8
or, 1.6x = 51.8
or, x = 32.4
Hence, the required value of x is 32.4.
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What is the slope of the line? m=
Answer: 11/12
Step-by-step explanation:
it is a rise over run so the slope would be 11/12
Solve for x.
− x/2<−1
Answer: x>2
Step-by-step explanation:
To solve for x, we want to use inverse operations to isolate x.
[tex]-\frac{x}{2} < -1[/tex] [multiply both sides by -2]
[tex]x > 2[/tex]
what is 4.24 divided by 4
Answer:
the answer is 1.06
Step-by-step explanation:
1.06 x 4 = 4.24
All i need is part C thank you so much brainiest to first person to answer!
Answer:
No, because with Y- represents the reading time & X-Representing the TV time. The two variables are being effected by the age of te viewer/reader
Step-by-step explanation:
Also show how you got the answer pls
Answer:
m² + 7m - 9
Step-by-step explanation:
[tex]\frac{m^3+16m^2+54m-81}{m+9}[/tex]
setting up the synthetic division
- 9 | 1 16 54 - 81
↓ - 9 - 63 81
-------------------------
1 7 - 9 0 ← remainder
quotient is m² + 7m - 9
since the remainder is zero then (m + 9) is a factor of the polynomial
Solve for x. Would appreciate the help thanks
The value of x in the triangles is 24.5
How to determine the value of x?From the question, we have the following parameters that can be used in our computation:
The triangle
On the triangle, we have the following ratio
7 + x : x = 27 : 27 - 6
Evaluate the difference
7 + x : x = 27 : 21
Express as fraction
This gives
x/(x + 7) =21/27
So, we have
27x = 21(x + 7)
Expand
27x - 21x = 147
So, we have
6x = 147
Evaluate
x = 24.5
Hence, the value of x is 24.5
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help meeeeeeeeeee pleaseee
The pressure of 28 inches of mercury occurs about 6 miles from the eye of the hurricane. We get this from the given algebraic expression.
What is an expression?An expression is formed by variables, constants, and algebraic operations. Since the operation among them is an algebraic or arithmetic operation, it is said to be an algebraic expression.
Calculation:It is given that the algebraic expression that relates the barometric pressure and the eye of the hurricane as
f(x) = 0.48 ln(x+2) + 27
Here x is the distance in miles from the eye of the hurricane.
f(x) is the pressure of the mercury in a barometer in inches
So, the required distance from the eye of the hurricane when the pressure of 28 inches of mercury in the meter is
(Here f(x) = 28)
f(x) = 0.48 ln(x+2) + 27
⇒ 28 = 0.48 ln(x+2) + 27
⇒ 0.48 ln(x+2) = 28 - 27
⇒ ln (x+2) = 1/0.48
⇒ ln(x+2) = 2.0833
Applying exponential base "e" on both sides, we get
(x+2) = [tex]e^{2.0833}[/tex]
⇒ x + 2 = 8.0309
⇒ x = 8.0309 - 2 = 6.0309
When the result is rounded to the nearest whole number, we get x = 6 miles.
Thus, for the pressure of 28 inches of mercury, the eye of the hurricane is 6 miles far from the barometer.
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Mrs. Jameson paid $22.50 for each student to visit an amusement park. Write an equation relating the total cost, y, and the number of students, x, attending the park.
The equation for the total cost is y = 22.5x, and the total cost of four more students joining the group is $292.50.
What is a linear equation?
It is defined as the relation between two variables, if we plot the graph of the linear equation, we will get a straight line.
If in the linear equation, one variable is present, then the equation is known as the linear equation in one variable.
Given:
Mrs. Jameson paid $202.50 for a group of 9 students to visit an amusement park.
Each student cost = 202.50/9 = $22.5
Total cost equation (y):
y = 22.5x
Here x is the number of students.
Plug x = 4 in the equation y = 22.5x
⇒ y = 22.5 (4)
y = $90
Total cost = 90 + 202.50 = $292.50
Hence, the equation for the total cost is y = 22.5x, and the total cost of four more students joining the group is $292.50.
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a least-squares fitting of a simple regression model minimizes the sum of the squares of the perpendicular distances from the data points to the regression line.
The "method is least square" is a least-squares fitting of a simple regression model minimizes the sum of the squares of the perpendicular distances from the data points to the regression line.
The term linear regression refers the predictive method to predict or estimate the values of a dependent quantitative variable denoted by y with the help of other independent variables denoted by x1,x2,x3 by using one of many methods like that of ordinary least squares.
Here we need to find what is the method for a least-squares fitting of a simple regression model minimizes the sum of the squares of the perpendicular distances from the data points to the regression line.
As per the definition of the linear regression, we have know that a least-squares fitting of a simple regression model which minimizes the sum of the squares of the perpendicular distances from the data points to the regression line.
Here we have to take the square for the number to fit into the given interval.
So, this is known as method for least square and it does not use the sum of perpendicular distances between the points and the ‘estimated regression equation’ does not used to minimize.
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Use the figures to answer the questions.
1. What is the diameter of circle A ?
2. What is the diameter of circle B ?
3. What is the radius of circle A ?
4. What is the radius of circle B?
5. Are the radius and diameter of the two circles proportional? Explain.
6. The circumference of circle Q is 21 centimeters.
a. Write a proportion to find the circumference of circle P
b. What is the circumference of circle P?
Answer:
1) 12ft
2) 8ft
3) 6 ft
4) 3ft
5)
Answer:
1. 12ft
2. 8ft
3. 6ft
4. 4ft
5. The diameter and radius of the circles are proportional by a factor of 1.5 because 12/8 = 1.5 and if I multiply circle B's radius by a factor of 1.5 I will get the radius of Circle A as is with the diameter.
6.
A- (3/7)(21) = x
B- x approximately equals 9
Step-by-step explanation:
Circumference formula
C= 2(pi)r
z equals circumference of circle q
Q = 7
P = 3
Z = 21
(P/Q)(Z) = x
Therefore (3/7)(21) = 9
We can prove this by using the Circumference formula
C= 2(pi)r
C=2(pi)1.5
C= 3(pi)
C= 9.4247...
Rounded = 9
hope this helps!!
There are 1,000 grams in 1 kilogram. Nate is mailing a package that has a mass of 12 kilograms. He wants to know the mass of the package in grams? Complete the statement to describe how to convert kilograms to grams.
12,000g
Step-by-step explanation:Here are the steps to figuring out your problem:Since there are 1,000 grams per kilogram, that means:
1 kg = 1,000 g
2kg = 2,000g
3kg = 3,000g
And so on until we get to 12 kilograms:
12kg = 12,000g
Solution:
12,000g
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Answer:
12000 gram
Step-by-step explanation:
step 1:
[tex]1 kg=1000g\\12kg = x[/tex]
step 2:
do a criscross multiplication to find x.
[tex]x*1kg=1000g*12 kg[/tex]
step 3:
devide both sides by 1kg
[tex]x=(1000g*12kg)/1kg[/tex]
step 4:
cancel units tha exist on both the numerator and the denominator.(in our case cancel kg)
[tex]x=(1000kg*12)/1[/tex]
Step 5: simplify
[tex]x=12000g[/tex]
PLS HELP ILL GIVE BRAINLIESTTTT Find the probability of rolling an odd sum and a sum less than or equal to four
(rolling two fair dice)
Answer:
The probability of rolling an odd sum and a sum less than or equal to four is 1/12
Step-by-step explanation:
3.4.5* At the end of the section, we let r=1/x-1 in x/x-1+2/3=2/x-1 to get x r+2/3=2 r
While at first we may seem stuck because we have two variables instead of one, we can still use this substitution to solve the problem!
(a) Solve the equation r=1/x-1 for x in terms of r. In other words, manipulate the equation until you have x equal to an expression with r' s in it, but no x's.
(b) Substitute this expression for x in the equation x r+2/3=2 r. Do you have a linear equation now? Solve that equation for r. Use your value of r to find x.
a) Making x the subject of the formula in terms of r is; x = (r + 1)/r
b) Solving for r and x gives; r = 5/3 and x = 8/5
How to change the subject of the formula?a) We are given that;
r = 1/(x - 1)
We are asked to make x the subject of the formula in terms of r. Thus;
Multiply both sides by (x - 1) to get;
rx - r = 1
r + 1 = xr
x = (r + 1)/r
b) We are given that;
xr + ²/₃ = 2r
Put (r + 1)/r for x to get;
r(r + 1)/r + ²/₃ = 2r
r + 1 + ²/₃ = 2r
1 + ²/₃ = r
r = 5/3
Put 5/3 for r into the x equation to get;
x = ((5/3) + 1)/(5/3)
x = (8/3)/(5/3)
x = 8/5
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To obtain information on the corrosion-resistance properties of a certain type of steel conduit, 45 specimens are buried in soil for a 2-year period. The maximum penetration (in mils) for each specimen is then measured, yielding a sample average penetration of x = 52.9 and a sample standard deviation of s = 4.2. The conduits were manufactured with the specification that true average penetration be at most 50 mils. They will be used unless it can be demonstrated conclusively that the specification has not been met. What would you conclude? (Use alpha = 0.05.) State the appropriate null and alternative hypotheses. H_0: mu = 50 H_a: mu notequalto 50 H_0: mu = 50 H_a: mu > 50 H_0: mu notequalto 50 H_a: mu > 50 H_0: mu > 50 H_a: mu = 50 Calculate the test statistic and determine the P-value. (Round your test statistic to two decimal places and your P-value to four decimal places.) z = P-value = State the conclusion in the problem context. Reject the null hypothesis. There is sufficient evidence to conclude that the true average penetration is more than 50 mils. Do not reject the null hypothesis. There is not sufficient evidence to conclude that the true average penetration is more than 50 mils. Reject the null hypothesis. There is not sufficient evidence to conclude that the true average penetration is more than 50 mils. Do not reject the null hypothesis. There is sufficient evidence to conclude that the true average penetration is more than 50 mils.
Statistic value t=2.27
p-value: < 0.00001
To obtain information on the corrosion-resistance properties of a certain type of steel conduit a random sample of 45 specimens was taken and buried for two years.
The study variable is:
X: Max. penetration of a steel conduit.
The data of the sample
n= 45
sample mean X[bar]= 52.9
sample standard deviation S= 4.2
The conduits are manufactured to have a true average penetration of at most 50 mills, symbolically: μ ≤ 50
The hypothesis is:
H₀: μ ≤ 50
H₁: μ > 50
α: 0.05
To choose the corresponding statistic to use to study the population mean, the variable must have a normal distribution. There is no available information to check this, so I'll just assume that the variable has a normal distribution and, since the population variance is unknown and the sample is small, the statistic to use is a Student t.
Under the null hypothesis, the critical region and the p-value are one-tailed.
Critical value: [tex]t_{n- 1;1-\alpha} =t_{44;0.95} =1.68[/tex]
Rejection rule:
Reject the null hypothesis when t ≥ 1.68
t= [tex]\frac{52.9-50}{\frac{4.2}{\sqrt{45} } }[/tex]
t= 2.9-0.626 = 2.27
If the calculated value is greater than the critical value, the decision is to reject the null hypothesis.
p-value:
P(t ≥ 2.27) = 1 - P(t < 2.27) = < 0.00001.
The p-value is less than α so the decision is to reject the null hypothesis.
Since the null hypothesis was rejected, then the population average of the penetration of the conduits specimens is greater than 50 mils. It is not recommendable to use these conduits.
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