Answers
Length of the side of square is x = 5.7 unit
What is Pythagoras theorem?The Pythagorean theorem states, "In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides." The sides of this triangle were called the perpendicular, the base and the hypotenuse. Here, the hypotenuse is on the opposite side of the 90° angle, so it is the longest side.
Given,
Figure of a rectangle and a right triangle share its hypotenuse with length of rectangle
diagonal of rectangle = 8
length of both sides of right triangle other than hypotenuse = 4
Let length of rectangle be l
By Pythagoras theorem
l² = 4² + 4²
l² = 32
l = 4√2
Now, In the rectangle
8² = l² + x²
64 = 32 + x²
x² = 32
x = 4√2
x = 5.7
rectangle is an square.
Hence value of x, length of the side of square is 5.7 unit.
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Related Questions
The Sugar Sweet Company will choose from two companies to transport its sugar to market. The first company charges $6500 to rent trucks plus an addl fee of $100.25 for each ton of sugar. The second company charges $4496 to rent trucks plus an additional fee of $225.50 for each ton of sugar.
1. For what amount of sugar do the two companies charge the same?
2.What is cost when the two companies charge the same?
Answers
Step-by-step explanation:
cost1(t) = 100.25t + 6500
cost2(t) = 225.5t + 4496
both companies charge the same for the amount of t (tons), when both functions deliver the same result :
100.25t + 6500 = 225.5t + 4496
2004 = 125.25t
t = 2004/125.25 = 16
1. for the transport of 16 tons of sugar they both charge the same.
2. that charge is
100.25×16 + 6500 = $8,104
ive been looking at this for hours sos
Answers
Answer: 36
Step-by-step explanation:
Which change, if any, is needed to the underlined text?
“Taking up to six months, hikers who plan to complete the entire
journey”
-A journey of up to six months, hikers planning to complete it entirely
-Taking up to six months to complete, hikers who plan the entire journey
-The entire journey takes up to six months to complete, so hikers who plan to
finish it
-No change
Answers
No change is needed. The underlined text is grammatically correct and conveys the intended meaning clearly.
What is the meaning of the underlined test?The underlined text "Taking up to six months, hikers who plan to complete the entire journey" means that the journey being referred to may take up to six months to complete, and it is being undertaken by hikers who have planned to complete the entire journey.
The text suggests that the journey is a long and challenging one that requires careful planning and preparation by the hikers.
Therefore, the correct answer is as given above. It could then be concluded that the underlined text does not need to change.
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For each of the following shape sequences:
i) draw a sequence table for the first six patterns, taking care to use the correct letter for the pattern number and the correct letter for the number of shapes
ii) find a formula for the number of shapes used in terms of the pattern number
iii) use your formula to find the number of shapes used in the 300th pattern.
Number of matches
n=1 n=2 n=3
m-4 m=7 m-10
Answers
Answer:
Step-by-step explanation:
It seems like the problem is asking about a sequence of shapes rather than a sequence of matches, so I will assume that the correct information for the problem is:
Number of shapes:
n=1 n=2 n=3
m-4 m=7 m+2
i) Sequence table for the first six patterns:
Pattern number Number of shapes
1 m-4
2 m
3 m+2
4 m+6
5 m+10
6 m+14
ii) Formula for the number of shapes in terms of the pattern number:
From the table, we can see that the number of shapes used in each pattern is increasing by a constant amount of 4. Therefore, we can write the formula as:
Number of shapes = (pattern number - 1) * 4 + m
where m is the number of shapes in the first pattern.
iii) Number of shapes used in the 300th pattern:
To find the number of shapes used in the 300th pattern, we can use the formula and substitute pattern number = 300:
Number of shapes = (300 - 1) * 4 + m
Since we don't know the value of m, we can't determine the exact number of shapes. However, we do know that the number of shapes in the first pattern is either m-4, m, or m+2, depending on the value of m. If we assume the smallest possible value of m, which is 1 (since the number of shapes can't be negative), then the number of shapes in the 300th pattern would be:
Number of shapes = (300 - 1) * 4 + 1-4 = 1195
Therefore, if m = 1, then the number of shapes used in the 300th pattern is 1195. However, if m is greater than 1, then the number of shapes in the 300th pattern would be higher.
Answer:
every other 3
Step-by-step explanation:
If someone could help me out on this!
Answers
The following are solutions to the given problems given f(x)=x²- 5x+7 and g(x)=3x² + 4x - 2;
(f- g)(x) = -2x² - 9x + 9
(g-f)(x) = 2x² + 9x - 9
(f + g)(x) = 4 x² - x + 5
What is a function?A function can be defined as a relation between a set of inputs having one output each. In other words, a function is a relationship between inputs where each input is related to exactly one output. Every function has a domain and co-domain (range). A function is generally denoted by f(x) where x is the input.
here, we have,
The general mathematical representation of a function is y = f(x).
f(x)=x²- 5x+7
g(x)=3x² + 4x - 2
so, we get,
(f- g)(x) = x²- 5x+7 - (3x² + 4x - 2)
(f- g)(x) = x²- 5x+7 - 3x² - 4x + 2
(f- g)(x) = -2x² - 9x + 9
(g-f)(x) = 3x² + 4x - 2 - (x²- 5x+7)
(g-f)(x) = 3x² + 4x - 2 - x² + 5x - 7
(g-f)(x) = 2x² + 9x - 9
(f + g)(x) = x²- 5x+7 + 3x² + 4x - 2
(f + g)(x) = 4 x² - x + 5
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The perimeter of a rectangular table is 18 feet. The table is 42 inches wide. Which of the following show the length of the table?
Answers
Answer:
First, we need to convert the width of the table from inches to feet to ensure that the units match. There are 12 inches in a foot, so:
42 inches ÷ 12 inches/foot = 3.5 feet
Let the length of the table be L feet. The perimeter is the sum of the lengths of all four sides of the rectangle, so:
2L + 2(3.5 feet) = 18 feet
Simplifying and solving for L:
2L + 7 feet = 18 feet
2L = 11 feet
L = 5.5 feet
Therefore, the length of the table is 5.5 feet.
Answer:
66
Step-by-step explanation:
the problem ask fr it draw out 2 rectangle
MODELING REAL LIFE You measure your distance from the base of the Citrus Tower and the angle of elevation from the ground to the top of the tower. Find the height $h$h of the Citrus Tower to the nearest hundredth.
A tower with height labeled h. The tower is the height of a right triangle with base of length 120 feet. The base angle of the right triangle is labeled 62 degrees.
Height: feet
Answers
Check the picture below.
[tex]\tan(62^o )=\cfrac{\stackrel{opposite}{h}}{\underset{adjacent}{120}}\implies 120\tan(62^o )=h\implies 225.69\approx h[/tex]
Make sure your calculator is in Degree mode.
The height of the tower is 226 feet.
What is Trigonometry?One area of mathematics known as trigonometry examines the relationship between the sides and angles of a right triangle. The relationship between sides and angles is defined for 6 trigonometric functions.
We have,
A tower with height labeled h.
The tower is the height of a right triangle with base of length 120 feet.
Now using Trigonometry
tan 62 = P / B
tan 62 = h / 120
h = 120 tan 62
h = 225.69
h = 226 feet
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Let N,p∈N, labeled data points (x1,y1),…,(xN,yN)∈Rp×{−1,1} and penalty parameter C∈R>0. A version of the SVM classification optimization problem is the following: minβ∈Rp,β0∈R,ξ≥0∥β∥22+C∑i=1Nξi s.t. ξi≥1−yi(xiTβ+β0). We denote β^,β^0,ξ^ as an optimal solution of the optimization problem. In general, it holds that... O If (Cn)n∈N is a positive, sequence of penalty parameters diverging to infinity, then the corresponding sequence (∥∥β^n∥∥22)n∈N of squared norms of optimal solutions also diverges to infinity. O Assume that for i,iˉ∈{1,…,N} it holds that yi=1 and yi~=−1. If we switch the labels, i.e., we replace yi by −yi and yi~ by −yi~, then β^,β^,ξ^ is not an optimal solution anymore. O If yi=1 for all i=1,…,N, then β^=0p. O If all features are scaled by the same factor λ∈R>0, i.e., xi is replaced by λxi for all i=1,…,N, then λβ^,β^,ξ^ solves sVM for penalty parameter λ2C.
Answers
To summarize, the correct answer is:
"If all features are scaled by the same factor λ∈R>0, i.e., xi is replaced by λxi for all i=1,…,N, then λβ^,β^,ξ^ solves SVM for penalty parameter λ2C."
The correct answer is "If all features are scaled by the same factor λ∈R>0, i.e., xi is replaced by λxi for all i=1,…,N, then λβ^,β^,ξ^ solves SVM for penalty parameter λ2C."
This is because scaling all features by the same factor λ does not change the relative importance of each feature in the classification problem. Therefore, the optimal solution for the scaled features will be the same as the optimal solution for the original features, but scaled by λ. The penalty parameter C will also need to be scaled by λ^2 to account for the scaling of the features.
To summarize, the correct answer is:
"If all features are scaled by the same factor λ∈R>0, i.e., xi is replaced by λxi for all i=1,…,N, then λβ^,β^,ξ^ solves SVM for penalty parameter λ2C."
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Adrien won the lottery! He’s buying a piece of land so that he can entertain his friends from Trig class. His new property is a triangular parcel of land that has 4 miles of lakefront, and the other boundaries have lengths of 2.8 miles and 1.8 miles. What angles, to the nearest degree, does the lakefront make with the other two boundaries and what is the size of the remaining angle?
Answers
The lakefront makes angles of approximately 30 degrees and 120 degrees with the boundaries of 2.8 miles and 1.8 miles, respectively. The remaining angle is approximately 30 degrees.
To understand why, we can use the law of cosines to find the angles of the triangle. Let's call the length of the lakefront side "c", and the lengths of the other two sides "a" and "b". Using the law of cosines, we can find the angles:
[tex]cos A = (b^2 + c^2 - a^2) / (2bc)[/tex]
[tex]cos B = (a^2 + c^2 - b^2) / (2ac)[/tex]
[tex]cos C = (a^2 + b^2 - c^2) / (2ab)[/tex]
Plugging in the values we know, we get:
[tex]cos A = (1.8^2 + 4^2 - 2.8^2) / (21.84) = 0.283[/tex]
[tex]cos B = (2.8^2 + 4^2 - 1.8^2) / (22.84) = 0.717[/tex]
[tex]cos C = (2.8^2 + 1.8^2 - 4^2) / (22.81.8) = -0.283[/tex]
Using the inverse cosine function, we can find the angles:
A ≈ 75 degrees
B ≈ 43 degrees
C ≈ 62 degrees
Therefore, the lakefront makes angles of approximately 30 degrees (180 - 75 - 75) and 120 degrees (180 - 43 - 17) with the other two boundaries, and the remaining angle is approximately 30 degrees (180 - 120 - 30).
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HELP NEEDED FROM ANYONE
Answers
The length of the diagonal is between 4 - 5 and closer to 4
How to determine the intervalFirst, we need to know the properties of a rectangle;
A rectangle is a quadrilateralThe opposite sides are equal to each otherEach interior angle is equivalent to 90 degreesThe sum of all the interior angles is equivalent to 360 degreesThe diagonals bisect each other at right anglesBoth the diagonals have the same lengthA rectangle with side lengths a and b has its perimeter as 2a+2b unitsA rectangle with side lengths a and b has its area as: ab square unitsA diagonal of a rectangle is the diameter of its circumcircleGiven that the diagonal measures;
√18
Find the value
4. 2
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vocabalary Is the
expression 3x + 2x - 4 in
simplest form? Explain.
The terms 3x and 2x are
terms and
<>
in simplest form.
長
be combined. Therefore, the expression
The expression in simplest form is.
Answers
Answer:
Step-by-step explanation:
This expression isn't in simplest form. The terms 3x and 2x can be added, since they are like terms.
In simplest form, this expression would be 5x - 4.
Hope this helps!
Given the function value of the acute angle, find the other five trigonometric function values. cosα=√3/3 sinα= (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) tanα= (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) cscα= (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) secα= (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) cotα= (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)
Answers
The values of the other trigonometric functions are:
sinα =√6/3
tanα=√2
cosecα=3/√6
secα=√3
cotα=1/√2
Given that cosα=√3/3
By definition cosine function , it is a ratio of adjacent side and hypotenuse.
So, adjacent side =√3
Hypotenuse = 3
Let us find the third side of a triangle, by using pythagoras theorem.
√3²+x²=3²
3+x²=9
x=√6
So opposite side is √6.
Now let us find the other trigonometric values
Sinα= opposite side/hypotenuse
Sinα=√6/3
Tanα=opposite side/adjacent side
=√6/√3
Tanα=√2
Cosecα = hypotenuse/opposite side
=3/√6
Cosecα=3/√6
secα=hypotenuse side/adjacent side
=3/√3
=√3
secα=√3
cotα=Adjacent side/opposite side
=√3/√6
cotα=1/√2
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The compete question is given in attachment.
Sometimes, the relationship between the variables doesn't stay the _______ This type of relation is called a piecewise linear relationship. The graph of a piecewise linear relationship is made of non-overlapping _______ straight lines, like this one.
Answers
The required fill-in-the-blanks are "same" and "line segments".
What is the linear relationship?A linear relationship is a connection that takes the shape of a straight line on a graph between two distinct variables - x and y. When displaying a linear connection using an equation, the value of y is derived from the value of x, indicating their relationship.
Sometimes, the relationship between the variables doesn't stay the "same". This type of relation is called a piecewise linear relationship.
The graph of a piecewise linear relationship is made of non-overlapping "line segments", like this one.
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The graph shows the relationship between the time Katrina spends jogging and her total distance traveled. Which function represents the relationship?
Answers
The function which represent the distance and time is y=6x-17.5.
What is function?A mathematical phrase, rule, or law that establishes the link between an independent variable and a dependent variable (the dependent variable). In mathematics, functions exist everywhere, and they are crucial for constructing physical links in the sciences.
Here according to the given data, let us take to points to find linear function then,
[tex](x_1,y_1)=(0.5,3)[/tex] and [tex](x_2,y_2)=(1,6)[/tex].
Now using slope formula, m = [tex]\frac{y_2-y_1}{x_2-x_1}[/tex]
=> Slope m = [tex]\frac{6-3}{1-0.5} = \frac{3}{0.5}[/tex] = 6
Now using equation formula , [tex]y-y_1=m(x-x_1)[/tex] then,
=> y-0.5=6(x-3)
=> y-0.5=6x-18
=> y=6x-18+0.5
=> y=6x-17.5
Hence the function which represent the distance and time is y=6x-17.5.
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simply the expression (6^2)4
Answers
Answer:
144
Step-by-step explanation:
6*6=36
36*4=144
Consider a circle whose equation is x2 + y2 – 2x – 8 = 0. Which statements are true? Select three options. The radius of the circle is 3 units. The center of the circle lies on the x-axis. The center of the circle lies on the y-axis. The standard form of the equation is (x – 1)² + y² = 3. The radius of this circle is the same as the radius of the circle whose equation is x² + y² = 9.
Answers
The three cοrrect statements fοr the given circle are:
1)The radius οf the circle is 3 units.
2)The centre οf the circle lies οn the x-axis.
3)The radius οf this circle is the same as the radius οf the circle whοse equatiοn is x² + y² = 9.
What is a circle?All pοints in a plane that are at a specific distance frοm a specific pοint, the centre, fοrm a circle. In οther wοrds, it is the curve that a mοving pοint in a plane draws tο keep its distance frοm a specific pοint cοnstant. Circle's basic equatiοn is represented by:
(x-h)² + (y-k)² = r²
where the radius is r and the circle's centre's cοοrdinates are (h,k). Hence, we can quickly determine the equatiοn οf a circle if we knοw its radius and centre cοοrdinates.
The equatiοn οf the circle is given as:
x² + y²– 2x – 8 = 0
This can be cοnverted tο the standard fοrm.
Fοr that, we add and subtract 1 in the abοve equatiοn.
x² + y²– 2x – 8 +1 - 1 = 0
Rearranging,
x² – 2x + 1 + y²– 8 -1 = 0
(x - 1)² + y² = 8+1 = 9
Sο the equatiοn οf the circle in the standard fοrm is:
(x - 1)² + y² = 9
Nοw we will lοοk intο the given statements.
1) The radius οf the circle is 3 units.
This is true because radius = √9 = 3 units.
2) The centre οf the circle lies οn the x-axis.
The centre οf the given circle is (1,0).
Sο it dοes lie οn the x-axis.
3) The centre οf the circle lies οn the y-axis.
The centre is (1,0).
Sο this is nοt true.
4)The standard fοrm οf the equatiοn is (x – 1)² + y² = 3.
This is false.
5) The radius οf this circle is the same as the radius οf the circle whοse equatiοn is x² + y² = 9
This is true because bοth equatiοns have 9 οn the right side οf the equatiοn.
Therefοre the three cοrrect statements fοr the given circle are:
1)The radius οf the circle is 3 units.
2)The centre οf the circle lies οn the x-axis.
3)The radius οf this circle is the same as the radius οf the circle whοse equatiοn is x² + y² = 9.
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What is the common ratio of this geometric sequence
Answers
Answer:
what about you read well and stop cheating
I really need the answers to these math equations
Answers
A correct 2-column proof include the following: B. table B.
What is a supplementary angle?In Mathematics, a supplementary angle simply refers to two (2) angles or arc whose sum is equal to 180 degrees.
Since line segment r is parallel to line segment s, the 2-column proof to justify that angle b and angle h are supplementary angles should be written as follows;
Statement Reasons_________________
r║s Given
∠b ≅ ∠c Vertical angles
∠c and ∠e are supplementary Same-side interior angles
∠e ≅ ∠h Vertical angles
∠b and ∠h are supplementary Substitution
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5. Jessica is building a model rocket for her physics class. After studying the flight path of her rocket, she has
concluded that she wants her rocket to achieve a maximum height of 50 ft. The equation for her rocket is
-3x² + 6x + 48. Will Jessica's rocket clear 50 ft?: (Hint Find the vertex of the equation to find the maximum
height of the rocket)
Answers
explanation:
the quadratic equation in standard form is
y=ax^2+bx+c
and to find the vertex the formula is
-b/2a
-3x^2+6x+48
a=-3
b=6
c=48
now u just substitute the numbers in the letters so
-b/2a=
-6/2(-3)=
-6/-6= 1
now u substitute 1 where x is
-3x^2+6x+48=
-3(1)^2+6(1)+48=
-3+6+48=51
so the maximum is 51.
I hope this helps!!!!
You are going to paint one side of a building. The wall is 105 feet long and 35 feet high. One gallon of paint covers 317 square feet with a single coat of paint. Find how many gallons should be purchased to apply one coat of paint to the side of the building.
Answers
Amount of paint that should be purchased to apply one coat of paint to the side of the building is 12 gallons.
What is area?Area is the amount of space occupied by a two-dimensional shape. That is, it is a quantity that measures the number of unit squares that cover the surface of a closed figure. The standard unit of area is the square unit, commonly expressed in square inches, square feet, and so on.
Given,
length of the wall = 105 feet
height of the wall = 35 feet
Area of the wall
= length × height
= 105 × 35
= 3675 square feet
One gallon paints 317 square feet
Number of gallons to paint 3675 square feet
= 3675/317
= 11.59 ≈ 12
Hence, 12 gallons should be purchased to apply one coat of paint to the side of building.
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What are the coordinates for the graph and what do I need to do?
Answers
not completely sure but i think you times the x and y numbers then draw a graph of the units
Did I get these Zeros right while factoring?
Answers
Answer:
You forgot the +- part when taking the square roots
Step-by-step explanation:
5 x^2 = -2
x^2 = -2/5
x = +- sqrt ( -2/5) <==== note sqrt(-2/5) NOT sqrt (-2) / 5
x = ± i sqrt(2/5)
x^2 = 8
x = ±sqrt 8
x = ±2 sqrt (2)
4. To pay for a trip to Machu Picchu in 4 years, Doug wants to deposit money into a savings account that earns2.9%annual interest compounded continuously. Doug calculates the amount he needs to deposit as shown below.A250025002012002226.19=Pe7=Petosice =Pe034=P=PSo, Doug needs to deposit$2226.19into his account.
Answers
Number of years = 4
Hi there! To answer your question, Doug needs to deposit $2226.19 into a savings account that earns 2.9% interest compounded continuously. This amount was calculated using the formula P = Pe^rt, where P is the final amount Doug will have after 4 years, e is the natural logarithm constant (2.71828), r is the interest rate (2.9%), and t is the number of years (4).
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F(x)=x^2+6x+8 What are the zeroes of the function? Write the smaller x first, and the larger x second
Answers
To find the zeros, set the function equal to 0. In other words, solve f(x)=0.
f(x) = 0
x^2 +6x + 8 = 0
(x+4)(x +2) = 0
x = -4 and x = -2
The vertex will happen when x = -b/2a:
[tex]x=\dfrac{-6}{2(1)} = -3[/tex]
Then use this x-value of –3 to find the y-value of the vertex:
y = (-3)^2 + 6(-3) + 8 = 9-18+8 = -1
The vertex is (-3,-1).
Side note: The x-value of the vertex can also be found by finding the average of the two zeros:
(-4 + (-2)) / 2 = -6/2 = -3
17. Find the coordinates of the image of EB given E(2, -1) and B(0, 5) after a dilation with center
(3,-1) and scale factor of 3.
Answers
The coordinates of the image of EB are (0, -1) and (-6, 18).
What is dilation?Dilation is a process for creating similar figures by changing the dimensions.
The center of dilation is given as (3, -1) and the scale factor is 3.
The distance between the center of dilation and point E is:
d = √((3-2)² + (-1-(-1))²) = 1
The distance between the center of dilation and point B is:
d = √((3-0)² + (-1-5)²) = sqrt(65)
To find the new distance, we multiply the distance by the scale factor:
For point E: 3 × 1 = 3
For point B: 3 × √(65)
Now, we can find the coordinates of the image:
For point E:
x = 3 × (2 - 3) + 3 = -3 + 3 = 0
y = 3 × (-1 -(-1)) -1 = 0 - 1 = -1
So the image of E is (0, -1).
For point B:
x = 3 × (0 - 3) + 3 = -9 + 3 = -6
y = 3 × (5 -(-1)) -1 = 18
So the image of B is (-6, 18).
Therefore, the coordinates of the image of EB are (0, -1) and (-6, 18).
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The volume of a cone is 196π cubic feet. Its radius is 7 feet. Find the height
Answers
[tex]\textit{volume of a cone}\\\\ V=\cfrac{\pi r^2 h}{3}~~ \begin{cases} r=radius\\ h=height\\[-0.5em] \hrulefill\\ V=196\pi \\ r=7 \end{cases}\implies 196\pi =\cfrac{\pi (7)^2 h}{3}\implies 196\pi =\cfrac{49\pi h}{3} \\\\\\ \cfrac{3}{49\pi }\cdot 196\pi =h\implies 12=h[/tex]
Select the correct answer.
What is the inverse of function f?
f (x) = X+7
Answers
Step-by-step explanation:
Inverse of a Function.
To find the inverse of the function f(x) = √(x+7), we need to interchange the roles of x and y and solve for y.
Let y = f(x) = √(x+7)
To find the inverse, we need to solve for x in terms of y.
Step 1: Square both sides of the equation to eliminate the square root:
y^2 = x + 7
Step 2: Solve for x:
x = y^2 - 7
So the inverse of the function f(x) is:
f^(-1)(y) = y^2 - 7
We can also express the inverse in terms of x:
f^(-1)(x) = x^2 - 7
Note that we changed the variable from x to y, and from y to x, when expressing the inverse function.
Colonial gas company charges its customers according to their usage of gas as follows: a $6.00 customer charge, &1.180 per unit (100 cubic feet) for the first 20 units, and $0.806 per unit for each over 20.
a) Determine the cost function and draw its graph. (Please draw the graph on a paper)
b) What are the total and average charges for using 15 units?
c) How many units were used if the total charge is $73.93?
Answers
a) The cost function can be represented as C(x) = 6 + 1.180x for the first 20 units and C(x) = 6 + 1.180(20) + 0.806(x-20) for units over 20.
b) The total charge for using 15 units is C(15) = 6 + 1.180(15) = $23.70.
c) If the total charge is $73.93, we can use the cost function to solve for the number of units used.
The graph of the cost function will have a linear portion for the first 20 units and then another linear portion with a different slope for units over 20.
The average charge is $23.70/15 = $1.58 per unit.
Since the total charge is greater than the cost for the first 20 units, we know that more than 20 units were used. We can use the second part of the cost function to solve for x:
73.93 = 6 + 1.180(20) + 0.806(x-20)
73.93 = 30.6 + 0.806x - 16.12
73.93 - 30.6 + 16.12 = 0.806x
59.44 = 0.806x
x = 73.69 units
Therefore, 73.69 units were used.
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Let N E N be such that N > 2. Let Ω be the set of non-empty subsets of {1,...,N}, i.e. Ω = {ω C {1,...,N}: ω ≠0}. Let F be the o-algebra on Ω formed by all subsets of Ω and P be the uniform probability measure on (Ω, F). For ω E Ω, let X and Y be the random variables defined as X(ω) = max(ω) and Y(ω) = min(ω) So that, for example, if w = {1, 2, N} then X(W)= N and Y (W) = 1. (a) Show that the probability mass function of X is px (n) = 2^n-1 / 2^N – 1 n € {1,...,N} and 0 otherwise. (b) For any t € R, compute the value of the function 0: R -> R defined as ɸ(t) = E [2^tx] [2 marks] (c) Show that the joint probability mass function of (X,Y) is 1 / 2^N -1, for m
Pxx^(n, m) = { 2^(n-m-1) / 2^N -1 for n=m, n, m{1,...,N}
0, otherwise [2 marks] (d) Determine the probability mass function of W - X - Y. [3 marks)
Answers
(a) The probability that X = n is 2^n-1 / 2^N - 1.
(b) ɸ(t) = E[2^tx] = ∑_{n=1}^N 2^tx P(X=n) = ∑_{n=1}^N 2^tx (2^n-1 / 2^N - 1) = (2^t / 2^N - 1) ∑_{n=1}^N 2^(n-1)t = (2^t / 2^N - 1) (2^Nt - 1) / (2^t - 1) = 2^Nt / (2^N - 1)
(c) The probability that X = n and Y = m is 2^(n-m-1) / 2^N - 1.
(d) This is the same as finding the probability that the difference between the maximum and minimum elements of a subset of {1,...,N} is k. If k = 0, then the only subsets with maximum and minimum elements differing by k are the single-element subsets, so P(W=k) = N / 2^N - 1. If k > 0, then there are (N-k) choices for the minimum element m and 2^(k-1) subsets of {m+1,...,m+k-1}, so P(W=k) = (N-k) 2^(k-1) / 2^N - 1.
To find the probability mass function of X, we need to find the probability that X = n for each n ∈ {1,...,N}. This is the same as finding the probability that the maximum element of a subset of {1,...,N} is n. There are 2^n-1 subsets of {1,...,n-1}, and each of these subsets can be combined with n to form a subset of {1,...,N} with maximum element n. Therefore, the probability that X = n is 2^n-1 / 2^N - 1.
To find the value of ɸ(t) for any t ∈ R, we need to compute the expected value of 2^tx. Using the formula for expected value, we get:
ɸ(t) = E[2^tx] = ∑_{n=1}^N 2^tx P(X=n) = ∑_{n=1}^N 2^tx (2^n-1 / 2^N - 1) = (2^t / 2^N - 1) ∑_{n=1}^N 2^(n-1)t = (2^t / 2^N - 1) (2^Nt - 1) / (2^t - 1) = 2^Nt / (2^N - 1)
To find the joint probability mass function of (X,Y), we need to find the probability that X = n and Y = m for each n,m ∈ {1,...,N}. If n = m, then the only subset of {1,...,N} with maximum element n and minimum element m is {n}, so P(X=n, Y=m) = 1 / 2^N - 1. If n ≠ m, then there are 2^(n-m-1) subsets of {m+1,...,n-1}, and each of these subsets can be combined with m and n to form a subset of {1,...,N} with maximum element n and minimum element m. Therefore, the probability that X = n and Y = m is 2^(n-m-1) / 2^N - 1.
To find the probability mass function of W = X - Y, we need to find the probability that W = k for each k ∈ {0,...,N-1}. This is the same as finding the probability that the difference between the maximum and minimum elements of a subset of {1,...,N} is k. If k = 0, then the only subsets with maximum and minimum elements differing by k are the single-element subsets, so P(W=k) = N / 2^N - 1. If k > 0, then there are (N-k) choices for the minimum element m and 2^(k-1) subsets of {m+1,...,m+k-1}, so P(W=k) = (N-k) 2^(k-1) / 2^N - 1.
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Please do the follo When the fraction rewrite it as a mixe integer. 0.6-:2(2)/(11)
Answers
The mixed integer would be 2 3/4.
A mixed integer is a number that includes both a whole number and a fraction. For example, 2 1/2 is a mixed integer because it includes the whole number 2 and the fraction 1/2.
To rewrite a fraction as a mixed integer, you need to divide the numerator (the top number) by the denominator (the bottom number) and find the quotient and remainder. The quotient will be the whole number part of the mixed integer and the remainder will be the numerator of the fraction part. The denominator will stay the same.
For example, if you have the fraction 11/4, you can divide 11 by 4 to get a quotient of 2 and a remainder of 3. So the mixed integer would be 2 3/4.
Here are the steps to rewrite a fraction as a mixed integer:
1. Divide the numerator by the denominator.
2. Write the quotient as the whole number part of the mixed integer.
3. Write the remainder as the numerator of the fraction part of the mixed integer.
4. Keep the denominator the same.
5. Simplify the fraction if necessary.
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