Dilation involves adjusting the size of an object or a figure, without altering its shape.
The object can be increased or decreased depending on its scale factor.
A scale factor less than 1 results in a figure of reduced dimensions whereas, a scale factor greater than 1 results in a figure or an object of enlarged dimensions.
In the ΔABC, a dilation of 4/3, which is greater than 1, will thus result into an enlargement.
The correct option is B.
are f(x) and g(x) inverse functions across the domain (5, + infinity)
Given:
[tex]\begin{gathered} F(x)=\sqrt{x-5}+4 \\ G(x)=(x-4)^2+5 \end{gathered}[/tex]Required:
Find F(x) and G(x) are inverse functions or not.
Explanation:
Given that
[tex]\begin{gathered} F(x)=\sqrt{x-5}+4 \\ G(x)=(x-4)^{2}+5 \end{gathered}[/tex]Let
[tex]F(x)=y[/tex][tex]\begin{gathered} y=\sqrt{x-5}+4 \\ y-4=\sqrt{x-5} \end{gathered}[/tex]Take the square on both sides.
[tex](y-4)^2=x-5[/tex]Interchange x and y as:
[tex]\begin{gathered} (x-4)^2=y-5 \\ y=(x-4)^2+5 \end{gathered}[/tex]Substitute y = G(x)
[tex]G(x)=(x-4)^2+5[/tex]This is the G(x) function.
So F(x) and G(x) are inverse functions.
[tex]\begin{gathered} G(x)-5=(x-4)^2 \\ \sqrt{G(x)-5}=x-4 \\ x=\sqrt{G(x)-5}+4 \end{gathered}[/tex]Final Answer:
Option A is the correct answer.
how do you find a point slope in geometry
see explanation below
Explanation:
To find the point slope form of an equation, we will apply the formula:
[tex]y-y_1=m(x-x_1)[/tex]Given two points, we will be able to find the slope = m
for example: (1, 2), (2, 4)
m = slope = change in y/ change in x
m = (4-2)/(2-1)
m = 2/1
m = 2
Then, we will pick any of the points and insert into the formula for the point slope.
Let's assume we are using point (1, 2) = (x1, y1)
inserting into the formula together with the slope gives:
y - 2 = 2(x - 1)
The above is a point slope for the points given.
Draw the angle 0=-pi/2 in standard position find the sin and cos
An angle in standard position has the vertex at the origin and the initial side is on the positive x-axis.
Thus, the initial side of the angle is:
Now, half the circumference measures pi, thus, pi/2 is a quarte of the circumference. As we want to find the angle -pi/2, then we need to rotate the terminal side clockwise:
Find the sine and the cosine.
The sine and the cosine in the unit circle are given by the coordinates as follows:
[tex](\cos\theta,\sin\theta)[/tex]As can be seen in the given unit circle, the terminal side is located at:
[tex](0,-1)[/tex]Thus, the values of cosine and sine are:
[tex]\begin{gathered} \cos\theta=0 \\ \sin\theta=-1 \end{gathered}[/tex]Evaluate 1312e 4 Sov? 3x²x3 dx (Type an exact answer.)
We have to solve the integral:
[tex]\int ^4_03x^2e^{x3}dx[/tex]We will apply a variable substitution in order to simplify the solution. We have a hint when we see that the derivative of x^3 is 3x^2, that is part of the factors.
[tex]\begin{gathered} u=x^3\Rightarrow du=(3x^2)dx \\ x=0\Rightarrow u=0^3=0 \\ x=4\Rightarrow u=4^3=64 \end{gathered}[/tex]Then, we can write:
[tex]\int ^4_03x^2e^{x3}dx=\int ^4_0e^{x3}(3x^2)dx=\int ^{64}_0e^udu[/tex]Then, we have a simpler integral to solve:
[tex]\int ^{64}_0e^udu=e^u+C=e^{64}-e^0=e^{64}-1[/tex]The exact solution is e^64-1.
find the solution to the following system by substitution x + y = 20 y = 3x 8
Based on the substitution method, the solution of the system of the equation is x = 3 and y = 17.
Substitution method:
Substitution method is the way of finding the value of any one of the variables from one equation in terms of the other variable.
Given,
Here we have the system of equations
x + y = 20
y = 3x + 8
Now we need to find the solutions for these equation using the substitution method.
From the given details we know that the value of y is defined as 3x + 8.
So, we have to apply these value on the other equation in order to find the value of x,
x + (3x + 8) = 20
4x + 8 = 20
4x = 20 - 8
4x = 12
x = 3
Now apply the value of x into the other equation in order to find the value of y,
y = 3(3) + 8
y = 9 + 8
y = 17
Therefore, the solution of the equation is x = 3 and y = 17.
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May I please get help with describing each or the math problems
From the given traingles, let's select the correct statements.
(a) Select all that describe BD.
Here, the line BD divides angle B into 2 equal parts. It means BD bisects ∠D.
An angle bisector is a line that divides an angle into two equal angles.
Hence, we can say BD is an angle bisector of ∠B.
(b) Select all that describe HI.
Since m∠FIH is a right triangle, it means ∠HIG is also a right triangle.
Also, the line HI originates from the vertex.
Since. the it forms a right angle, we can say HJ is an altitude of the triangle FGH.
Hence, HJ is an altitude of ΔFGH.
(c) Select all that describe MN.
Here, we can see that line MN divides the line segment KL into two equal parts, it means that point M is the median of the line segment KM and the pperpendicular bisector of line segment KL.
A perpendicular bisector is a line segment that divides another line segement into two equal parts.
KM = LM
Hence, MN is the perpendicular bisector of KL.
ANSWER:
• (a) Angle bisector of ∠B.
,• (b) Altitude of ΔFGH.
,• (c) Perpendicular bisector of KL.
14. Construction workers are laying out the rectangular foundation for a new building.They want to check that the corner is 90°. They measure the diagonal as shown to be 9.5 m. Is the angle 90° Explain your reasoning.
Explanation: We can see on the image that the two sides and the diagonal represent a triangle. We also know that this triangle to have a 90 degrees angle is will be called a right triangle. Finally, all right triangles obey the Pythagorean equation
[tex]h^2=a^2+b^2[/tex]NOTE:
h = hypotenuse
a and b = other sides
Step 1: Once we know the length of the two sides we can use the Pythagorean equation to find the length of the hypotenuse for the triangle to be a right triangle and consequently have an angle that measures 90 degrees.
Step 2: Let's calculate as follows
[tex]\begin{gathered} h^2=a^2+b^2 \\ h=\sqrt[]{8^2+6^2} \\ h=10 \end{gathered}[/tex]Step 3: We can see above, that to have an angle that measures 90 degrees (right triangle) the triangle have to have a hypotenuse = 10 which is different from 9.5.
Final answer: So the angle does not measure 90°.
For the bird, determine the following: The maximum height The axis of symmetry The total horizontal distance travelled A quadratic equation written in vertex form
Explanation:
The table of values is given below as
Using a graphing tool, we will have the parabola represented below as
A car is traveling at a speed of 70 kilometers per hour. What is the car's speed in miles per hour? How many miles will the car travel in 5 hours? In your computations, assume that 1 mile is equal to 1.6 kilometers. Do not round your answers.
What is the car's speed in miles per hour?
Let's make a conversion:
[tex]\frac{70\operatorname{km}}{h}\times\frac{1mi}{1.6\operatorname{km}}=\frac{43.75mi}{h}[/tex]How many miles will the car travel in 5 hours?
1h---------------------->43.75mi
5h---------------------> x mi
[tex]\begin{gathered} \frac{1}{5}=\frac{43.75}{x} \\ x=5\times43.75 \\ x=218.75mi \end{gathered}[/tex](statistics) solve part A, B, and C in the question on the picture provide, in 1-3 complete sentences each.
(a.) First let's define the terms;
Population - it is the pool of individual in which a statistical sample is drawn.
Parameter - it is a measure of quantity that summarizes or describes a Population.
Sample - is a smaller and more managable version of a group or population.
Statistics - same with parameter but rather than the population, it summarizes or describes
the sample.
Now that we know the definitions we can now answe the letter a;
Population: Students
Parameter: the population portion of the new students that like the new healthy choices (p)
Sample: 150 students
Statistics: estimated propotion of the students that like the new healthy choices (p-hat)
(b) P-hat = 0.6267 simply means that 62.67% of the 150 sample students like the new healthy choices.
(c) The answer for that is NO, because the simulated propotion which is shown by the graph seems to be equally distributed below and above 0.7. To support the claim of the manager most of the dots should be below 0.7 to show support to his claim that 70% of the new students like the new healthy choices.
if q(x)= int 0 ^ x^ 3 sqrt 4+z^ 6 dz then
Solution:
Given that:
What's the divisor, dividend, Quotient, and reminder in a long divison problem
In a long division problem, say 8/5:
[tex]\frac{8}{5}\text{ is the quotient}[/tex]• 8 is the divisor
,• 5 is the dividend
[tex]\frac{8}{5}=1\frac{3}{5}[/tex]• 3 is the remainder.
Give the following numberin Base 2.7710 = [ ? ] 2Enter the number that belongs in the green box.
To convert a number on base 10 to binary(base 2), we use the following steps
1 - Divide the number by 2.
2 - Get the integer quotient for the next iteration.
3 - Get the remainder for the binary digit.
4 - Repeat the steps until the quotient is equal to 0.
Using this process in our number, we have
Then, we have our result
[tex]77_{10}=1001101_2[/tex]Question 17
2(h - 6) + 20 = -4
Graph A) -f(x) B) f(x+2) -4Then find the domain and range of each
a. Graph -f(x):
By the transformations rules for functions, the graph of -f(x) is equal to a reflection over the x-axis, and a change of the y-coordinates:
[tex](x,y)\rightarrow(x,-y)[/tex]Then, given the function:
[tex]f(x)=\sqrt[]{x}[/tex]The graph of -f(x) is:is
The domain of the function is the set of all possible x-values, then it is:
[tex]\lbrack0,+\infty)[/tex]The range is the set of all possible values of the function, then it is:
[tex]\lbrack0,-\infty)[/tex]b. Graph f(x+2)-4:
The transformation f(x+2) is an horizontal translation left 2 units.
And the transformation f(x+2)-4 is a vertical translation down 4 units.
Then, the coordinates of this graph in comparison to the given graph are:
[tex](x,y)\rightarrow(x-2,y-4)[/tex]Then for the point (1,1) the new coordinates are (1-2,1-4)=(-1,-3).
For (4,2): the new coordinates (4-2,2-4)=(2,-2)
For (9,3): the new coordinates (9-2,3-4)=(7,-1)
The graph is:
The domain of this function is:
[tex]\lbrack-2,+\infty)[/tex]And the range is:
[tex]\lbrack-4,+\infty)[/tex]The coordinate pairs for triangle ABC are A(1,2), B(4,5), C(2,2). It undergoes a translation of 2 units right and 1 unit 1 up. Write down the coordinates of A'
We will have the transformation rule (x, y) -> (x+2, y+1)
Then, for A' we will have:
A'(3, 3)
B'(6, 6)
C'(4, 3)
Express $20.35 as an equation of working h hours, when I equals income
Let
I ------> income in dollars
h -----> number of hours
$20.35 is the hourly pay
so
the linear equation that represent this situation is
I=20.35*hFind the y-intercept and slope of the line below. Then write the equation is slope intercept form (y=mx+b).
The y-intercept is the value of y when x = 0
To identify y-intercept on a graph, we will check for the the value of y when the line crosses the y axis
From the graph, the line crosses the y axis at y = 6
Hence, the y-intercept is 6
To get the slope, we will pick any two points on the line.
Using points (0, 6) and (4, 0)
Applying the slope formula:
[tex]m\text{ = }\frac{y_2-y_1}{x_2-x_1}[/tex][tex]\begin{gathered} x_1=0,y_1=6,x_2=4,y_2\text{ = }0 \\ m\text{ = }\frac{0\text{ - 6}}{4\text{ - 0}} \\ m\text{ = }\frac{-6}{4} \\ m\text{ = slope = -3/2} \end{gathered}[/tex]NOTE: the slope is negative because it is going from up to down (moving downwards)
The equation of slope in intercept form: y = mx + b
m = slope = -3/2
b = y-intercept = 6
The equation in y-intercept becomes:
[tex]y\text{ = }\frac{-3}{2}x\text{ + 6}[/tex]Find the perimeter of the square.
Width = 4x
Length = 36 – 5x
Answer:
The perimeter of the square is 64 units===========================
GivenA square with dimensions:
Width = 4x,Length = 36 - 5x.To findThe perimeterSolutionSquare has all sides equal:
width = length4x = 36 - 5x4x + 5x = 369x = 36x = 4Each side is:
4*4 = 16 unitsPerimeter:
P = 4*16 = 64 unitsThe perimeter of the square is found as 64 units.
What is defined as the perimeter of the square?The perimeter of such a square is indeed the total length of all of its sides. As a result, we can calculate the perimeter of the a square besides adding its four sides.A square's sides are all equal. As a result, the perimeter of such a square is determined by multiplying the side of a square by four.For the given question,
The dimension of the square are given as;
Width = 4xLength = 36 – 5xFor square, as all sides are equal.
Then,
Width = Length
Put the values.
4x = 36 – 5x
9x = 36
x = 4
Put in dimensions.
Width = 4×4 = 16 unitsLength = 36 – 5×4 = 16 units.The perimeter of square is;
Perimeter = 4 × side
Perimeter = 4 × 16
Perimeter = 64 units.
Thus, the perimeter of the square is found as 64 units.
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The oldest child in a family of four children is three times as old as the youngest. The two middle children are 19 and 23 years old. If the average age of the children is 28.5, how old is the youngest child?
Answer:
18 years old
Solution:
Let x represent the age of the youngest child.
So the age of the oldest = 3x
If the ages of the two middle children are 19 and 23, and the average age of the four children is 28.5, let's go ahead and find x;
[tex]\begin{gathered} \frac{(x+19+23+3x)}{4}=28.5 \\ 4x+42=114 \end{gathered}[/tex]Let's go ahead and subtract 42 from both sides;
[tex]4x=72[/tex]Dividing both sides by 4, we'll have;
[tex]x=\frac{72}{4}=18[/tex]Therefore, the youngest is 18 years old.
how much cardboard is needed to make the single slice pizza box shown
We must find the amount of cardboard needed to make a slice of pizza box which basically means finding the surface area of the piece of box shown. This is composed of five faces divided in three groups:
- Two equal triangular faces with a base of 6.7 in and a height of 11 in.
- Two equal rectangular faces with a base of 11.5 in and a height of 1 in.
- A single rectangular face with a base of 6.7 in and a height of 1 in.
The area of the piece of box is given by the sum of the areas of the 5 faces so let's find the area of the faces of each group.
The area of a triangle is given by half the product of the length of its base and its height. Then the area of each triangular face is:
[tex]A_t=\frac{6.7\times11}{2}=36.85[/tex]So each triangular face has an area of 36.85 in².
The area of a rectangle is given by the product of its base and height. Then for the pair of equal rectangular faces we have:
[tex]A_{r1}=11.5\times1=11.5[/tex]So each of these two faces has an area of 11.5 in².
The area of the remaining rectangular face is then given by:
[tex]A_{r2}=6.7\times1=6.7[/tex]So the area of the last face is 6.7 in².
Then the total surface area is given by the sum of the areas of the 5 faces. Then we get:
[tex]A=2A_t+2A_{r1}+A_{r2}=2\times36.85+2\times11.5+6.7=103.4[/tex]AnswerThen the answer is 103.4
pleaseee help meeee For questions 9 - 10, answer the question about inverses. 9. The function m(d) below relates the miles Bob can drive his rental car and the numbers of dollars it will cost. 10. The function a(h) below relates the area of a triangle with a given base 7 and the height of the triangle. It takes as input the number of dollars spent and returns as output the number of miles. It takes as input the height of the triangle and returns as output the of the triangle. m(d) = 40(d- 35) ain= Write the equation that represents the inverse function, d(m), which takes the number of miles driven, m, as input and returns the number of dollars owed, d. Write the equation that represents inverse function, h(a), which takes triangle's area as input and returns height of the triangle.
First problem:
Find the inverse of the function
m = 40 (d - 35)
Recall that for the inverse function we need to solve for d in terms of m (reverse the dependence), so we proceed to isolate d on the right hand side of the equation:
divide both sides by 40
m/40 = d - 35
now add 35 to both sides:
m/40 + 35 = d
The inverse function (dollars in terms of miles) is given then by:
d(m) = 1/40 m + 35
Second problem:
a = 7 * h / 2
in order to find the inverse function (as h in terms of a) we solve for h on the right hand side of the equation as shown below:
multiply both sides by 2:
2 * a = 7 * h
now divide both sides by 7 in order to isolate h on the right
2 a / 7 = h
So our inverse function of height in terms of area is given by:
h(a) = (2 a) / 7
how do you find the sale price of the item if original price $71 and mark down to 34% the sale price is
Answer:
The sale price is $46.86
Explanation:
Given an original price of $71, and a markdown of 34%
The sale price is:
$71 - (34% of $71)
= $71 - (0.34 * $71)
= $71 - $24.14
= $46.86
Answer:
The sale price is $46.86
Explanation:
Given an original price of $71, and a markdown of 34%
The sale price is:
$71 - (34% of $71)
= $71 - (0.34 * $71)
= $71 - $24.14
= $46.86
Which of the following points is in the solution set of y < x2 - 2x - 8? O 1-2. -1) O 10.-2) 0 (4.0)
Given the functon
[tex]yExplanation
To find the points that lie in the solution set we will lot the graph of the function and indicate the ordered pirs.
From the above, we can see that the right option is
Answer: Option 1
On the planet Alaber, there are 15 dubbles to every 13 rews. If farmer Mimstoon has 100 rews on his frent farm, how many dubbles are on the farm?
You have that on planet Alaber, there are 15 dubbles to every 13 rews. This proportion can be wrtten as 15:13, or 15/13.
In order to calculate how many dubbles are on the farm, while there are 100 rews. You use the previous ratio and proceed as follow:
15/13 = x/100 where x is the unknown number of dubbles
This is because the ratio between dubbles and rews must be the same.
You solve the previous equation as follow:
15/13=x/100 multiply both sides by 100 to cancel the denomitaro 100 right side
15/13(100) = x/100(100)
1500/13 = x
In order to write the previous result as a mixed number you divide numerator and denominator:
1500 | 13
143 115
70
65
5
Then, x = 1500/13 is also equal to:
x = 115 13/5
This means there are approximately 115 dubbles for 100 rews
Solve for the dimensions of the rectangle. Area= length•widthThe length of a rectangle is 2cm greater than the width. The area is 80cm2. Find the length and width.
The length of a rectangle is 2cm greater than the width. The area is 80cm2. Find the length and width.
L=W+2
W=W
[tex]\begin{gathered} A=L\cdot W \\ A=(W+2)\cdot W \\ A=W^2+2W \\ A=80\operatorname{cm} \\ Then, \\ 80=W^2+2W \\ W^2+2W-80=0 \end{gathered}[/tex][tex]\Delta=4+320=324[/tex][tex]\begin{gathered} W=\frac{-2\pm\sqrt[]{324}}{2}=\frac{-2\pm18}{2} \\ W_1=\frac{-20}{2}=-10 \\ W_2=\frac{16}{2}=8 \end{gathered}[/tex]The width should be positive, therefore W=8
L=W+2
L=8+2=10
The length is L=10
Which form most quickly reveals the vertex? choose one answer: a. m(x)=2(x+4)^2-8 b. m(x)=2(x+6)(x+2)c. m(x)=2x^2+16x+24what is the vertex? vertex=(___,___)
The vertx from of the quadratic function is
[tex]f(x)=a(x-h)^2+k[/tex]Where
(h, k) are the coordinates of the vertex
a is the coefficient of x^2
By comparing this form with the answers
a.
[tex]m(x)=2(x+4)^2-8[/tex]a = 2
h = -4
k = -8
The vertex point is (-4, -8)
The quickly reveals the vertex is answer a
Given l//m//n find the value of x (5x)° (6x-13)°
The line l and the transversal line are intersecting each other.
So, from the theorem of Vertically Opposite angle
A pair of vertically opposite angles are always equal to each other.
thus, 5x = 6x - 13
Simplify the expression :
5x = 6x -13
6x-5x =13
x = 13
Answer : x = 13
Covert1 1/4 percent to a decimal 5 bill has received a wage increased. His new hourly wage is $14.30 compared to previous wage of $12.95 find the percentage increase in bill hourly wage. Round it off to 2 decimal places
The percentage increase in bill hourly wage is 10.42%
Given,
Bill has received a wage increased.
His new hourly wage is $14.30
and, compared to old wage of $12.95
To find the percentage increase in bill hourly wage.
Now According to the question:
New bill is = $14.30
Old bill is = $ 12.95
= ($14.30 - $12.95) / $12.95
= $1.35 / $12.95
= 10.42%
Hence, The percentage increase in bill hourly wage is 10.42%
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How can you use transformations to verify that the triangles are similar?
We need to know about congruency to solve the problem. Two pairs of congruent angles prove that the triangles are similar.
We can define similarity of two geometrical objects on a plane as possibility to transform one into another using dilation optionally combined with congruent transformations of parallel shift, rotation and symmetry. We need to use transformation to verify whether the triangles in the diagram are similar. The two triangles have a common angle D and angles ABD and ECD are equal. Thus we can say that we have two pairs of congruent angles in the two triangles, so the two triangles are similar.
Therefore the triangles are similar since they have two pair of congruent angles.
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