Answers
From the line theorems explained below and when we apply it to the given image, we have; X║Y, A║B, Z ⊥ A
What is the transverse line theorem?The perpendicular transversal theorem states that in a plane, if a line is perpendicular to one of two parallel lines, then it is also perpendicular to the other line.
Now, since we are told that Line X is perpendicular to Line A, then it means that Line X is also perpendicular to line B.
The inverse of perpendicular transversal theorem states that if there are two perpendicular lines to a straight line, they're parallel to each other. Thus, it means that Line X is parallel to Line Y and similarly, line A is parallel to Line B.
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Related Questions
Resuelve los siguientes ejercicios:
Hallar la ecuación de la circunferencia:
a) Centro (2,-1) y radio 4
b) Centro C (1,3) y que pasa por el punto P (4, 6)
3. Hallar la ecuación de la parábola, las coordenadas de su foco y la longitud de su lado recto, si el vértice es el origen y pasa por el punto P:
a) Eje focal coincidente con el eje coordenado en x, P(2,4)
b) Eje focal coincidente con el eje coordenado en, P (6,3)
4. Hallar el foco, la ecuación de la directriz y la longitud del lado recto de las siguientes parábolas.
a) 4 x2 = 32y
b) 2y2 = -3x
5. Dadas las ecuaciones de las elipses, hallar las longitudes del semieje mayor y semieje menor, las coordenadas de los focos, los vértices y la longitud del lado recto.
a) 81 x2 + 144 y2 = 11664
b) 36x2+25y2 = 3600
Answers
Por restricciones de longitud no es posible resumir las respuestas asociadas a esta pregunta, invitamos cordialmente a leer la explicación para mayores detalles sobre el análisis de secciones cónicas.
¿Cómo analizar ecuaciones de secciones cónicas?
Según la geometría analítica, existen cinco tipos de secciones cónicas: (i) Circunferencia, (ii) Parábola, (iii) Elipse, (iv) Hipérbola, (v) Recta. 2) a) La ecuación estándar de la circunferencia se caracteriza con el centro (h, k) y la longitud del radio (r):
(x - h)² + (y - k)² = r²
(x - 2)² + (y + 1)² = 4²
b) La longitud del radio de la circunferencia se obtiene por el teorema de Pitágoras sobre la longitud del segmento CP:
r = √[(4 - 1)² + (6 - 3)²]
r = √(3² + 3²)
r = 3√2
(x - 1)² + (y - 3)² = 18
3) a) El eje focal forma parte del eje de simetría de la parábola. La ecuación estándar de la parábola es:
4 · p · x = y²
Donde p es la distancia entre el foco y el vértice.
Si tenemos que (x, y) = (2, 4), entonces la ecuación de la parábola es:
4 · p · 2 = 4²
p = 2
8 · x = y²
Las coordenadas del foco de la parábola son de la forma (x, y) = (h + p, k):
F(x, y) = (2, 0)
Ahora se determinan los extremos del lado recto: (x = 2)
8 · 2 = y²
y = ± 4
Los extremos del lado recto son (2, 4) y (2, - 4), cuya longitud de segmento es 8 unidades.
b) El eje focal forma parte del eje de simetría de la parábola. La ecuación estándar de la parábola es:
4 · p · x = y²
Si tenemos que (x, y) = (6, 3), entonces la ecuación de la parábola es:
4 · p · 6 = 3²
p = 8 / 3
(32 / 3) · x = y²
Las coordenadas del foco son F(x, y) = (8 / 3, 0).
Ahora se determinan los extremos del lado recto: (x = 6)
(32 / 3) · 6 = y²
y = ± 8
Los extremos del lado recto son (6, 8) y (- 6, - 8), cuya longitud de segmento es 16 unidades.
4) a) Tenemos una ecuación estándar de la forma 4 · p · y = x². A continuación, hallamos todas las variables requeridas:
x² = 8 · y
p = 2
Directriz: y = - 2, Foco: F(x, y) = (0, 2), Longitud del lado recto: 4
b) Tenemos una ecuación estándar de la forma 4 · p · x = y². A continuación, hallamos todas las variables requeridas:
y² = - (3 / 2) · x
p = - 3 / 8
Directriz: x = 3 / 2, Foco: F(x, y) = (- 3 / 2, 0), Longitud del lado recto: 3.
5) En esta parte debemos manipular algebraicamente las ecuaciones hasta su forma estándar para determinar los datos requeridos de cada caso. La ecuación estándar de la elipse tiene el siguiente problema:
(x - h)² / a² + (y - k)² / b² = 1
Donde:
(h, k) - Centro de la elipse.a, b - Longitudes de los semiejes.a) 81 · x² + 144 · y² = 11664
x² / 144 + y² / 81 = 1
x² / 12² + y² / 9² = 1
Longitud del semieje mayor: 12
Longitud del semieje menor: 9
c = √(12² - 9²)
c ≈ 7.937
Coordenadas de los focos: F₁ (x, y) = (- 7.937, 0), F₂ (x, y) = (7.937, 0)
Vértices: V₁ (x, y) = (- 12, 0), V₂ (x, y) = (12, 0)
Longitud del lado recto
144 · y² = 11664 - 81 · x²
y² = (11664 - 81 · x²) / 144
y = ± (1 / 12) · √(11664 - 81 · x²)
y = ± (1 / 12) · √(11664 - 81 · 7.937²)
y = ± 6.750
La longitud del lado recto es 13.5.
b) 36 · x² + 25 · y² = 3600
x² / 10² + y² / 12² = 1
Longitud del semieje mayor: 12
Longitud del semieje menor: 10
c = √(12² - 10²)
c ≈ 6.633
Coordenadas de los focos: F₁ (x, y) = (0, - 6.637), F₂ (x, y) = (0, 6.637)
Vértices: V₁ (x, y) = (0, - 12), V₂ (x, y) = (0, 12)
Longitud del lado recto
36 · x² = 3600 - 25 · y²
x² = 100 - (25 / 36) · y²
x = √[100 - (25 / 36) · y²]
x = √[100 - (25 / 36) · 6.637²]
x = ± 8.331
La longitud del lado recto es 16.662.
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In Fig. 6.39, sides QP and RQ of ΔPQR are produced to points S and T respectively. If ∠SPR = 135° and ∠PQT = 110°, find ∠PRQ.
[tex] \: [/tex]
[tex] \leadsto \sf{ \pink{Pls \: give \: me \: correct \: answer!!}}[/tex]
Thank u :)
Answers
[tex] \huge{↬ \boxed{ \sf{ \pink{A\green{n \blue{s \color{yellow}w \red{e \orange{r}}}}}}}}[/tex]
Given: ∠SPR = 135° and ∠PQT = 110°To find: ∠PRQ[tex] \leadsto[/tex]According to Angle sum property of a triangle , sum of the interior angles of a triangle is 180°.
➱ ∠SPR + ∠QPR = 180° [Linear pair]
➱ 135° + ∠QPR = 180°
➱ ∠QPR = 180° - 135°
➱ ∠QPR = 45°.....(i)
➱ ∠PQT + ∠PQR = 180° [Linear pair]
➱ 110° + ∠PQR = 180°
➱ ∠PQR = 180° - 110°
➱ ∠PQR = 70°.....(ii)
Now,➛ ∠PQR + ∠QPR + ∠PRQ = 180° [Angle sum property of a triangle]
➱ 70°+ 45° + ∠PRQ = 180° [from (i) and (ii)]
➱ ∠PRQ = 180° - 115°
➥ ∠PRQ = 65°
hope it's help u!! :D
Solve the following
8903=e
Answers
[tex]\huge\text{Hey there!}[/tex]
[tex]\huge\textbf{Equation:}[/tex]
[tex]\mathsf{8,903 = e^{5x}}}[/tex]
[tex]\huge\textbf{Simplify:}[/tex]
[tex]\mathsf{8,903 = e^{5x}}}[/tex]
[tex]\mathsf{e^{5x} = 8,903}}[/tex]
[tex]\huge\textbf{Solve for the exponent:}[/tex]
[tex]\large\textsf{We get: }\downarrow\\\\\mathsf{2.718282^{5x}=8903}[/tex]
[tex]\huge\textbf{Take the logarithm from both sides:}[/tex]
[tex]\mathsf{log(2.718282^{5x}) = log(8,903x)}[/tex]
[tex]\large\textsf{We get:}\\\\\mathsf{5x \times log(2.718282)=log(8903)}[/tex]
[tex]\huge\textbf{Simplify it:}[/tex]
[tex]\mathsf{5x = \dfrac{log(8903)}{log(2.718282)}}[/tex]
[tex]\large\textsf{We get: }\\\\\mathsf{5x = 9.094143}[/tex]
[tex]\huge\textbf{Divide 5 to both sides:}[/tex]
[tex]\mathsf{\dfrac{5x}{5} = \dfrac{9.094143}{5}}[/tex]
[tex]\huge\textbf{Simplify it:}[/tex]
[tex]\mathsf{x= \dfrac{9.094143}{5}}[/tex]
[tex]\mathsf{x = 1.818829}[/tex]
[tex]\mathsf{x \approx 2}[/tex]
[tex]\huge\textbf{Therefore, your answer should be:}[/tex]
[tex]\huge\boxed{\mathsf{x =}\frak{\ 1.818829}}\huge\checkmark[/tex]
[tex]\large\textbf{Or if you're estimating your answer}\downarrow[/tex]
[tex]\huge\boxed{\mathsf{x \approx }\frak{\ 2}}\huge\checkmark[/tex]
[tex]\huge\text{Good luck on your assignment \& enjoy your day!}[/tex]
~[tex]\frak{Amphitrite1040:)}[/tex]
n this equation, one-fourth is added to the variable y.
y+14=34
What is the value of y?
The value of y is = -------------
Answers
Answer:
I think its 19.75
A national chain of department stores ranks its 1,000,000 salespeople by the monetary value of their sales. Karen's sales are at the 26th percentile. Brian's sales
are at the 33rd percentile.
(If necessary, consult a list of formulas.)
(a) Which of the following must be true about Karen's sales?
The value of Karen's sales were about 26% of the chain's total.
O The value of Karen's sales was in the top half of all of the salespeople.
O Karen sold $2600 in merchandise.
O Karen had sales lower in value than about 74% of the salespeople.
(b) Which of the following must be
true about Karen's and Brian's sales?
The value of Brian's sales were $700 more than Karen's.
O The value of Karen's sales were $700 more than Brian's.
O Both Karen and Brian had sales higher in value than the median.
O Both Karen and Brian had sales lower in value than the median.
Answers
(a) The truth about Karen's sales is that Karen had sales lower in value than about 74% of the salespeople. So, the correct option is D.
(b) The truth about Karen's and Brian's sales is that both Karen and Brian had lower sales in value than the median. So, the correct option is D.
Given that department stores rank their 1,000,000 salespeople by the monetary value of their sales. Karen's sales are at the 26th percentile and brian's sales are at the 33rd percentile.
A percentile is a measure used in statistics that indicates the value below which a certain percentage of observations in a group of observations falls.
(a) Karen had sales lower in value than about 74% of the salespeople because the 26% percentile represents his relative position among all the employees in terms of sales.
(b) Both Karen and Brian had sales lower in value than the median because the Median represents the 50th percentile. Since Karen's sales are at the 26th percentile and Brian's sales are at the 33rd percentile it implies their sales are less than the median value.
Hence, the truth about Karen's sales is Karen had sales lower in value than about 74% of the salespeople, and true about Karen's and Brian's sales is both Karen and Brian had lower sales in value than the median.
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Triangles L M N and P O N connect at point N. Angles L M N and N O P are congruent.
Why is the information in the diagram enough to determine that △LMN ~ △PON using a rotation about point N and a dilation?
because both triangles appear to be equilateral
because∠MNL and ∠ONP are congruent angles
because one pair of congruent corresponding angles is sufficient to determine similar triangles
because both triangles appear to be isosceles, ∠MLN ≅ ∠LMN, and ∠NOP ≅ ∠OPN
Answers
The information is enough because: C. because one pair of congruent corresponding angles is sufficient to determine similar triangles.
What are Similar Triangles?Similar triangles can be formed during dilation. All corresponding angles of similar triangles are usually congruent to each other.
From the image given, the using rotation about point N and then dilation by a scale factor will make both triangles similar. Thus, with the given information, we can determine both triangles are similar because: C. because one pair of congruent corresponding angles is sufficient to determine similar triangles.
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Answer:
C
Step-by-step explanation:
Took the test
A Web music store offers two versions of a popular song. The size of the standard version is 2.8 megabytes (MB). The size of the high-quality version is 4.4 MB. Yesterday, the high-quality version was downloaded three times as often as the standard version. The total size downloaded for the two versions was 3840 MB. How many downloads of the standard version were there?
Answers
Step-by-step explanation:
s = number of standard downloads
2s = number of high-quality downloads
2.8s + 4.6(2s) = 2760
2.8s + 9.2s = 2760
12s = 2760
s = 230
The standard version was downloaded 230 times and the high-quality version was downloaded 460 times.
mangleR = 120° and mangleS = 100°. Find mangleT. The diagram is not drawn to scale.
Answers
Explain how to draw a line segments that measures 2 7/16 inches.
Answers
The line segment which represents 2 7/16 inches should first be drawn with the use of a ruler.
What is a Ruler?This is referred to an instrument which is used to measure distances between points or draw straight lines and also usually graduated in inches.
On the inches graduation, the zero mark should be located and the 1/16 inch which is found before the half-inch mark between 2 and 3 inches. When this is done, a line should be drawn to meet both points which will therefore measure 2 7/16 inches long.
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You get a raise of 3.5% each year.
a. What is the approximate doubling time of your salary?
b. If you are currently 25 years old and are making $40,000 per year, use the approximate doubling time to find your annual salary when you retire at the age of 65.
Answers
Using an exponential function, it is found that:
a) The doubling time of the salary is of approximately 20 years.
b) The salary will be of $160,000.
What is an exponential function?An increasing exponential function is modeled by:
[tex]A(t) = A(0)(1 + r)^t[/tex]
In which:
A(0) is the initial value.r is the growth rate, as a decimal.The growth rate for this problem is:
r = 0.035.
The doubling time is t for which A(t) = 2A(0), hence:
[tex]A(t) = A(0)(1 + r)^t[/tex]
[tex]2A(0) = A(0)(1.035)^t[/tex]
[tex](1.035)^t = 2[/tex]
[tex]\log{(1.035)^t} = \log{2}[/tex]
[tex]t\log{1.035} = \log{2}[/tex]
[tex]t = \frac{\log{2}}{\log{1.035}}[/tex]
t = 20 years.
You retire in 40 years, which is 2 doubling periods, hence the salary will be of:
40000 x 2 x 2 = $160,000.
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Planes A and B intersect.
n
772
W
k
Mark this and return
V
19
Z
X
2
Which describes the intersection of line m and line nº
O point W
point X
point Y
O point Z
Answers
The point that describes the intersection of line m and line n is; Point W
How to interpret Intersection of Planes?
Intersection of the two lines is defined as the point where the two lines cross or meet each other.
It is given that the lines m and n intersect each other, which means that they must be intersecting each other at some point.
From the figure, it can be seen that in plane A, the lines m and n intersect each other at point W, thus point W is the point of intersection of the two line m and n.
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Two sides of a four-sided figure have negative slopes. Which are the endpoints of the sides of this figure?
(–4, –4), (–4, –1), (–1, –4), (–1, –1)
(–2, –4), (–1, –1), (1, –1), (2, –4)
(1, 1), (2, 4), (5, 4), (4, 1)
(1, 4), (2, 1), (5, 1), (4, 4)
Answers
The endpoints of the sides of the quadrilateral are; (1, 4), (2, 1), (5, 1), (4, 4)
How to calculate the slope of sides of a quadrilateral?To get the endpoints of the quadrilateral that has 2 sides with negative slope, we will use the formula for slope;
m = (y2 - y1)/(x2 - x1)
Option A; Coordinates are (–4, –4), (–4, –1), (–1, –4), (–1, –1). Slopes are;
m1 = (-1 + 4)/(-4 + 4) = undefined
m2 = (-4 + 1)/(-1 + 4) = -1
m3 = (-1 + 4)/(-1 + 1) = undefined
m4 = (-4 + 1)/(-4 + 1) = 1
No two slopes are negative and this is not correct.
Option B; Coordinates are (–2, –4), (–1, –1), (1, –1), (2, –4). Slopes are;
m1 = (-1 + 4)/(-1 + 2) = 3
m2 = (-1 + 1)/(1 + 1) = 0
m3 = (-4 + 1)/(2 - 1) = -3
m4 = (-4 + 4)/(-2 - 2) = 0
No two slopes are negative and this is not correct.
Option C; Coordinates are (1, 1), (2, 4), (5, 4), (4, 1). Slopes are;
m1 = (4 - 1)/(2 - 1) = 3
m2 = (4 - 4)/(5 - 2) = 0
m3 = (1 - 4)/(4 - 5) = 3
m4 = (1 - 1)/(1 - 4) = 0
No two slopes are negative and this is not correct.
Option D; Coordinates are (1, 4), (2, 1), (5, 1), (4, 4). Slopes are;
m1 = (1 - 4)/(2 - 1) = -3
m2 = (1 - 1)/(5 - 2) = -0
m3 = (4 - 1)/(4 - 5) = -3
m4 = (4 - 4)/(1 - 4) = -0
Two slopes are negative and this is correct.
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What is the simplified form of the expression the quantity x squared minus 4x minus 21 end quantity divided by 4 times the quantity x plus 3 end quantity?
Answers
The simplified form of the given expression [tex]\frac{x^2-4x-21}{4(x+3)}[/tex] is [tex]\frac{x-7}{4}[/tex].
How to simplify a fractional expression?The steps to simplify a fractional expression are:
Factorize the expressions both in the numerator and the denominator using different methodsRemove the common terms in the numerator and denominatorThus, the simplified expression will be obtained.Calculation:The given expression is [tex]\frac{x^2-4x-21}{4(x+3)}[/tex]
Factorizing the numerator:
x² - 4x - 21 = x² - 7x + 3x - 21
= x(x - 7) + 3(x - 7)
= (x - 7)(x + 3)
Then,
[tex]\frac{x^2-4x-21}{4(x+3)}[/tex] = [tex]\frac{(x-7)(x+3)}{4(x+3)}[/tex]
common factor (x + 3) is canceled out from both the numerator and the denominator
So,
[tex]\frac{x^2-4x-21}{4(x+3)}[/tex] = [tex]\frac{x-7}{4}[/tex]
Therefore, the simplified expression is [tex]\frac{x-7}{4}[/tex].
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Answer:x-3/4
Step-by-step explanation:
(x-3)(x+7)/4(x+7)
=x-3/4
The prep club and honor society are joining forces
Answers
Given that the inequalities and that
x ⇒ pep club donations
y ⇒ represent donations raised by the honor society
The inequality is represented by x + y≥ 500
The solution for this is the shaded area above the solid line.
Given x ≥ 100, this is also indicated by the shaded region to the right
What does point (300, 100) represent? is it a viable Solution? why?The point (300,100) indicates that the contributions made for the pep club were $300 and the donations raised by the honor society were $100.
The point (300,100) is NOT a feasible solution since it does not conform to the gray area of the system of inequalities solution.
To proof the above,
Let x = 300; and y = 100
Plugin in both values in the inequalities we have:
300 + 100 [tex]\ngeq[/tex] 500
400 [tex]\ngeq[/tex] 500 ; Hence it is NOT true; also
300 [tex]\ngeq[/tex] 500.
What does (500, 200) represent? Is it a viable Solution? why?The point (500,200) indicates that the contributions raised for the pep club were $500 and the donations raised by the honor society were $200.
The point (500,200) IS A possible solution since it falls inside the darkened area of the system of inequalities solution.
To proof the above,
Let x = 500; and y = 200
Plugin in both values in the inequalities we have:
500 + 200 ≥ 500
700 ≥ 500; hence, this is true and it is safe to indicate that the ordered pair satisfies the inequality A.
Conclusion:
The point (500, 200) is a viable solution.
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Easy points because I got points to blow,
Answers
Answer:
Did you know that the first person convicted of speeding was going eight mph.
Step-by-step explanation:
The height of a door is 1.5 feet longer than its width, and its front area is 1516.5 square feet. Find the width and height of the door.
Answers
Answer:
For an exact answer the width would be the square root of 1011 and the length would be 1.5x the square root of 1011.
Step-by-step explanation:
A = lw
1516.5 = w(1.5)w
1516.5 = 1.5w^2 Divide both sides by 1.5
1011 = w^2 Take the square root of each side
Square root of 1011 = w.
If R={(1,2),(2,3),(3,9),(4,5)} find R-1
Answers
Answer:
[tex]R^{-1}[/tex]={(2,1), (3,2), (9,3), (5,4)}
Step-by-step explanation:
Solution Given:
To find the inverse we need to exchange Domain to Range and vice-versa.
so
[tex]R^{-1}[/tex]={(2,1), (3,2), (9,3), (5,4)}
The temperature was 56 degrees this morning, but dropped one degree per hour through the rest of the day. The equation is y=-x+56. You may usey=56-x . They are the same equation. Use x-values: 0, 5, and 10.
Answers
At x=0
y=56-0y=56°CAt x=5
y=56-5y=51°CAt x=10
y=56-10y=46°Cshow a graph for the following: f(x)=3^2
Answers
The blue curve of the image attached below represents the graph of the function f(x) = 3 · x².
How to determine the graph of a given function
In this question we have a power function formed by the product of a primitive function g(x) = x² and vertical dilation h(x) = 3, then:
f(x) = h(x) · g(x)
f(x) = 3 · x²
Now we proceed to graph both the primitive function and the transformed function with a graphing tool. Please notice that the primitive function is the red curve.
Remark
The statement presents typing mistakes. Correct form is shown below:
Show a graph for the following: f(x) = 3 · x².
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Multiply –8m(–6m – 7).
–14m2 – 15m
48m2 + 56m
–14m – 15
48m + 56
Answers
Answer: 48m2 + 56m
Step-by-step explanation:
Answer:
Answer B
Step-by-step explanation:
Tell wether 3 is a solution of each inequality
x-7> -5; x = 3
-3x< - 12.5 ; x=3
Answers
Second inequality, 3 is not a solution.
Hope it helps : )
A teacher designs a test so that 93% of students who study will pass and 9% of students who don't study will pass. 88%
of students study for a test. What is the probability that a randomly selected student passes?
A. 1.057
B. 0.833
C. 0.084
D. 0
E. 0.829
Answers
Answer:
Step-by-step explanation:
First find the percentage of the ones that pass that study
.93 *(88)=88.84%
next find the ones that pass that did not study
.09*(100-88)=1.08
sum these to get the total passing percentage
88.84+1.08=82.92%
The probability is 82.9 per 100 or .8292 for student
E is the answer
Nan is 20 year old. In 8 years,she will be twice as old as Clarisse.how old is clarisse now?
Answers
Answer:
6
Step-by-step explanation:
The equation for this problem would be c = (20 + 8) / 2 - 8. Add 20 and 8 to get 28, divide 28 by 2 to get 14, then subtract 8 to get 6.
The graphs of f(x) = 5* and its translation, g(x), are
shown on the graph.
30 1x
30 25 20 15 10 f
goxx)
40
f(x)
-6-5-4-3-2-15-
(1,5)
(0
(2,25)
(2, 15)
2 3 4 5
(1.-5)
-10-0.9)
-15-
-20
-25
30
X
What is the equation of g(x)?
Og(x) = 5x-9
Og(x) = 5x-10
Og(x) = 5-9
Og(x) = 5-10
Answers
g(x) is just a translation of 10 units downwards, this means that the equation for g(x) is:
[tex]g(x) = f(x) - 10 = 5^x - 10[/tex]
Which is the equation of the translation?Here we know that:
[tex]f(x) = 5^x[/tex]
And we want to find the equation for g(x), which is a translation of f(x).
If you look at the graph of f(x), you can see that it passes through:
(0, 1), (1, 5), (2, 25), etc.
And the graph of g(x) passes through:
(0, -9), (1, -5), (2, 15)
So in each point, the y-component is 10 units smaller than the correspondent for f(x).
Then g(x) is just a translation of 10 units downwards, this means that the equation for g(x) is:
[tex]g(x) = f(x) - 10 = 5^x - 10[/tex]
So the correct option is the last one.
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I need help determining how many roots
Answers
By graphing the polynomial, we conclude that there is only one real root, so the correct option is A.
How many roots has the given polynomial?
We have the polynomial:
[tex]y = -3x^2 + 12x - 12[/tex]
To see how many real roots this polynomial has, we can graph it and see how many times the graph intercepts the x-axis.
The graph can be seen below:
There we can see that there is only one intercept, so there is only one real root.
So the correct option is A.
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Which of the following functions has an initial value of -1/2 and a rate of change of 0?
A.
2
x
B. y=−12x
y
=
−
1
2
x
C. y=−12x
y
=
−
1
2
x
D. y=−12
Answers
The function that has the given initial value and rate of change is: D. y = -1/2.
What is the Initial Value and Rate of Change of a Function?A function is given as, y = mx + b, where, m is the rate of change, and b, which is a constant is the initial value.
Given the following:
b = -1/2
m = 0
Plug in the values into y = mx + b
y = 0(x) + (-1/2)
y = -1/2
Therefore, the function is: D. y = -1/2.
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Consider the following graph
a) What is the equation of the axis?
b) What is the period?
c) What is the amplitude?
d) Is this a sine graph or a cosine graph? Explain
Answers
a) The equation of axis is the middle y-coordinate, which is y = -1.5.
b) The period is the amount of time it takes for the graph to repeat, which is 6.
c) The amplitude is the vertical distance from an extreme point to the midpoint, which is 3.
d) This is a cosine graph since when x=0, y is not equal to 0.
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Which expression represents the following statement? Add 8 to 6 times the quotient of 21 and 3.
6 x (21 ÷ 3) + 8
6 + 8 − (21 ÷ 3)
21 ÷ 3 − (8 + 6)
6 x (21 + 3) + 8
Answers
Answer:
(a) 6 x (21 ÷ 3) + 8
Step-by-step explanation:
The first step in translating English to math is to understand the English.
Parsing itThe quotient of 21 and 3 is written (21 ÷ 3). Six times that quotient is ...
6×(21÷3)
When 8 is added to this, it becomes ...
6 × (21 ÷ 3) + 8
Dave leaves his office in Padelford Hall on his way to teach in Gould Hall. Below are several different scenarios. Take distance units to be “feet” and time units to be “minutes.” Assume Dave’s path to Gould Hall is along a straight line which is 2400 feet long.
I. Dave leaves Padelford Hall and walks at a constant spend until he reaches Gould Hall 10 minutes later.
II. Dave leaves Padelford Hall and walks at a constant speed. It takes him 6 minutes to reach the halfway point. Then he gets confused and stops for 1 minute. He then continues on to Gould Hall at the same constant speed he had when he originally left Padelford Hall.
III. Dave leaves Padelford Hall and walks at a constant speed. It takes him 6 minutes to reach the half-way point. Then he gets confused and stops for 1 minute to figure out where he is. Dave then continues on to Gould Hall at twice the constant speed he had when he originally left Padelford Hall.
g. Using all three scenarios, represent each scenario as an algebraic function.
Answers
See below for the algebraic expressions of the scenarios.
How to represent the scenarios?The given parameters are:
Distance = 2400 feetUnit of time = MinutesScenario 1
Represent the speed with x.
So, we have:
Speed = Distance/Time
The time is given as:
Time = 10 minutes
Since he did not stop at all;
The speed is
x = 2400/10
Multiply both sides by 10
10x = 2400
The above expression represents the scenario 1.
Scenario 2
Represent the speed with x, and time with t
So, we have:
Speed = Distance/Time
The time to reach halfway is given as:
Time = 6 minutes
He stopped for 1 minute, before continuing at the same initial speed
So, the total time it 13 minutes
The speed is represented as:
x = 2400/13
Multiply both sides by 13
13x = 2400
The above expression represents the scenario 2.
Scenario 3
Represent the speed with x, and time with t
So, we have:
Speed = Distance/Time
The time to reach halfway is given as:
Time = 6 minutes
He stopped for 1 minute, before continuing at twice the initial speed
So, the total time is
t = 6 + 1 + 1/2 * 6 minutes
t = 10 minutes
The speed is represented as:
x = 2400/10
Multiply both sides by 10
10x = 2400
The above expression represents the scenario 3.
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What is the area of an equilateral triangle having side 'a' units?
Answers
Answer:
[tex]A=\frac{\sqrt{3}}{4}a^2[/tex]
Step-by-step explanation:
Since an equilateral has all of the sides equal, we can find the height of triangle using: [tex]a^2+b^2=c^2[/tex]. I attached a diagram which should explain how I got the dimensions of the three sides. Using the information from the diagram we get the equation:
[tex]h^2+(\frac{a}{2})^2=a^2[/tex]
Subtract a^2 from both sides
[tex]h^2=a^2-(\frac{a}{2})^2[/tex]
Take the square root of both sides
[tex]h = \sqrt{a^2-(\frac{a}{2})^2}[/tex]
If you know the area of a triangle, it's: [tex]\frac{1}{2}bh[/tex]. In this case the base=a, and the height is what we defined above. Using this we get:
[tex]A = \frac{a}{2}*\sqrt{a^2-(\frac{a}{2})^2}[/tex]
We can distribute the exponent over the division to get:
[tex]A = \frac{a}{2}*\sqrt{a^2-(\frac{a^2}{4})[/tex]
Now we can rewrite a^2 as 4a^2/4
[tex]A = \frac{a}{2}*\sqrt{\frac{4a^2}{4}-(\frac{a^2}{4})[/tex]
Now add the two fractions:
[tex]A = \frac{a}{2}*\sqrt{\frac{3a^2}{4}[/tex]
We can distribute the square root the division just like how we distributed the exponent 2, since the square root can be expressed as an exponent (1/2)
[tex]A = \frac{a}{2}*\frac{\sqrt{3a^2}}{\sqrt{4}}[/tex]
There's a radical identity that states: [tex]\sqrt[n]{a} * \sqrt[n]{b} = \sqrt[n]{a*b}[/tex]. We can use this to rewrite one radical as multiple radicals to simplify it:
[tex]A = \frac{a}{2}*\frac{\sqrt{a^2}*\sqrt{3}}{2}[/tex]
Simplify:
[tex]A = \frac{a}{2}*\frac{a*\sqrt{3}}{2}[/tex]
Now multiply the two fractions
[tex]A = \frac{a^2*\sqrt{3}}{4}[/tex]
This is the formula for the area of an equilateral triangle, but it is also often written as:
[tex]A=\frac{\sqrt{3}}{4}a^2[/tex]