It depends on how many different options there are on the spinner
9 different sums are possible from the given data
To calculate the number of different sums possible, let's break down the problem step by step:
1. Determine the number of cards in each set:
- Set 1: 1, 2, 3, 4, 5 (5 cards)
- Set 2: 6, 7, 8, 9, 10 (5 cards)
2. Find the total number of combinations by multiplying the number of cards in each set:
- Total combinations = 5 (cards in Set 1) × 5 (cards in Set 2) = 25 combinations
Now, let's consider all the possible sums:
- The lowest possible sum is when we pick the lowest card from each set: 1 + 6 = 7.
- The highest possible sum is when we pick the highest card from each set: 5 + 10 = 15.
To find all the possible sums, we consider the range from the lowest sum to the highest sum, inclusive:
7, 8, 9, 10, 11, 12, 13, 14, 15
There are 9 different sums in total.
So, the correct answer is: 9 different sums are possible.
The key to solving this problem lies in understanding the concept of combinations and the range of possible sums. We find the number of combinations by multiplying the number of cards in each set. Then, by considering the lowest and highest cards from each set, we determine the range of possible sums. Finally, by listing out all the sums within that range, we find that there are 9 different sums possible. This method ensures a systematic approach to solving the problem, providing a clear and accurate answer.
Complete question:
You pick one card from each set, spin the spinner, and find the sum. How many different sums are possible
Two friends leave there houses and go towards each others, one friend runs 8 miles a hour, the other runs 7 miles an hour, there houses are 20 miles apart they both leave at 6:00 pm, what time do they meet each other?
hi there the answer is 7:30 see the picture
At each point (x,y) on the curve the slope is 2x+3 the curve passes through the point (-3,0)
Answer:
The equation of the curve is y = x² + 3x
Step-by-step explanation:
I get your drift, but wish you had stated specifically that you wanted to find the equation of this curve.
Slope of tangent line to curve is dy/dx = 2x + 3.
Integrating that with respect to x yields y = x² + 3x + C.
The point (-3, 0) must satisfy this equation:
0 = (-3)² + 3(-3) + C, so C = 0.
The equation of the curve is y = x² + 3x
The equation of the curve where the slope at any point is 2x + 3 and it passes through the point (-3, 0) is y = x² + 3x.
The student's question involves finding the equation of a curve given that at each point (x, y) on the curve, the slope is 2x + 3, and the curve passes through the point (-3, 0). This is a problem that involves differential equations, as it asks us to find a function y(x) whose derivative with respect to x is given.
To find the equation of the curve, we integrate the slope function. Since the slope at any point is 2x + 3, we have:
∂y/∂x = 2x + 3
Integrating both sides with respect to x gives:
y = x² + 3x + C
To find the constant of integration C, we use the fact that the curve passes through (-3, 0). Substituting these values into the equation gives:
0 = (-3)² + 3(-3) + C
0 = 9 - 9 + C
Thus, C = 0, and the equation of the curve is:
y = x² + 3x.
What is the volume of the cylinder below?
For this case we have that, by definition, the volume of the cylinder is given by:
[tex]V = \pi * r ^ 2 * h[/tex]
Where:
h: It's the height
A: It's the radio
We have to:
[tex]h = 15\\r = 12[/tex]
Substituting:
[tex]V = \pi * (12) ^ 2 * 15\\V = \pi * 144 * 15\\V = 2160 \pi \ units ^ 3[/tex]
Answer:
Option B
Answer: 2160
Step-by-step explanation:
apex
Which of the following would you do if you were to write the expression 3 * 3 * 3 *3 * 3 * 3 *3 * 3 * 3 *3 * 3 * 3 using an exponent?
Check al that apply.
A. You would write 12^3
B. You would use 12 as the base and 3 as the exponent.
C. You would write 3^12
D. You would use 3 as the base and 12 as the exponent.
Answer:
C and D
Step-by-step explanation:
3 is the base because its the number being multiplied repeatedly while the exponent would be the number of times its being multiplied.
Given the scale drawing of a one bedroom apartment, what is the actual area of the bedroom?
A) 27 ft2
B) 54 ft2
C) 96 ft2
D) 108 ft2
Answer: Option D.
Step-by-step explanation:
You can observe that the dimensions of the bedroom in the scale drawing are:
[tex]length=2"\\width=1.5"[/tex]
You know that [tex]\frac{1}{2}"=3feet[/tex]
Then you need to convert these dimensions to the actual dimensions. So, you get:
[tex]actual\ lenght=\frac{(2")(3ft)}{\frac{1}{2}" }=12ft\\\\actual\ width=\frac{(1.5")(3ft)}{\frac{1}{2}" }=9ft[/tex]
Therefore, you can calculate the actual area of the bedroom. This is:
[tex]Area=(actual\ lenght)(actual\ width)\\Area=(12ft)(9ft)\\Area=108ft^2[/tex]
Answer:
Option D.
Step-by-step explanation:
From the given figure it is clear that
Length of the bedroom in scale drawing = 2''
Width of the bedroom in scale drawing = 1.5''
It is given that
Scale [tex]\frac{1}{2}''=3feet[/tex]
[tex]1''=6feet[/tex]
Using this conversion we get
[tex]2''=12feet[/tex]
[tex]1.5''=9feet[/tex]
Actual length of the bedroom = 12 feet
Actual width of the bedroom = 9 feet
Area of a rectangle is
[tex]Area=length\times width[/tex]
Actual area of the bedroom is
[tex]Area=12\times 9[/tex]
[tex]Area=108[/tex]
The actual area of the bedroom is 108 ft2.
Therefore, the correct option is D.
What is the approximate volume of the oblique cone? Use π ≈ 3.14 and round to the nearest tenth. 117.8 cubic units 141.3 cubic units 235.5 cubic units 282.6 cubic units
The volume of an oblique cone, to the nearest tenth is: 235.5 cubic units.
What is the Volume of an Oblique Cone?Volume of an oblique cone = (1/3) × π × r² × h, where r is radius and h is height of the oblique cone.
Given:
Radius = 5Height = 9π ≈ 3.14Thus:
Volume of an oblique cone = (1/3) × 3.14 × 5² × 9
Volume of an oblique cone = 235.5 cubic units.
Learn more about volume of oblique cone on:
https://brainly.com/question/26524753
The approximate volume of the oblique cone is V = 235.5 cubic units
What is a Cone?A cone is a three-dimensional shape in geometry that narrows smoothly from a flat base (usually circular base) to a point(which forms an axis to the center of base) called the apex or vertex.
The volume of a cone is given by the equation
Volume of Cone = ( 1/3 ) πr²h
where r is the radius of the cone
h is the height of the cone
Surface area of the cone = πr ( l + r )
where l = √ ( b² + r² )
Given data ,
Let the radius of the cone be r = 5 units
Let the height of the cone be h = 9 units
And , Volume of Cone = ( 1/3 ) πr²h
On simplifying , we get
V = ( 1/3 ) ( 3.14 ) ( 5 )² ( h )
V = 1.0467 x 25 x 9
V = 235.5 units³
Hence , the volume of cone is V = 235.5 units³
To learn more about cone click :
https://brainly.com/question/1984638
#SPJ5
The complete question is attached below :
What is the approximate volume of the oblique cone? Use π ≈ 3.14 and round to the nearest tenth.
a) 117.8 cubic units
b) 141.3 cubic units
c) 235.5 cubic units
d) 282.6 cubic units
Solve the system of linear equations below.
2x + y = 5
x + y = 4
I don't just want the answer can someone explain how to do this?
Answer:
[tex]x=1\\y=3[/tex]
Step-by-step explanation:
You can use the Method of Elimination:
Multiply the first equation by -1, add both equations and then solve for the variable "x":
[tex]\left \{ {{(-1)(2x + y )= 5(-1)} \atop {x + y = 4}} \right.\\\\\left \{ {{(-2x - y = -5} \atop {x + y = 4}} \right. \\.........................\\-x=-1\\\\(-1)(-x)=(-1)(-1)\\\\x=1[/tex]
Now substitute [tex]x=1[/tex] into any of the original equations and solve for the variable "y":
[tex]x + y = 4\\1 + y = 4\\y=4-1\\y=3[/tex]
f(x)= 7x^5+35x^4-2x^3-4x^2+30x+8
find f(-5)
Answer:
8
Step-by-step explanation:
When you are told to find f(-5), that means that you are to sub in a -5 for every x you see. That -5 could be any number or expression. Subbing in a -5 looks like this:
[tex]7(-5)^5+35(-5)^4-2(-5)^3-4(-5)^2+30(-5)+8[/tex]
Be careful with all those negatives. Doing the math on that gives you that f(-5) = 8
Use the divergence theorem to find the outward flux of the vector field F(xyz)= 4x^2 i + 4y^2 j + 3z^2 k across the boundary of the rectangular prism: 0
[tex]\vec F(x,y,z)=4x^2\,\vec\imath+4y^2\,\vec\jmath+3z^2\,\vec k\implies\nabla\cdot\vec F=8x+8y+6z[/tex]
Let [tex]S[/tex] be the surface of the rectangular prism bounded by the planes [tex]x=0[/tex], [tex]x=a[/tex], [tex]y=0[/tex], [tex]y=b[/tex], [tex]z=0[/tex], and [tex]z=c[/tex]. By the divergence theorem, the integral of [tex]\vec F[/tex] over [tex]S[/tex] is given by the integral of [tex]\nabla\cdot\vec F[/tex] over the interior of [tex]S[/tex] (call it [tex]R[/tex]):
[tex]\displaystyle\iint_S\vec F\cdot\mathrm d\vec S=\iiint_R(\nabla\cdot\vec F)\,\mathrm dV[/tex]
[tex]=\displaystyle\int_0^c\int_0^b\int_0^a(8x+8y+6z)\,\mathrm dx\,\mathrm dy\,\mathrm dz=\boxed{abc(4a+4b+3c)}[/tex]
Answer:
[tex]\displaystyle \iiint_D {\nabla \cdot \textbf{F}} \, dV = \boxed{6875}[/tex]
General Formulas and Concepts:
Calculus
Differentiation
DerivativesDerivative NotationDerivative Property [Multiplied Constant]:
[tex]\displaystyle (cu)' = cu'[/tex]
Derivative Rule [Basic Power Rule]:
f(x) = cxⁿf’(x) = c·nxⁿ⁻¹Integration
IntegralsIntegration Rule [Reverse Power Rule]:
[tex]\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C[/tex]
Integration Rule [Fundamental Theorem of Calculus 1]:
[tex]\displaystyle \int\limits^b_a {f(x)} \, dx = F(b) - F(a)[/tex]
Integration Property [Multiplied Constant]:
[tex]\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx[/tex]
Integration Property [Addition/Subtraction]:
[tex]\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx[/tex]
Multivariable Calculus
Partial Derivatives
Vector Calculus (Line Integrals)
Del (Operator):
[tex]\displaystyle \nabla = \hat{\i} \frac{\partial}{\partial x} + \hat{\j} \frac{\partial}{\partial y} + \hat{\text{k}} \frac{\partial}{\partial z}[/tex]
Div and Curl:
[tex]\displaystyle \text{div \bf{F}} = \nabla \cdot \textbf{F}[/tex][tex]\displaystyle \text{curl \bf{F}} = \nabla \times \textbf{F}[/tex]Divergence Theorem:
[tex]\displaystyle \iint_S {\big( \nabla \times \textbf{F} \big) \cdot \textbf{n}} \, d\sigma = \iiint_D {\nabla \cdot \textbf{F}} \, dV[/tex]
Step-by-step explanation:
*Note:
Your question is incomplete, but I have defined the missing portions of the question below.
Step 1: Define
Identify given.
[tex]\displaystyle \textbf{F} (x, y, z) = 4x^2 \hat{\i} + 4y^2 \hat{\j} + 3z^2 \hat{\text{k}}[/tex]
[tex]\displaystyle \text{Region (Boundary de} \text{fined by rectangular prism):} \left\{ \begin{array}{ccc} 0 \leq x \leq 5 \\ 0 \leq y \leq 5 \\ 0 \leq z \leq 5 \end{array}[/tex]
Step 2: Find Flux Pt. 1
[Vector Field] Find div F:Step 3: Find Flux Pt. 2
[Flux] Define:We can evaluate the Flux integral (Divergence Theorem integral) using basic integration techniques listed under "Calculus":
[tex]\displaystyle \begin{aligned}\int\limits^5_0 \int\limits^5_0 \int\limits^5_0 {\big( 8x + 8y + 6z \big)} \, dx \, dy \, dz & = \int\limits^5_0 \int\limits^5_0 {4x^2 + x \big( 8y + 6z \big) \bigg| \limits^{x = 5}_{x = 0}} \, dy \, dz \\& = \int\limits^5_0 \int\limits^5_0 {\big( 40y + 30z + 100 \big)} \, dy \, dz \\& = \int\limits^5_0 {20y^2 + y \big( 30z + 100 \big) \bigg| \limits^{y = 5}_{y = 0}} \, dz \\\end{aligned}[/tex]
[tex]\displaystyle\begin{aligned}\int\limits^5_0 \int\limits^5_0 \int\limits^5_0 {\big( 8x + 8y + 6z \big)} \, dx \, dy \, dz & = \int\limits^5_0 {\big( 150z + 1000 \big)} \, dz \\& = \big( 75z^2 + 1000z \big) \bigg| \limits^{z = 5}_{z = 0} \\& = \boxed{6875} \\\end{aligned}[/tex]
∴ [tex]\displaystyle \Phi = \boxed{6875}[/tex]
---
Learn more about Divergence Theorem: https://brainly.com/question/2407209
Learn more about multivariable calculus: https://brainly.com/question/7964566
---
Topic: Multivariable Calculus
Unit: Stokes' Theorem and Divergence Theorem
Determine if (-2,6) and (4,12) are solutions to the system of equations:
y = x + 8
2y = x2 + 8
A) Both are solutions.
B) Neither is a solution.
C) It cannot be determined
D) Only (-2,6) is a solution.
The given points (-2,6) and (4,12) are the solution of the given equations. Option A is correct.
What is simplification?The process in mathematics to operate and interpret the function to make the function or expression simple or more understandable is called simplifying and the process is called simplification.
Here,
Given a system of equations,
y = x + 8 - - - - (1)
2y = x² + 8 - - - (2)
In order to determine that the points (-2,6) and (4,12) both are solutions of the given system of equations we must put the points in the equations,
So,
Put (-2, 6) in both equations,
Equation 1
put x = -2,
y = -2 + 8
y = 6
Equation 2
2y = [-2]² + 8
y = 4 + 8 / 2
y = 6
So points (-2, 6) is the solution of the given system of equation,
Similarly, by assigning the other point in the equation gives the same solutions
Thus, the given points (-2,6) and (4,12) are the solution of the given equations. Option A is correct.
Learn more about simplification here:
https://brainly.com/question/12501526
#SPJ5
mary took a trip to orlando that took 3 hours long. mary forgot to check the time as she left, but when she got to orlando, her phone read 10:00 am. what time did mary start her trip?
Where was Mary leaving from to get to Orlando? : )
because it's still in the morning so e you can easily subtract 3 hours from
10:00 am.
10 - 3 = 7:00 am
HELLPPPPPPPPPPPPPPPPPP PLEASE WILL GIVE MORE THAN 5 POINTSSSSSSSS
In this triangle, which of the following is true?
**APEX**
| Multiple Choice |
The answer to the question is D
Answer:
The measure of x is 35 and a = 8.19
Step-by-step explanation:
Write the sentence as a conditional statement. Two parallel lines lie in the same plane. If two lines are parallel, then the lines lie in the same plane. If two lines lie in the same plane, then the lines are parallel.
The conditional statement is:
If two lines are parallel then they lie in the same plane.Step-by-step explanation:We know that a conditional statement is the statement which is written in the form of:
If p then q i.e. p → q
where p is the hypothesis of the statement.
and q is the conclusion of the statement which is based on the hypothesis.
Here we are given a statement as:
Two parallel lines lie in the same plane.
i.e. the hypothesis of the conditional statement is:
Two lines are parallel.
and conclusion is: They will lie in the same plane
( Because if two lines lie in the same plane then we can't conclude that they will be parallel )
What graph shows the solution set of the inequality x^2+10x+16/x-3 >0
[tex]x < - 10.434462 \\ x > 0[/tex]
Answer:
D
Step-by-step explanation:
Which of the following is the formula for the volume of a pyramid with base area B and height h?
A. [tex]V=\frac{1}{3} Bh[/tex]
B. [tex]V=Bh[/tex]
C. [tex]V=\frac{1}{3} Bh^2[/tex]
D.[tex]V=-\frac{1}{3} Bh[/tex]
Answer:
Correct choice is A.
Step-by-step explanation:
We have to find about which of the given choices is the formula for the volume of a pyramid with base area B and height h.
We know that formula of volume of pyramid is given by:
[tex]volume = \frac{1}{3}\left(Base\ area\right)\left(Height\right)[/tex]
Plug the given values
[tex]volume = \frac{1}{3}\left(B\right)\left(h\right)[/tex]
So the correct choice is A.
Answer: A. [tex]\dfrac{1}{3}\times\text{B}\times\text{h}[/tex]
Step-by-step explanation:
We know that the volume of a pyramid is given by :-
[tex]\text{Volume}=\dfrac{1}{3}\times\text{Base Area of the pyramid}\times\text{Height of pyramid}[/tex]
Let B be the Base Area of the pyramid and h be the Height of pyramid, then the formula for the volume of a pyramid with base area B and height h will become :-
[tex]\text{Volume}=\dfrac{1}{3}\times\text{B}\times\text{h}[/tex]
Hence, the formula for the volume of a pyramid with base area B and height h =[tex]\dfrac{1}{3}\times\text{B}\times\text{h}[/tex]
Julio and his family ordered 6 pizzas for dinner. His parents ate 3/7 of one pizza, Julio's brothers ate 2 3/4 pizzas and his sisters ate 2 1/2 pizzas. How much pizza was left for Julio to eat?
approximately 0.3 or 3/10 pizza was left for Julio to eat.
mind you some numbers were rounded. if you want the exact number then it's 9/28 which is equal to 0.32.
hope this helps
After accounting for the amounts of pizza eaten by Julio's family (parents, brothers, and sisters), you find that there was approximately 0.36 pizzas left for Julio to eat.
Explanation:This question deals with some basic arithmetic and fractions. Start by counting how many pizzas Julio's family ate altogether. His parents ate 3/7 of one pizza, his brothers ate 2 3/4 pizzas, and his sisters ate 2 1/2 pizzas.
To add these amounts together, you might find it easiest to convert all the fractions to sevenths. The brothers ate 2 4/7 pizzas (as 4/7 is equivalent to 1/2) and the sisters ate 2 3.5/7 pizzas (as 3.5/7 is equivalent to 1/2). Add these to the 3/7 pizza the parents ate, you find the family ate a total of approximately 5.64 pizzas.
Since Julio's family ordered 6 pizzas in total, you then subtract the total amount eaten by the family from 6, which is approximately 0.36 pizzas. Therefore, Julio was left with 0.36 pizzas to eat.
Learn more about sequence here:
https://brainly.com/question/33564650
#SPJ2
What can you say about the end behavior of the function [tex]f(x) = log_{10} (5x-1)[/tex]?
A. One end decreased and one end approaches a constant
B. One end increase and one end decreases
C. Both ends increase
D. Both ends decrease
For the given logarithmic function, it is undefined as x approaches negative infinity, and increases without bound as x approaches positive infinity. The correct answer is B. One end increases and one end decreases.
Explanation:The function [tex]f(x) = log_{10} (5x-1)[/tex] is a logarithmic function with a base of 10. Its end behavior is determined by the properties of logarithmic functions.
Logarithmic functions have one side that increases without bound (to positive infinity) and one side that approaches a certain limit, but never reaches it, hence getting infinitely close to it. In this case, as x approaches negative infinity, the function becomes undefined as (5x-1) becomes a negative number and logarithm is undefined for negative values. On the other end, as x approaches positive infinity, the function will also approach positive infinity because the log of a large number gets larger.
Thus, the correct answer would be B. One end increases and one end decreases (becomes undefined).
Learn more about End Behavior of Logarithmic Function here:https://brainly.com/question/30284289
#SPJ12
The end behavior of the function is that A. one end decreases and one end approaches a constant.
Explanation:The given function is f(x) = log10 (5x-1).
To determine the end behavior of the function, we can consider the limit of the function as x approaches positive and negative infinity.
As x approaches positive infinity, the argument of the logarithm, 5x-1, will become larger and larger, resulting in a logarithm that approaches positive infinity. Therefore, the right end of the graph increases.
Similarly, as x approaches negative infinity, the argument of the logarithm will become smaller and smaller, resulting in a logarithm that approaches negative infinity. Therefore, the left end of the graph decreases.
Thus, the correct answer is A. One end decreases and one end approaches a constant.
Learn more about End behavior of a function here:
https://brainly.com/question/32061836
#SPJ12
A pitcher contains 4.32-pint of lemon syrup. 12.6 pint of water are added to the syrup to make lemonade. How much lemonade is made
Answer:
16.92 pints of lemonade was made
Step-by-step explanation:
A teacher instituted a,new reading program at school. After 10 weeks in the program it was found that the mean reading speed of a random same 19second grade students was 94.6 wpm. What might you conclude based on this result?
The reading program really improved some of the student's reaping speeds.
Estimate the percent of each number. 47% of 77
To estimate 47% of 77, multiply 77 by 0.47. The estimated value is approximately 36.19.
Explanation:To estimate the percent of a number, we multiply the number by the percentage as a decimal. In this case, to estimate 47% of 77, we multiply 77 by 0.47. This gives us an estimated value of 36.19. Therefore, 47% of 77 is approximately 36.19.
graph the line with slope 1/2 passing through the point (4,4)
Using the point slope formula (y-y1=m(x-x1)) it is found to be y=1/2x+2.
The range of F(x) = logb x is the set of all real numbers. True or false
The range of F(x) = logb x is the set of all real numbers.
TRUE
The range of F(x) = logb x is the set of all real numbers since the correct option is True.
How to find a Domain?The domain of this function is R+ or a positive real number For range if b>1 function will be an increasing function and the graph will be asymptotic to the negative y-axis. which implies if x goes to infinity f(x) goes to infinity and vice versa if 0<b<1 function will be a decreasing and graph will be asymptotic to the positive y-axis. which implies if x goes to infinity f(x) goes to 0 and vice-versa the range of the above function is all real numbers.
Learn more about the domain here: https://brainly.com/question/13113489
#SPJ2
What number should both sides of the following equation be multiplied by in order to solve for x?
x/7.25= 4
4
7.25
7
29
Answer:
7.25
Step-by-step explanation:
Multiplying both sides of this equation by 7.25 will both eliminate the fraction and isolate (solve for) x;
7.25(x/7.25= 4) → x = 4(7.25)
x = 29
The answer is 29
U Multiply 7.25 and 4 and you get 29
Check your a answer by putting 29 in for x and it's correct
I hope this helped you guys
The stilts increase a persons heart by 1.5 yards.If Bonnie who is 5 feet and 7 inches tall walks on stilts how tall will be.
Bonnie will be 10 feet and 1inch. Because 1.5 yards is actually 4.5 feet, since 3 feet make up 1 yard. 5 feet + 4 feet= 9 feet. 7 inches plus 6 inches(which is half a foot) = 13 inches or 1 foot and 1inch. 9 feet + 1 foot and 1 inch = 10 feet and 1 inch.
Answer:
[tex]\boxed{\text{10 ft 1 in}}[/tex]
Step-by-step explanation:
1. Convert the height of stilts to inches
1.5 yd × (36 in/1 yd) = 54 in
2. Convert Bonnie's height to inches
5ft 7in = 5 ft × (12 in/1 ft) + 7 in = 60 in + 7 in = 67 in
3. Calculate Bonnie's height on stilts
67 in + 54 in = 121 in
4. Convert inches back to feet and inches
121 in × (1 ft/12 in) = 10R1 ft = 10 ft 1 in
Bonnie will be [tex]\boxed{\text{10 ft 1 in}}[/tex] tall on stilts.
Which graph represents the function f(x) = 1/x - 1?
Answer:
To quickly solve this problem, we can use a graphing tool or a calculator to plot the equation.
Please see the attached image below, to find more information about the graph
The equation is:
f(x) = 1/x - 1
Domain
All real numbers except for {0}
Answer:
It would be the fourth graph, D
Explanation:
When faced with these problems, you can either plot it on your own or you can use a graph generator online.
What are the answers and why?
Answer:
[tex]\boxed{\text{(v) and (viii)}}[/tex]
Step-by-step explanation:
The three step test for continuity states that a function ƒ(x) is continuous at a point x = a if three conditions are satisfied:
f(a) is defined. The limit of ƒ(x) as x approaches a exists. The limit of ƒ(x) as x approaches a is equal to f(a).(i) Left-hand limit = right-hand limit.
Pass. The limit from either side is 8.
(ii) Left-hand limit = limit.
Pass. If the limits from either direction exist, the limit exists.
(iii) Limit as x ⟶ ∞ is not part of the three-step test.
(iv) Limit as x ⟶ 1 exists. Pass.
(v) f(1) is defined.
FAIL. f(1) is not defined.
(vi) Limit as x ⟶ ∞ is not part of the three-step test.
(vii) Passing the three-step test is not a step in the test.
(viii) The limit as x ⟶ 1 does not equal f(1).
FAIL. f(1) is undefined.
The steps in the three-step test for which the function fails are [tex]\boxed{\textbf{(v) and (viii)}}[/tex].
Write the equation of a horizontal line that passes through the point (–2, 2).
x = –2
y = –2
y = 2
x = 2
Answer:
option C
y = 2
Step-by-step explanation:
Given in the question,
a co-ordinate = (-2,2)
x = -2
y = 2
Equation of the straight line
y = mx + c
here m = gradient of the line
c = y - intercept
we know that gradient of the horizontal line = 0
plug value in the equation above
y = (0)x + 2
y = 2
The equation of a parabola is given.
y=34x2−6x+15
What are the coordinates of the vertex of the parabola?
Enter your answer in the boxes.
(
,
)
ANSWER
[tex]( \frac{3}{34} , \frac{501}{34} )[/tex]
EXPLANATION
The given parabola has equation;
[tex]y = 34 {x}^{2} - 6x + 15[/tex]
Comparing this equation to
[tex]y = a{x}^{2} + bx + c[/tex]
we have
a=34, b=-6 and c=15
The x-coordinate of the vertex is given by:
[tex]x = \frac{ - b}{2a} [/tex]
[tex]x = \frac{ - - 6}{2(34)} [/tex]
[tex]x = \frac{ 6}{2(34)} [/tex]
[tex]x = \frac{ 3}{34} [/tex]
The y-coordinates of the vertex is obtained by substituting the x-value of the vertex into the equation:
[tex]y = 34( { \frac{3}{34} })^{2} - 6( \frac{3}{34}) + 15[/tex]
[tex]y = { \frac{9}{34} } - \frac{18}{34}+ 15[/tex]
[tex]y = \frac{501}{34} [/tex]
The vertex is
[tex]( \frac{3}{34} , \frac{501}{34} )[/tex]
Answer: (4,3)
He meant 3/4 not 34
Help please simplify: 4(3b)
Answer: 12b
Step-by-step explanation: you multiply 4 by 3 but keep the b
Answer:
C ---> 12b
Step-by-step explanation:
First you would distribute the 4 to the 3b which will give you 12b.
4(3b)
4*3=12
12b
Classify each conic section and write its equations in standard form. Show work.
[tex]4x^2+2y^2-8x+16y-52=0[/tex]
Answer:
The conic is ellipse of equation (x - 1)²/22 + (y + 4)²/44 = 1
Step-by-step explanation:
* Lets revise how to identify the type of the conic
- Rewrite the equation in the general form,
Ax² + Bxy + Cy² + Dx + Ey + F = 0
- Identify the values of A and C from the general form.
- If A and C are nonzero, have the same sign, and are not equal
to each other, then the graph is an ellipse.
- If A and C are equal and nonzero and have the same sign, then
the graph is a circle
- If A and C are nonzero and have opposite signs, and are not equal
then the graph is a hyperbola.
- If either A or C is zero, then the graph is a parabola
* Now lets solve the problem
The equation is 4x² + 2y² - 8x + 16y - 52 = 0
∴ A = 4 and C = 2 ⇒ same sign and different values
∴ The equation is ellipse
* The standard form of the ellipse is
(x - h)²/a² + (y - k)²/b² = 1
- Lets try to make this form from the general form
- Group terms that contain the same variable, and move the
constant to the opposite side of the equation
∴ (4x² - 8x) + (2y² + 16y) = 52
- Factorize the coefficients of the squared terms
∴ 4(x² - 2x) + 2(y² + 8y) = 52
- Complete the square for x and y
# To make completing square
- Divide the coefficient of x (or y) by 2 and then square the answer
- Add and subtract this square number and form the bracket of
the completing the square
# 2 ÷ 2 = 1 ⇒ (1)² = 1 ⇒ add and subtract 1
∴ 4[(x² - 2x + 1) - 1] = 4(x² - 2x + 1) - 4
- Rewrite as perfect squares ⇒ 4(x -1)² - 4
# 8 ÷ 2 = 4 ⇒ (4)² = 16 ⇒ add and subtract 16
∴ 2[(y² + 8y + 16) - 16] = 2(y² + 8y + 16)² - 32
- Rewrite as perfect squares ⇒ 2(y + 4)² - 32
∴ 4(x - 1)² - 4 + 2(y + 4)² - 32 = 52
∴ 4(x - 1)² - 4 + 2(y + 4)² = 32 + 4 + 52
∴ 4(x - 1)² + 2(y + 4)² = 88 ⇒ divide all terms by 88
∴ (x - 1)²/22 + (y + 4)²/44 = 1