To make the function differentiable at x = 2, we must ensure continuity by matching function values and differentiability by equating derivatives from both sides at x = 2. Solving the system of equations obtained from these conditions will give the values of a and b.
Explanation:To find the values of a and b that make the function f(x) = { x² if x ≤ 2, ax³ + bx if x > 2 } differentiable at x = 2, we need to ensure both continuity and differentiability of f(x) at this point. First, continuity at x = 2 requires that the limits from the left and the right are the same, meaning f(2) = 2² = 4 should equal a(2)³ + b(2). Second, for differentiability, the derivatives from the left and right at x = 2 must also be equal. The derivative of x² is 2x, so f'(2) = 4. Differentiating ax³ + bx gives 3ax² + b, so f'(2) = 12a + b must also equal 4. Solving these equations:
4 = 8a + 2b4 = 12a + bgives us a system of equations that when solved, will provide the exact values of a and b required.
Seorang ayah memberikan sebuah tantangan kepada anaknya untuk i menghitung jumlah uang koin yang diperlukan untuk memenuhi papan catur. I Pada kotak pertama diberi I uang koin, kotak kedua 2 uang koin, 4 uang koin untuk kotak ketiga, 8 koin untuk kotak keempat demikian berlanjut sampai memenuhi 64 kotak. A. Bantu anak tersebut menentukan auaunan banyak koin pada tiap tiap kotak papan catur tersebut.Nyatakan dalam bentuk perpangkatan
The total number of coins required to fill all the [tex]64[/tex] boxes are [tex]\boxed{\bf 18446744073709551615}[/tex].
Further explanation:
In a chessboard there are [tex]64[/tex] boxes.
The objective is to determine the total number of coins required to fill the [tex]64[/tex] boxes in chessboard.
In the question it is given that in the first box there is [tex]1[/tex] coin, in the second box there are [tex]2[/tex] coins, in the third box there are [tex]8[/tex] coins and it continues so on.
A sequence is formed for the number of coins in different boxes.
The sequence formed for the number of coins in different boxes is as follows:
[tex]\boxed{1,2,4,8,...}[/tex]
The above sequence can also be represented as shown below,
[tex]\boxed{2^{0},2^{1},2^{2},2^{3},...}[/tex]
It is observed that the above sequence is a geometric sequence.
A geometric sequence is a sequence in which the common ratio between each successive term and the previous term are equal.
The common ratio [tex](r)[/tex] for the sequence is calculated as follows:
[tex]\begin{aligned}r&=\dfrac{2^{1}}{2^{0}}\\&=2\end{aligned}[/tex]
The [tex]n^{th}[/tex] term of a geometric sequence is expressed as follows:
[tex]\boxed{a_{n}=ar^{n-1}}[/tex]
In the above equation [tex]a[/tex] is the first term of the sequence and [tex]r[/tex] is the common ratio.
The value of [tex]a[/tex] and [tex]r[/tex] is as follows:
[tex]\boxed{\begin{aligned}a&=1\\r&=2\end{aligned}}[/tex]
Since, the total number of boxes are [tex]64[/tex] so, the total number of terms in the sequence is [tex]64[/tex].
To obtain the number of coins which are required to fill the [tex]64[/tex] boxes we need to find the sum of sequence formed as above.
The sum of [tex]n[/tex] terms of a geometric sequence is calculated as follows:
[tex]\boxed{S_{n}=a\left(\dfrac{r^{n}-1}{r-1}\right)}[/tex]
To obtain the sum of the sequence substitute [tex]64[/tex] for [tex]n[/tex], [tex]1[/tex] for [tex]a[/tex] and [tex]2[/tex] for [tex]r[/tex] in the above equation.
[tex]\begin{aligned}S_{n}&=1\left(\dfrac{2^{64}-1}{2-1}\right)\\&=\dfrac{18446744073709551616-1}{1}\\&=18446744073709551615\end{aligned}[/tex]
Therefore, the total number of coins required to fill all the [tex]64[/tex] boxes are [tex]\boxed{\bf 18446744073709551615}[/tex].
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Answer details:
Grade: High school
Subject: Mathematics
Chapter: Sequence
Keywords: Series, sequence, logic, groups, next term, successive term, mathematics, critical thinking, numbers, addition, subtraction, pattern, rule., geometric sequence, common ratio, nth term.
Coins on the chessboard follow a doubling pattern. In the nth box, the coins can be expressed as [tex]\(2^{(n-1)}[/tex]. The total coins for all 64 boxes is [tex]2^{63}[/tex].
Certainly, let's break down the doubling pattern of coins in each chessboard box, expressed in exponential form:
1. **First Box (kotak pertama):
- Number of coins: [tex]\(2^0 = 1\)[/tex] (2 raised to the power of 0).
2. **Second Box (kotak kedua):
- Number of coins: [tex]\(2^1 = 2\)[/tex] (2 raised to the power of 1).
3. **Third Box (kotak ketiga):
- Number of coins: [tex]\(2^2 = 4\)[/tex] (2 raised to the power of 2).
4. **Fourth Box (kotak keempat):
- Number of coins: [tex]\(2^3 = 8\)[/tex] (2 raised to the power of 3).
The pattern continues, doubling the number of coins with each subsequent box.
For the n-th box, the number of coins is given by [tex]\(2^{(n-1)}[/tex], where n is the box number.
So, the exponential form for the number of coins in each chessboard box is [tex]\(2^{(n-1)}[/tex], where n is the box number ranging from 1 to 64.
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Que. A father challenges his child to calculate the total number of coins needed to fill a chessboard. In the first box, 1 coin is placed, 2 coins in the second box, 4 coins in the third, and so on, up to the 64th box. Help the child determine the doubling pattern of coins in each chessboard box, expressed in exponential form.
A chocolate company makes chocolate malt balls that are 0.75 inches in diameters.The carton they are to be packed inis apoximately a rectangular prism with the dimension of 3 inches by 3 inches by7.How many malt balls will fit in the carton
Answer:
144 malt balls will fit in the carton
Step-by-step explanation:
* Lets explain how to solve the problem
- To solve the problem we must to know each dimensions of the
cartoon will fit how many balls
- To do that divide each dimension by the diameter of the ball
∵ The diameter of the chocolate malt ball is 0.75 inches
∵ The dimensions of the carton are 3 inches , 3 inches , 7 inches
* Lets find how many balls will fit in the side of 3 inches
∵ 3 ÷ 0.75 = 4
∴ There are 4 balls will fit in the side of 3 inches
∵ Two dimensions of the carton are 3 inches
∴ There 4 × 4 balls fit in the base of the carton
∵ The height of the carton is 7
* Lets find how many balls can fit in the height
∵ 7 ÷ 0.75 = 9.3333
∴ 9 balls can fit the height of the carton
∴ There are 4 × 4 × 9 balls will fit in the carton
∴ The number of the balls = 4 × 4 × 9 = 144 balls
* 144 malt balls will fit in the carton
I need the graph for the equation:
y=9.50x-3
Answer:
see below
Step-by-step explanation:
It's a little tough to draw on regular graph paper because the slope is so steep and the slope is not an integer. Shown below is the graph with a couple of the points labeled.
In kite WXYZ, the measure of x=z=86° and y=72°
What is the measure of w?
Answer:
The measure of angle W is 116°.
Step-by-step explanation:
Given information: WXYZ is a kite, X=Z=86° and Y=72°.
According to the angle sum property of a kite, the sum of all interior angles of a kite is 360°.
In kite WXYZ,
[tex]\angle W+\angle X+\angle Y+\angle Z=360[/tex]
[tex]\angle W+86+72+86=360[/tex]
[tex]\angle W+244=360[/tex]
Subtract 244 from both sides.
[tex]\angle W+244-244=360-244[/tex]
[tex]\angle W=116[/tex]
Therefore, the measure of angle W is 116°.
A group consists of four men and six women. Five people are selected to attend a conference.
a. In how many ways can five people be selected from this group of ten?
b. In how many ways can five women be selected from the six women?
c. Find the probability that the selected group will consist of all women.
Answer:
10C5=252
6C5=6
6C5/10C5= 1/42
What is the probability that a King is drawn from a deck of 52 cards, without replacement, and then a second King is drawn?
Answer: 12/2652 or 1/221
Step-by-step explanation:
There are 4 kings in a deck
So the probability of getting a king would be
4/52 then after receiving a king and not replacing you will the have a 3/51 chance
So all together you will have a:
4/52 * 3/51 = 12/2652 or simplified 1/221
Hope this helps
The probability of drawing a King from a deck of 52 cards without replacement and then drawing a second King is 1/221.
We have,
To find the probability of drawing a King from a deck of 52 cards without replacement, and then drawing a second King, we can calculate it as follows:
The probability of drawing a King as the first card is 4/52 since there are 4 Kings in a deck of 52 cards.
After removing one King from the deck, there are now 51 cards left, including 3 Kings.
The probability of drawing a second King, given that a King has already been drawn, is 3/51.
To find the overall probability of both events occurring, we multiply the individual probabilities:
(4/52) * (3/51) = 12/2652
Simplifying the fraction, we have:
12/2652 = 1/221
Therefore,
The probability of drawing a King from a deck of 52 cards without replacement and then drawing a second King is 1/221.
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Apply the square root property of equality
Answer:
First blank: 1/4
Second blank: 2/3
Step-by-step explanation:
[tex](x+\frac{1}{4})^2=\frac{4}{9}[/tex]
Applying the square root of both sides gives:
[tex](x+\frac{1}{4})=\pm \sqrt{\frac{4}{9}}[/tex]
[tex]x+\frac{1}{4}=\pm \frac{\sqrt{4}}{\sqrt{9}}[/tex]
[tex]x+\frac{1}{4}=\pm \frac{2}{3}[/tex]
The blanks are 1/4 and 2/3.
What is the square root property of equality?When we take the square root on both sides of the equation, then the whole square term becomes its square root, but the constant term on the other side has a ± sign as the square root of n can be -√n as well as √n, because the square of a negative number is also a positive number.
The solution to the problemSo the given equation is (x+1/4)² = 4/9
Taking square root on both sides we get
(x+1/4) = ±2/3 using the square root property of equality.
Hence the blanks are 1/4 and 2/3 of the given question.
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Please please help me with this
Answer: y= -1/2x + 3/4
Step-by-step explanation: For a line in the for of y=mx + b, the slope is m and y intercept is b.
Answer:
y = - [tex]\frac{1}{2}[/tex] x + [tex]\frac{3}{4}[/tex]
Step-by-step explanation:
The equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
Rearrange 6x + 12y = 9 into this form
Subtract 6x from both sides
12y = - 6x + 9 ( divide all terms by 12 )
y = - [tex]\frac{6}{12}[/tex] x + [tex]\frac{9}{12}[/tex], that is
y = - [tex]\frac{1}{2}[/tex] x + [tex]\frac{3}{4}[/tex]
PLEASE HELP!!!
Aleko’s Pizza has delivered a beautiful 16 inch diameter pie to Lee dorm room. The pie is slice into 8 equal sizes pieces, but Lee is such a non-conformist he cuts off an edge as pictured. John then takes on e of the remaining triangular slices. Who has more pizza and by how much?
Answer:
Lee has more pizza
Lee has 2.24 in^2 more than John
Step-by-step explanation:
step 1
Find the area of each slice of pizza
[tex]A=\frac{1}{8}\pi r^{2}[/tex]
we have
[tex]r=16/2=8\ in[/tex] ----> the radius is half the diameter
substitute
[tex]A=\frac{1}{8}\pi 8^{2}[/tex]
[tex]A=8\pi\ in^{2}[/tex]
step 2
Find the area of John's part (area of shaded triangle)
The measure of the central angle of each slice of pizza is equal to
[tex]360\°/8=45\°[/tex]
so
the height of triangle is equal to the base
Let
x ---->the base of the shaded triangle
[tex]cos(45\°)=\frac{x}{r}[/tex]
[tex]cos(45\°)=\frac{x}{8}[/tex]
Remember that
[tex]cos(45\°)=\frac{\sqrt{2}}{2}[/tex]
substitute
[tex]\frac{\sqrt{2}}{2}=\frac{x}{8}[/tex]
solve for x
[tex]x=4\sqrt{2}\ in[/tex]
Find the area of shaded triangle
[tex]A=(1/2)(4\sqrt{2})(4\sqrt{2})=16\ in^2[/tex]
step 3
Find the area of Lee's part
The area of Lee's part is equal to the area of two slices of pizza minus the area of two triangles
so
[tex]2(8\pi)-2(16)=(16\pi-32)\ in^2[/tex]
assume
[tex]\pi =3.14[/tex]
[tex](16(3.14)-32)=18.24\ in^2[/tex]
so
Lee's part is greater than John part's
Find the difference
[tex]18.24-16=2.24\ in^2[/tex]
therefore
Lee has more pizza
Lee has 2.24 in^2 more than John
Without exact dimensions of Lee's cut, we can't calculate the precise difference in area after he cuts off an edge, but John likely has more pizza since his slice is unmodified.
Explanation:The student's question involves comparing areas of pizza slices after one has been modified by cutting off an edge. To answer who has more pizza and by how much, we need to calculate the area of the pizza slices. The original pizza is 16 inches in diameter, and when divided into 8 equal slices, each slice is a sector of a circle with a central angle of 45 degrees. If Lee cuts off an edge and John takes an unmodified triangular slice, John would likely have more pizza because Lee's slice has been reduced in size. However, without knowing the exact dimensions of the removed edge, we can't calculate the precise difference in area.
]
What is the slope of the following linear function?
Answer:
The answer to your question is: m = -1/3
Step-by-step explanation:
First, we look for 2 points in the graph
A (0, -3)
B (3, -4)
Then find the slope
m = (y2 - y1) / (x2 - x1)
m = (-4 - - 3) / ( 3 - 0) Substitution
m = (-4 + 3) / 3 Simplify
m = -1 /3
Rachel scored 670 on the analytic portion of the GRE (Graduate Record Exam). GRE scores are normally distributed with a mean of 600 and a standard deviation of 30. How many standard deviations is Rachel's score above the mean?
Answer:
Rachel's score is 2.3333 standard deviations above the mean
Step-by-step explanation:
GRE scores are normally distributed
Let be G the random variable ''Gre scores''
G ~ N (mean,standard deviation)
G ~ N (600,30)
Rachel scored 670 on the analytic portion of the GRE.
670 - 600 will be the score above the mean
670 - 600 = 70
To find this in terms of standard deviation we divide by the standard deviation
70/standard deviation = 70/30 = 7/3 = 2.33333333 standard deviations
Rachel's GRE score is approximately 2.33 standard deviations above the mean. The calculation is made by subtracting the mean from the observed score and dividing this by the standard deviation.
Explanation:The subject of this question pertains to the mathematical concept of Z-scores, used in statistics to measure how many standard deviations an element is from the mean. In the case of Rachel's GRE score, we can calculate the number of standard deviations her score is above the mean using the formula z = (X - μ) / σ where:
X is Rachel's score, which is 670μ is the mean score, which is 600σ is the standard deviation, which is 30
By substituting these values into the formula, we get:
z = (670 - 600) / 30 = 70 / 30 = 2.33
This means that Rachel's score is approximately 2.33 standard deviations above the mean.
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Samuel has to sell concert tickets worth at least $90. The price of a child ticket is $8, and the price of an adult ticket is $15. Let y be the number of child tickets sold and x be the number of adult tickets sold. Which of the following graphs best models this situation?
Answer:
The correct graph is the second one, that the line intersects x at 6 and y at 11.5
Step-by-step explanation:
Samuel has to sell at least $90. So, in this graph if he sell only child ticket, he will have to sell 11.5 tickets. Or if he sell only adult tickets, he will have to sell at least 6.
Answer:
The last graph is the best models this situation.
Step-by-step explanation:
First we need to find the equation of ticket selling. To not loss any money from this business Samuel need to sell at least 6 adult or 11.25 child tickets. I know ticket number must be integer but those numbers are x and y values that line crosses through axes. The equation is:
[tex]8x+15y\geq 90[/tex]
and the graph of this equation is attached.
What is the midpoint M of that line segment?
Answer:
Midpoint = ( (x1+x2)/2 , (y1+y2)/2 )
Step-by-step explanation:
Please upload the line segment otherwise, you can use the equation above to solve for it.
Let g(x)=5x-1 and h(x)=X^2-1
Solve:
g(h(x))=74
Answer:
Let's replace the h(x) function in g(x) and then use 74 as a result on the axis y. The correct answer is 4 .
Identify the sampling technique used, and discuss potential sources of bias (if any). Explain. After a hurricane, a disaster area is divided into 200 equal grids. Thirty of the grids are selected, and every occupied household in the grid is interviewed to help focus relief efforts on what residents require the most.
Answer:
The sampling technique used here is Cluster Sampling.
Step-by-step explanation:
Since here Population is divided into different parts called grid and whole elements of some selected grid is taken as sample. So, Cluster Sampling is used here.
Further the different types of sampling we have are:
Simple Random Sampling is the sampling where samples are chosen randomly, where each unit has an equal chance of being selected in a sample.
If the population is divided into a different group called cluster and all elements of clusters are selected as a sample then it is Cluster Sampling.
In Convenience sampling, observers collect the sample as his\her convenience.
In Systematic Sampling sample is chosen by some criteria like he\she is taken every 10th unit as a sample from the population.
In Stratified Sampling population is divided into several groups such that within the group it is homogeneous and between the group it is heterogeneous. And now a selection of each stratum and unit has an equal chance of selection.
In automobile mileage and gasoline-consumption testing, 13 automobiles were road tested for 300 miles in both city and highway driving conditions. The following data were recorded for miles-per-gallon performance.City: 16.2 16.7 15.9 14.4 13.2 15.3 16.8 16.0 16.1 15.3 15.2 15.3 16.2 Highway: 19.4 20.6 18.3 18.6 19.2 17.4 17.2 18.6 19.0 21.1 19.4 18.5 18.7 Use the mean, median, and mode to make a statement about the difference in performance for city and highway driving.
Answer:
Looking at the mean, the median and the mode, cars are more efficient on a highway than in a city
Step-by-step explanation:
First, we calculate the average (mean) performance by adding all values and dividing the sum by the number of values added.
[tex]Mean_{city} =\frac{(16.2+16.7+15.9+14.4+13.2+15.3+16.8+16.0+16.1+15.3+15.2+15.3+16.2)mpg }{13} =15.6 mpg[/tex]
[tex]Mean_{highway} =\frac{(19.4+20.6+18.3+18.6+19.2+17.4+17.2+18.6+19.0+21.1+19.4+18.5+18.7 )mpg }{13} =18.9 mpg[/tex]
Then, to know what the median is, we have to order from least to greatest and look the middle value, i.e. half of the values will be higher than the median and half will be lower.
For the mode, we have to look up what is the most repeated value in our list.
For city performances:
13.2 14.4 15.2 15.3 15.3 15.3 15.9 16 16.1 16.2 16.2 16.7 16.8The median value is 15.9 miles per gallon, and the mode is 15.3 miles per gallon.
For highway performances:
17.2 17.4 18.3 18.5 18.6 18.6 18.7 19 19.2 19.4 19.4 20.6 21.1The median value is 18.7 miles per gallon, and the mode is 18.6 and 19.4 miles per gallon.
We can say then, that looking at the mean, the median and the mode, cars are more efficient on a highway than in a city and that the least-consuming car in a city still is worst in terms of efficiency than the worst-performing in a highway.
Final answer:
The mean, median, and mode can be used to compare the performance of automobiles in city and highway driving conditions in terms of miles per gallon (mpg). Based on these measures, we can say that the performance of automobiles is generally better in highway driving conditions compared to city driving conditions.
Explanation:
The mean, median, and mode can be used to compare the performance of automobiles in city and highway driving conditions in terms of miles per gallon (mpg).
The mean is calculated by summing up all the mpg values and dividing it by the number of values. For city driving, the mean is 15.66 mpg, and for highway driving, the mean is 18.81 mpg.
The median is the middle value in a set of ordered numbers. For city driving, the median is 15.3 mpg, and for highway driving, the median is 18.6 mpg.
The mode is the value that appears most frequently in a set of numbers. For both city and highway driving, the mode is 15.3 mpg.
Based on these measures, we can say that the performance of automobiles is generally better in highway driving conditions compared to city driving conditions, as the mean and median mpg values are higher for highway driving.
Express the negations of each of these statements so that all negation symbols immediately precede predicates. a) ∀x∃y∀zT(x, y, z) b) ∀x∃yP(x, y) ∨ ∀x∃yQ(x, y) c) ∀x∃y(P(x, y) ∧ ∃zR(x, y, z)) d) ∀x∃y(P(x, y) → Q(x, y))
Answer:
a) ∀x∃y ¬∀zT(x, y, z)
∀x∃y ∃z ¬T(x, y, z)
b) ∀x¬[∃y (P(x, y) ∨ Q(x, y))]
∀x∀y ¬ [P(x, y) ∨ Q(x, y)]
∀x∀y [¬P(x, y) ^ ¬Q(x, y)]
c) ∀x ¬∃y (P(x, y) ^ ∃zR(x, y, z))
∀x ∀y ¬(P(x, y) ^ ∃zR(x, y, z))
∀x ∀y (¬P(x, y) v ¬∃zR(x, y, z))
∀x ∀y (¬P(x, y) v ∀z¬R(x, y, z))
d) ∀x¬∃y (P(x, y) → Q(x, y))
∀x∀y ¬(P(x, y) → Q(x, y))
∀x∀y (¬P(x, y) ^ Q(x, y))
Answer:
a) ∃x∀y∃z~T(x, y, z)
b) ∃x∀y~P(x, y) ∧ ∃x∀y~Q(x, y)
c) ∃x∀y(~P(x, y) ∨ ∀z~R(x, y, z))
d) ∃x∀y(P(x, y) → ~Q(x, y))
Step-by-step explanation:
The negation of a is written as ~a.
Note the following properties that are going to be applied in the problems here :
~(P → Q) = P → ~Q
De Morgan's Laws
~(P ∨ Q) = ~P ∧ ~Q
~(P ∧ Q) = ~P ∨ ~Q
~∃xP = ∀xP
~∀xP = ∃xP
So back to the original problem.
a) ∀x∃y∀zT(x, y, z)
We have the negation as
~[∀x∃y∀zT(x, y, z)]
= ∃x~∃y∀zT(x, y, z)
= ∃x∀y∀~zT(x, y, z)
= ∃x∀y∃z~T(x, y, z)
b) ∀x∃yP(x, y) ∨ ∀x∃yQ(x, y)
Negation is:
~[∀x∃yP(x, y) ∨ ∀x∃yQ(x, y)]
= ~∀x∃yP(x, y) ∧ ~∀x∃yQ(x, y)
= ∃x~∃yP(x, y) ∧ ∃x~∃yQ(x, y)
= ∃x∀y~P(x, y) ∧ ∃x∀y~Q(x, y)
c) ∀x∃y(P(x, y) ∧ ∃zR(x, y, z))
Negation is:
~[∀x∃y(P(x, y) ∧ ∃zR(x, y, z))]
= ~∀x∃y(P(x, y) ∧ ∃zR(x, y, z))
= ∃x~∃y(P(x, y) ∧ ∃zR(x, y, z))
= ∃x∀y~(P(x, y) ∧ ∃zR(x, y, z))
= ∃x∀y(~P(x, y) ∨ ~∃zR(x, y, z))
= ∃x∀y(~P(x, y) ∨ ∀z~R(x, y, z))
d) ∀x∃y(P(x, y) → Q(x, y))
Negation is:
~[∀x∃y(P(x, y) → Q(x, y))]
= ~∀x∃y(P(x, y) → Q(x, y))
= ∃x~∃y(P(x, y) → Q(x, y))
= ∃x∀y~(P(x, y) → Q(x, y))
= ∃x∀y(P(x, y) → ~Q(x, y))
Given that events "A" and "B" are independent, P(A)= 0.80 and P(A and B) = 0.24, what is P (B)?
Group of answer choices
0.104
0.192
0.56
0.30
Answer:
0.30
Step-by-step explanation:
They are independent, so:
P(A and B) = P(A) P(B)
0.24 = 0.80 P(B)
P(B) = 0.30
Identify all values, if any, in the data set that would be considered outliers when creating a modified boxplot. 4 5 7 9 10 10 12 13 15 16 16 17 18 23 31
Answer:
31
Step-by-step explanation:
Please please help me out!!!!!!
Answer:
see explanation
Step-by-step explanation:
Inequalities of the type | x | > a, always have solutions of the form
x < - a or x > a
This can be extended to expressions, that is
14 - 5x < - 8 OR 14 - 5x > 8 ( subtract 14 from both sides of both inequalities )
- 5x < - 22 OR - 5x > - 6
Divide both sides by - 5 , reversing the inequality sign as a consequence
x > [tex]\frac{22}{5}[/tex] OR x < [tex]\frac{6}{5}[/tex]
That is the solution is
x < [tex]\frac{6}{5}[/tex] OR x > [tex]\frac{22}{5}[/tex]
Answer:
Step-by-step explanation:
Inequalities of the type | x | > a, always have solutions of the form
x < - a or x > a
This can be extended to expressions, that is
14 - 5x < - 8 OR 14 - 5x > 8 ( subtract 14 from both sides of both inequalities )
- 5x < - 22 OR - 5x > - 6
Divide both sides by - 5 , reversing the inequality sign as a consequence
x > OR x <
That is the solution is
x < OR x >
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The formula v=r2h gives the volume of a cylinder with a radius r and height h. Find the volume of cylinder with radius (x+4) cm and height 5 cm. Write your answer in standard form.
Final answer:
To calculate the volume of a cylinder with a radius of (x+4) cm and height of 5 cm, substitute the values into the formula V = πr²h, resulting in V = 5π(x² + 8x + 16), which gives the volume in standard form.
Explanation:
The question is about finding the volume of a cylinder with a given radius of (x+4) cm and a height of 5 cm. The formula to calculate the volume of a cylinder is V = πr²h, where 'V' is the volume, 'r' is the radius, and 'h' is the height of the cylinder.
To find the volume with the given dimensions, we substitute 'r' with (x + 4) and 'h' with 5. This results in:
V = π(x + 4)² × 5 = π(x² + 8x + 16) × 5
Simplifying this expression gives us:
V = 5π(x² + 8x + 16)
This is the volume of the cylinder in standard form, expressed as a function of x. Therefore, the volume depends on the value of x, and this expression allows us to calculate it for any given 'x'.
The following were the recorded birth weights for babies born July 16, 2011: 8.1 lbs., 6.0 lbs., 4.7 lbs., 6.9 lbs., 5.6 lbs., 7.7 lbs., 6.3 lbs., 7.8 lbs., 6.1 lbs., and 9.2 lbs. What was the average birth weight on the day? Round to two decimal places.
Tom has 2 more than 5 times the number of CD’s that Jane has. Jane has 5 CD’s. Write an
expression to express this.
Step-by-step explanation:
t=2+5×j
j=5
t=2+5×5
t=27
Tom has 2 more than 5 times what Jane has, so you would multiply the amount Jane has by 5, then add 2 to that:
Tom = 5(5) +2
Tom = 25 +2
Tom = 27 CD's.
Please help asap and give the function.
Answer:
see below for the stretched graph; see the second attachment for the functions
Step-by-step explanation:
The transformation ...
g(x) = f(x/a)
represents a horizontal stretch of f(x) by a factor of "a". You want a stretch by a factor of 4, so you can use a=4:
g(x) = f(x/4)
_____
Horizontal stretch by a factor of 4 means all the points on the graph of g(x) are 4 times as far from they y-axis as they are on the graph of f(x). That is, x must be 4 times as large to give the same y-value.
The graphs of f(x) and g(x) are shown in the second attachment, along with their equations.
A researcher uses a repeated-measures design to compare individuals’ performance before treatment with their performance after treatment. If all the participants show improved performance of 8 or 9 points after treatment, what should the researcher find _______
a) a sample mean difference near zero.
b) the statistic near zero.
c) the variance of the difference scores is near zero.
d) none of the other options is correct.
Answer:
c. the variance of the difference scores is near zero
Step-by-step explanation:
If all the participants show improved performance of 8 or 9 points after treatment, what should the researcher find - the variance of the difference scores is near zero.
But this can be true only when the original scores had a low variance.
If all participants in a repeated-measures design show improvement of 8 or 9 points after treatment, the researcher should find that the variance of the difference scores is near zero because all the scores improved by a similar amount.
Explanation:In a repeated-measures design, the same subjects are tested before and after an intervention. If all the participants show improved performance of 8 or 9 points after treatment, the researcher should find that the variance of the difference scores is near zero. This is because the variance - the measure of how spread out a group of numbers are from the mean - would be narrow since all the scores improved by almost the same amount (8 or 9). Hence, option c) is the correct one.
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Find the equation in slope/intercept form that is perpendicular to 2x - 3y = 4 and passing through (-1/7, 4).
Answer:
y = -1½x + 3 11⁄14
Step-by-step explanation:
First convert from Standard Form to Slope-Intercept Form:
2x - 3y = 4
-2x - 2x
____________
-3y = -2x + 4
___ _______
-3 -3
y = ⅔x - 1⅓ >> Slope-Intercept Form
↑
slope
Now, Perpendicular Lines have OPPOSITE MULTIPLICATIVE INVERSE Rate of Changes [Slopes], so since the slope is ⅔, the opposite multiplicative inverse of that would be -1½, or -3⁄2. Anyway, do the following:
4 = -1½[-⅐] + b
3⁄14
-3⁄14 - 3⁄14
_______________
3 11⁄14 = b
y = -1½x + 3 11⁄14 >> New equation
I am joyous to assist you anytime.
Answer:
Step-by-step explanation:
So a few things to know before hand. Slope ntercept form is y = mx + b where m is the slope, and in this form that will always make b the y intercept.
A perpendicular slope is pretty easy to find. Of course, perpendicular means it intersects that first line at a 90 degree angle. So the x and y axis themselves are perpendicular. Anyway, if you know the slope of the line, a perpendicular slope is -1/m where m is the slope. S taking the simplest example, in the graph of just x, the slope is 1, so the perpendicular slope is -1/1 or just -1.
The last thing is to know how to write the equation of a linear function when you know a point and its slope. if you have the slope m and a point on the graph (a,c) you can use the point slope form which is this. y - c = m(x - a) where you solve for y. x an y stay as variables here.
Now knowing all that we can start. First we want to put the original graph into slope intercept form, which is pretty easy. Just manipulate the equation.
2x - 3y = 4
2x -4 = 3y
y = (2x - 4)/3
y = 2/3 x - 4/3
so m = 2/3 and b = -4/3
Now, we have the slope of this line and want the slope of a perpendicular line. Like I mentioned before the slope is -1/m so in this case that's -1/(2/3) = -3/2 Let me know if you don't get how that was gotten.
Now that we kno the perpendicular slope, we can make a perpendicular line. How do we make a line when we know the slope and a point? Keep in mind the point is (-1/7, 4)
y - c = m(x-a)
y - 4 = -3/2(x + 1/7)
y - 4 = -3/2 x - 3/14
y = -3/2 x + 53/14
If it weren't in slope intercept form you'd have to put it in that, but I took care of it in the process, so here's the answer. Let me know if there's anything you don't understand.
Lena's mother asked her to count the number of pennies in the penny jar. Her mother said I made seven stacks of six pennies each and there were four leftover pennies. When Lena counted she made nine stacks of five pennies each and two left.
Answer:
Part a) 9*5+2
Part b) 7*6+4
Part c) Lena is correct
Part d) see the explanation
Step-by-step explanation:
The complete question in the attached figure
Part a) Write a numerical expression to represent Lena’s way of counting
To represent Lena’s way of counting, multiply the number of stacks by the number of pennies in each stack plus the number of pennies left over
Let
x -----> the number of stacks
y ----> the number of pennies in each stack
z ----> the number of pennies left over
so
[tex]xy+z[/tex]
we have
x=9 stacks
y=5 pennies
z=2 pennies
substitute
[tex]9*5+2[/tex]
Part b) Write a numerical expression to represent her mother's way
To represent her mother’s way of counting, multiply the number of stacks by the number of pennies in each stack plus the number of pennies left over
Let
x -----> the number of stacks
y ----> the number of pennies in each stack
z ----> the number of pennies left over
so
[tex]xy+z[/tex]
we have
x=7 stacks
y=6 pennies
z=4 pennies
substitute
[tex]7*6+4[/tex]
Part c) Lena thinks her mother must have been working with fewer pennies than she was. Is Lena correct?
we have that
Lena’s expression
[tex]9*5+2[/tex]
Simplify
[tex]9*5+2=47[/tex]
Her mother’s expression
[tex]7*6+4[/tex]
Simplify
[tex]7*6+4=46[/tex]
therefore
Lena’s expression is more.
Lena is correct
Part d) Use a < , > , or = symbol to show how the two expressions compare
[tex]9*5+2 > 7*6+4[/tex]
[tex]47 > 46[/tex]
The number 47 is greater than the number 46
therefore
The symbol is " >"
Let (x1, y1),(x2, y2),(x3, y3) be points with distinct x-values. Prove there exists a polynomial p(x) of degree at most 2 passing through these points. State and prove a similar result for four points?
Answer: we can use the folowing polynomial.
P(x) = [tex]\frac{y1 (x - x2)(x -x3)}{(x1 - x2)(x1-x3)}[/tex] + [tex]\frac{y2 (x - x1)(x -x3)}{(x2 - x1)(x1-x3)}[/tex] + [tex]\frac{y3 (x - x2)(x -x1)}{(x3 - x2)(x3-x1)}[/tex]
you can see that P(x1) = y1
P(x2) = y2
P(x3) = y3
this is a Lagrange polynomial.
Sketch the graph of f(t) = 5/(2+3e^-t), t>=0
Could someone explain why the graph looks the way it does and the method to reach the sketch of the graph?
Explanation:
The term containing the variable, e^-t has a range from 0 to infinity, as all exponential terms do.
For t → -∞, e^-t → ∞ and the value of the rational expression becomes 5/∞ ≈ 0. That is, there is a horizontal asymptote at f(t)=0 for large negative values of t.
For t → ∞, e^-t → 0 and the value of the rational expression becomes approximately 5/2. That is, there is a horizontal asymptote at f(t) = 5/2 for large positive values of t.
Essentially, the curve is "S" shaped, with a smooth transition between 0 and 5/2 for values of t that make 3e^-t have values within an order of magnitude of the other term in the denominator, 2.
At t=0, 3e^-t = 1 and the denominator is 2+3=5. That is, f(0) = 5/5 = 1. Of course, the curve will cross the line f(t) = 5/4 (halfway between the asymptotes) when 3e^-t = 2, or t=ln(3/2)≈0.405. The curve is symmetrical about that point.
You can sketch the graph by finding values of t that give you points on the transition. Typically, you would choose t such that 3e^-t will be some fraction or multiple of 2, say 1/10, 1/3, 1/2, 1, 2, 3, 10 times 2.
___
f(t) is called a "logistic function." It models a situation where growth rate is proportional both to population size and the difference between population size and carrying capacity. In public health terms, it models the spread of disease when that is proportional to the number of people exposed and to the number not yet exposed.
NEED HELP ANSWERING ASAP !!
Answer:
sorry can you write it down and take a pic of it
please that is the only way i can answer it
Step-by-step explanation: