Use spherical coordinates.
Find the volume of the part of the ball rho ≤ 10 that lies between the cones
ϕ = π/6
and
ϕ = π/3.
Answers
Volume ≈ 289.667 × π cubic units of the part of the ball rho ≤ 10 that lies between the cones.
To find the volume of the part of the ball ρ ≤ 10 between the cones ϕ = π/6 and ϕ = π/3 using spherical coordinates, we need to use the triple integral in spherical coordinates:
Volume = ∫∫∫ (ρ^2 sinϕ) dρ dθ dϕ
We have the limits of integration as:
ρ: 0 to 10
θ: 0 to 2π
ϕ: π/6 to π/3
Now, we can set up the integral:
Volume = ∫(0 to 10)∫(0 to 2π)∫(π/6 to π/3) (ρ^2 sinϕ) dρ dθ dϕ
Evaluating the integral, we get:
Volume = (1/3)(1000)(2π)[cos(π/6) - cos(π/3)]
Volume ≈ 289.667 * π cubic units
Using spherical coordinates, find the volume of the part of the ball rho ≤ 10 that lies between the cones ϕ = π/6 and ϕ = π/3.The volume of the ball whose radius is rho = 10 can be calculated as:V = (4/3)πr³where r = 10, so V = (4/3)π(10)³ = 4000π/3 cubic units. The problem restricts the region of the ball by the cones with the following limits: ϕ
= π/6 to ϕ = π/3.The limits of ϕ are from π/6 to π/3, and the limits of ρ are from 0 to 10. When the three limits are combined and the volume integral is set up, the volume of the restricted region is obtained as follows:
V = ∫∫∫ f (ρ, ϕ, θ) ρ² sin ϕ dρ dϕ dθwhere f (ρ, ϕ, θ)
= 1ρ³ V
= ∫(π/6)^(π/3)∫0^10∫0^2π 1ρ³ ρ² sin ϕ dθ dρ dϕV
= ∫(π/6)^(π/3)∫0^10 2π [1/ρ] sin ϕ dρ dϕV
= 2π ∫(π/6)^(π/3) sin ϕ [ln 10 - ln 0] dϕV
= 2π (ln 10 - ln 1/2) ∫(π/6)^(π/3) sin ϕ dϕV
= 2π (ln 10 - ln 1/2) [-cos π/3 + cos π/6]V
= π/3 (ln 10 - ln 1/2) (1 + √3) cubic units or approximately 45.082 cubic units.
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Related Questions
PLEASE HELP .!!! ILL GIVE BRAINLIEST.. *EXRTA POINTS* .. DONT SKIP :(( !
ILL GIVE 40 POINTS .
Answers
Answer:
d
Step-by-step explanation:
if you go up by 4 years it goes up by 200 therefore dividing that by 4 (to get one year) would result in 50 dollars
Answer:
The Answer is A
Step-by-step explanation: All you have to do is calculate how far the dotted line is from beginning point
Draw a line representing the "rise" and a line representing the "run" of the line. State the slope of the line in simplest form.
Answers
Answer:
Rise is vertical, run is horizontal
Step-by-step explanation:
a sample of the salaries of assistant professors on the business faculty at a local university revealed a mean income of $124,000 with a standard deviation of $11,800. assume that salaries follow a bell-shaped distribution. use the empirical rule to answer the following questions. approximately what percentage of the salaries fall between $112,200 and $135,800? approximately what percentage of the salaries fall between $100,400 and $147,600? approximately what percentage of the salaries are greater than $147,600?
Answers
Approximately 68% of salaries fall between $112,200 and $135,800, 95% between $100,400 and $147,600, and less than 2.5% are greater than $147,600.
The empirical rule, also known as the 68-95-99.7 rule, states that in a bell-shaped distribution, approximately 68% of the data falls within one standard deviation of the mean, approximately 95% within two standard deviations, and approximately 99.7% within three standard deviations.
Using this rule, we can answer the following questions:
Approximately 68% of the salaries fall between $112,200 and $135,800, which is one standard deviation below and above the mean, respectively.Approximately 95% of the salaries fall between $100,400 and $147,600, which is two standard deviations below and above the mean, respectively.Since $147,600 is more than two standard deviations above the mean, we can estimate that approximately 2.5% of the salaries are greater than $147,600.Learn more about mean :
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Express 29 out of 40 as a percentage
Answers
Answer:
72.5%
Step-by-step explanation:
To express a value as a %:
value/whole x 100 = 29 / 40 x 100
This gives you 72.5%
Hope this helps
The rate at which a body cools also depends on its exposed surface area S. If S is a constant, then a modification of (2) from page 86 (section 3.1) is dT/dt=kS(T−Tm ) where k<0 and Tm is a constant. Suppose that two cups A and B are filled with coffee at the same time. Initially, the temperature of the coffee is 150∘F. The exposed surface area of the coffee in cup B is twice the surface area of the coffee in cup A. After 30 min the temperature of the coffee in cup A is 100∘F. If Tm=70∘, then what is the temperature of the coffee in cup B after 30 min ?
Answers
The temperature of coffee cup B after 30 mins is 81.25° F .
Given,
T (0) = 150 0 F
Tm = 70 0 F
Surface area of Coffee in cup B is twice the surface area of Coffee in cup A.
Now,
Let exposed surface area of Coffee in cup A be S' then surface area of coffee in cup B is 2S'.
Consider the equation:
\(\frac{dT}{dt} = kS(T-T_{m})\)
\(\frac{dT}{dt} = kS(T-T_{m})\\ \frac{dT}{dt} = kS(T-70)\\ \int \frac{dT}{(T-70)} = kS dt\\ log (T -70) = kSt + C\)
at t = 0, T = 150
log (150 -70) = C
log 80 = C
Substituting this in the main equation we get,
log(T - 70 ) = kSt + log 80
log (T - 70) - log 80= kSt
log [(T - 70)/80] = kSt
Now in case of Cup A , we shall first find the constant k
log [(T - 70)/80] = k S't
T(30) = 100
log[(100-70)/80] = k S' 30
\(k = \frac{log (3/8)}{30S'}\)
Equation for cup B is
log[(T-70)/80] = k 2S' t
After 30 min
\(log \frac{ (T-70)}{80'} =\frac{ log( 3/8 )}{30 S '} * 2S' * 30\)
\(log \frac{ (T-70)}{80'} =2[log (3/8)]\\ log \frac{ (T-70)}{80'} = 2log \frac{3}{8}\\ log \frac{ (T-70)}{80'} = log (\frac {3}{8} )^{2}\\ log \frac{ (T-70)}{80'} = log ( \frac {9}{64})\\\)
Taking antilog on both the sides we get
\(\frac{(T-70)}{80'} = \frac {9}{64}\\ T - 70 = \frac {9}{64}* 80 = \frac {45}{4} = 11.25\\ T = 11.25 + 70 = 81.25\)
Thus temperature of Coffee in cup B is 81.25° F .
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LAST TIME TODAY BUT 5TH GRADE MATH
Answers
Answer: 1) 109.5 or 109 1.2
2) 136 r2
Step-by-step explanation:
Answer : 109 1/2 and 136 R 2
Step - by - step explanation :
First, convert the improper fraction into a mixed number . . . what is 438 divided by 4 ? 109.5 or 109 1/2 !
Second, divide . . . what is 1634 divided by 12 ? 136 with a remainder of 2 !
( sorry if any of these are wrong )
Consider a transformation that takes any vector in R^2 as its input and always outputs the vector [2 1]. Give one (and only one) reason why this transformation is not a linear transformation.
Answers
The given transformation is not a linear transformation because it violates the property of homogeneity.
For a transformation to be linear, it must satisfy two properties: homogeneity and additivity. Homogeneity states that if we scale the input vector by a scalar k, then the output vector should also be scaled by the same k. In other words, T(kv) = kT(v), where T is the transformation, v is the input vector, and k is a scalar.
Let's consider the given transformation that always outputs the vector [2 1] for any input vector in R^2. Now, let's take an arbitrary vector v = [x y] in R^2 and scale it by a scalar k. Therefore, kv = [kx ky].
If we apply the transformation T to kv, we get T(kv) = T([kx ky]) = [2 1].
On the other hand, if we first apply the transformation T to v and then scale it by k, we get kT(v) = kT([x y]) = k[2 1] = [2k k].
Since [2 1] is not equal to [2k k], the homogeneity property is not satisfied. Therefore, the given transformation is not a linear transformation.
Therefore, the given transformation violates the homogeneity property, and it is not a linear transformation.
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g The sides of a rhombus are 12 units long, and one of its angles has a measure of 60 degrees. How long is the other diagonal
Answers
The length of the other diagonal of the rhombus is 24 units.
A rhombus is a parallelogram with four equal sides. Since the sides of the rhombus are 12 units long, all the sides are equal in length. One of the angles of the rhombus measures 60 degrees.
In a rhombus, the diagonals bisect each other at right angles, dividing the rhombus into four congruent right-angled triangles. The angle between the two diagonals is 60 degrees, which means that each right-angled triangle in the rhombus has a 30-degree angle.
To find the length of the other diagonal, we can use trigonometry. In a right-angled triangle with a 30-degree angle, the ratio of the length of the side opposite the angle to the length of the hypotenuse is 1/2. In this case, the side opposite the 30-degree angle is half the length of the diagonal we are trying to find.
Let's denote the length of the other diagonal as "d". Using trigonometry, we can set up the following equation:
sin(30 degrees) = (d/2) / 12
Simplifying the equation, we have:
1/2 = (d/2) / 12
Cross-multiplying, we get:
d/2 = 12 * 1/2
d/2 = 6
Multiplying both sides by 2, we find:
d = 12
Therefore, the length of the other diagonal of the rhombus is 24 units.
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factors other than the independent variable may also act on the dependent variable. if these factors vary systematically with the independent variable, they are called:
Answers
Factors other than independent variables which may act on the dependent variable , when these factors vary systematically with independent variable they are known as confounds variables or extraneous variable.
Factor which is other than the independent variable and might produce an effect in the given experiment.Manipulation of independent variable which has an effect on the expressions dependent variable.Other variables which has impact on changing the dependent variables are known as extraneous variable.Extraneous variables are also known as confounding variables.Confounding variables or extraneous variables these are the factors which does not manipulate and are part of the given experiment.Therefore, factors which vary systematically with the given independent variable is called confounds variables or extraneous variable.
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Roberta works in a home appliance store. She is paid $195 per week and 3% on all sales over $1,000. During one year, her average weekly sales were $7,800. About how much did she earn weekly?
Format your answer with a dollar sign and comma (ex. $10,000).
Answers
Answer:
300,750
Step-by-step explanation:
Roberta earn $429 weekly.
What is percentage?Percentage, a relative value indicating hundredth parts of any quantity. One percent (symbolized 1%) is a hundredth part; thus, 100 percent represents the entirety and 200 percent specifies twice the given quantity.
Given:
Roberts paid = $195
percent rate = 3%
she will earn weekly
= 195 + 7800 x 3/100
=195 + 7800 x 0.03
= 429
Hence, she will earn $429 weekly.
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Was the World Trade Center destroyed by a nuclear explosion?
Answers
Answer:
The answer is no it was destroyed by terrorists on 09/11/01
A tent is in the shape of the picture shown below, calculate the total of the tent, ROUND TO THE NEAREST HUNDREDTH
Answers
Answer: The volume of the tent is 293.33 ft.
Explanation: 11 x 10 x 8 = 880
The formula for the volume of a right rectangular pyramid is lwh/3.
880/3 = approximately 293.33
13 - 8m = 21
por favor ayuda
Answers
13-8m=21
8m=8
M=1
What is the equation of the line?
Answers
Answer:
y=0
Step-by-step explanation:
What is the value of -2 to the power of 4
Answers
We can find the value if we write the complete expression:
\((-2)^4=(-2)\cdot(-2)\cdot(-2)\cdot(-2)\)Considering only two factors at a time, and knowing that - * - = +, we have:
\((-2)\cdot(-2)\cdot(-2)\cdot(-2)=4\cdot4=16\)Therefore, (-2)^4 = 16
Answer:
-16!!
-2x-2x-2x-2= -16.
a die is tossed 180 times with the following results: x 1 2 3 4 5 6 f 28 36 36 30 27 23 is this a balanced die? use a 0.01 level of significance
Answers
Based on the chi-square test, there is no significant evidence to suggest that the die is not balanced.
We have,
To determine if the die is balanced, we can perform a chi-square test of goodness of fit.
The null hypothesis is that the die is balanced, and the alternative hypothesis is that the die is not balanced.
First, let's calculate the expected frequencies for each outcome assuming the die is balanced. Since there are 180 tosses in total, each outcome is expected to have an equal probability of 1/6.
Expected frequency for each outcome
= (Total tosses) x (Probability of each outcome)
Expected frequency for each outcome = (180) x (1/6)
Expected frequency for each outcome = 30
Now, we can calculate the chi-square test statistic using the formula:
χ² = Σ [(Observed frequency - Expected frequency)² / Expected frequency]
Let's calculate the chi-square test statistic:
χ² = [(28 - 30)² / 30] + [(36 - 30)² / 30] + [(36 - 30)² / 30] + [(30 - 30)² / 30] + [(27 - 30)² / 30] + [(23 - 30)² / 30]
χ² = [(-2)² / 30] + [(6)² / 30] + [(6)² / 30] + [(0)² / 30] + [(-3)² / 30] + [(7)² / 30]
χ² = 4/30 + 36/30 + 36/30 + 0/30 + 9/30 + 49/30
χ² = 134/30
χ² ≈ 4.467
Next, we need to compare the calculated chi-square value to the critical chi-square value at a significance level of 0.01 and degrees of freedom equal to the number of outcomes minus 1 (6 - 1 = 5).
Looking up the critical chi-square value in a chi-square distribution table with 5 degrees of freedom and a significance level of 0.01, we find it to be approximately 15.086.
Since the calculated chi-square value (4.467) is less than the critical chi-square value (15.086), we fail to reject the null hypothesis.
Therefore, we do not have sufficient evidence to conclude that the die is not balanced at a 0.01 level of significance.
Thus,
Based on the chi-square test, there is no significant evidence to suggest that the die is not balanced.
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Add.
(6x³ + 3x² − 2) + (x³ - 5x² − 3)
Express the answer in standard form. (Please and thank you)
Answers
Answer:
\(\\\sf7x^3 - 2x^2 - 5\)
Step-by-step explanation:
\(\\\sf(6x^3 + 3x^2 - 2) + (x^3 - 5x^2 - 3)\)
Remove parenthesis.
6x^3 + 3x^2 - 2 + x^3 - 5x^2 - 3
Rearrange:
6x^3 + x^3 + 3x^2 - 5x^2 - 2 - 3
Combine like terms to get:
7x^3 - 2x^2 - 5----------------------------------------
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Hope this helps! :)
Answer:
7x³ - 2x² - 5
Step-by-step explanation:
(6x³ + 3x² - 2) + (x³ - 5x² - 3)
Remove the round brackets.
= 6x³ + 3x² - 2 + x³ - 5x² - 3
Put like terms together.
= 6x³ + x³ + 3x² - 5x² - 2 - 3
Do the operations.
= 7x³ - 2x² - 5
____________
hope this helps!
The records at a particular two-year college classify a student as a freshman, a sopho- more, a student that has transferred out, or a student that has graduated. Each year 25% of students transfer, 15% have to repeat the year, and 60% complete the year. Once a student has transferred or graduated they have a 100% chance of staying clas- sified that way. Currently there are 2,000 freshmen, 1,500 sophomores, 500 students that have transferred, and 3,000 graduates. Determine the number of students in each category one year from now. You may use a calculator for any computations in this problem only.
Answers
One year from now, there are 300 freshmen, 1425 sophomores, 1375 students that transferred out, and 3900 students graduated.
Each category is based on the percentages given, 25% of students transfer, 15% have to repeat the year, and 60% complete the year and the fact that once a student has transferred or graduated they have a 100% chance of staying classified that way.
Percentage, which was adapted from the Latin word "per centum", which means "by the hundred", is a fraction or a ratio in which the value of whole is always 100. It is represented by the symbol "%".
If currently there are 2,000 freshmen, 1,500 sophomores, 500 students that have transferred, and 3,000 graduates,
Among the 2000 freshmen, 25% of students transfer, 15% have to repeat the year, and 60% complete the year.
students who transfer = 25% of 2000 = 500
freshmen(repeater) = 15% of 2000 = 300
sophomore = 60% of 2000 = 1200
Among the 1,500 sophomores, 25% of students transfer, 15% have to repeat the year, and 60% complete the year.
students who transfer = 25% of 1500 = 375
sophomore(repeater) = 15% of 1500 = 225
students who graduated = 60% of 1500 = 900
Adding the total number of sophomores(freshmen who completed the year and sophomores that have to repeat the year)
sophomores = 1200 + 225 = 1425
Adding the total number of students who transfer,
500 + 500 + 375 = 1375
Adding the total number of student that graduated,
3000 + 900 = 3900
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You pay $3 per square foot for carpeting. Which equation can be solved to find the
amount you will pay to cover a triangular section of the floor with a base of 2 yards and
a height of 5 yards?
Answers
Answer:
$135
Step-by-step explanation:
Step 1
Conversion
Base = 2 yards
1 yard = 3 feet
2 yards = x
Cross Multiply
x = 2 × 3 feet = 6 feet
Height = 5 yards
1 yard = 3 feet
5 yards = x
Cross Multiply
x = 5 × 3 feet = 15 feet
Step 2
Find the area of the triangle
Area of a triangle = 1/2 × base × height
Base = 6 feet
Height = 15 feet
Area = 1/2 × 6 × 15
= 45 square feet
Step 3
Find the cost of the carpet
1 Square foot = $3
45 square feet = x
Cross Multiply
x = 45 × $3
x = $135
Therefore, the amount you will pay to cover a triangular section of the floor is $135
Darryl is finding the product of -5y(-4y^3+6y^2-5) Is he correct? If not, what changes should be made?
Answers
A. Yes, Darryl’s work is correct.
Step-by-step explanation:
-5y(-4y^3 + 6y^2 -5)
Darryl used distributive property
Where the outside number (-5y) is being multiplied by all of the numbers in parentheses
-5y • -4y^3 = 20y^4
Two negative numbers multiplied makes a positive number and an extra y on -5y makes y^4
-5y • 6y^2 = -30y^3
Since -5y is being multiplied by a positive number, the product becomes negative. Extra y^3 comes from the y in -5
-5y • -5 = 25y
Two negatives are being multiplied causing the product to be a positive number and keeping the variable the same value.
Finally, add all of the results!
20y^4 + -30y^3 + 25y =
20y^4 - 30y^3 + 25y
An iphone costs $1000 and is on sale for 20% off
Answers
In AHIJ, HJ is extended through point J to point K, mZIJK = (7x – 15)°,
mZJHI
(3x + 18), and mZHIJ (x + 6)". Find mZJHI.
Answers
Answer:
3x + 18), and mZHIJ (x + 6)". Find mZJHI.
Step-by-step explanation:
it is 86[9=10+3]
E allowed
A metal sculpture has a total volume of 1250 cm³ and a mass of 8.1 kg.
Work out its density, in grams per cubic centimetre (g/cm³).
Give your answer to 2 d.p.
Answers
The density is 6.48 (g/cm³).
What is density?Density (vοlumetric mass density οr specific mass) is the substance's mass per unit οf vοlume. The symbοl mοst οften used fοr density is ρ (the lοwer case Greek letter rhο), althοugh the Latin letter D can alsο be used. Mathematically, density is defined as mass divided by vοlume:
\({\displaystyle \rho ={\frac {m}{V}}}\)
Here the given Vοlume = 1250 cm³, Mass = 8.1 kg.
We knοw that 1kg = 1000 g. Then,
Mass = 8.1*1000 = 8100 gram.
Nοw density = Mass/ Vοlume
=> Density = \(\frac{8100}{1250}\) = 6.48 (g/\(cm^3\))
Hence the density is 6.48 (g/\(cm^3\)).
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The Simple Linear Regression Analysis For The Home Price (Y) Vs. Home Size (X) Is Given Below. Regression Summary Price = 97996.5 + 66.445 Size R^2= 51% T-Test For (Beta) 1 (Slope): TS= 14.21, P<0.001 95% Confidence Interval For Beta1 (Slope) (57.2, 75.7) 1. Use The Equation Above To Predict The Sale Price Of A House That Is 2000 Sq Ft A. $190,334 B.
The simple linear regression analysis for the home price (y) vs. home size (x) is given below.
Regression summary
Price = 97996.5 + 66.445 size
R^2= 51%
t-test for (beta) 1 (slope): TS= 14.21, p<0.001
95% confidence interval for beta1 (slope) (57.2, 75.7)
1. Use the equation above to predict the sale price of a house that is 2000 sq ft
A. $190,334
B. $97996.50
C. $660,445
D. $230,887
Answers
The predicted sale price of a house that is 2000 sq ft is $230,887. The simple linear regression analysis shows that there is a significant linear relationship between the sale price and the size of a house.
Simple linear regression analysis is a statistical tool that is used to study the relationship between two variables. It involves determining the equation of a straight line that best fits the data points on a scatter plot. This line is known as the regression line, and it is used to predict the value of the dependent variable (y) for a given value of the independent variable (x). In this case, we are interested in predicting the sale price (y) of a house based on its size (x).
The equation of the regression line is given by Price = 97996.5 + 66.445 size. Given a home size of 2000 square feet, we can use this equation to predict the sale price of the house. The predicted sale price is obtained by plugging in the value of 2000 square feet for size in the equation. This gives us:
Price = 97996.5 + 66.445 × 2000
Price = 97996.5 + 132890
Price = 230886.5
Therefore, the predicted sale price of a house that is 2000 sq ft is $230,887.
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Sandra is asked to report on the recent increase in popularity of an application based on the number
of times it is downloaded. Which best describes how Sandra should report this information?
downloads per day
total number of days with downloads
downloads per person
total number of downloads
Answers
Answer:
Step-by-step explanation:
Report the number of downloads per day. By determining how this number is changing / increasing it will be possible to spot and describe an upward trend if there is one.
Solve for r. r - 76 = 9
Answers
Answer:
r=85
Step-by-step explanation:
r-76=9
add 76 to both sides
r=85
hope this is helpful
Answer: 85
Step-by-step explanation:
Since we're finding the placeholder of r and we're already subtracting, we need to add! What we need to do is add 76 + 9 which equals 85!
Hope this helps you! Good luck with your future work!
A card is drawn from a standard deck, put aside, and then another card is drawn. What is the probability that the first card is a king and that the second card is a queen?
FRACTION FORM
Answers
Answer:
4/663
Step-by-step explanation:
Since the first card is set aside after being drawn, this is dependent probability.
There are 4 kings in 1 deck of 52 cards.
Therefore the probability of choosing a king is 4/52 = 1/13.
But after choosing 1 card from the deck, there are now 51 cards in the deck. There are 4 queens in the deck of 51 cards. Therefore the probability of choosing a queen is 4/51.
To determine the probability of choosing both, multiply the 2 probabilities.
\(\frac{1}{13} * \frac{4}{51} = \frac{4}{663}\)
Answer:
1/13
Step-by-step explanation:
once you simplify and finish that is the simplified answer
The graph of a function is shown below. What is its range?
O (1, 2, 4)
O (1, 2, 3, 5)
O All real numbers.
O (1, 2, 3, 4)
Answers
(1,2,4)
Step-by-step explanation:Range describes the y-values of a graph.
Range
Range is the y-values that a graph covers. Remember that the y-values are found on the vertical axis. If the graph is not continuous, then the values between the points are not included in the range. Similar to the range, the domain of a graph is the x-values that a graph covers. If there is a coordinate point with a y-value, then that y-value should be included in the range.
Finding Range
In order to find the range, we need to find all the unique y-values of the graph. Additionally, the range is given in numerical order. This means starting from the least value and going up to the greatest. The lowest y-value is 1, then 2, and finally 4. Even though there are two points where y = 2, we are only looking for unique values. This means that the range is (1,2,4).
i need help pls..........
Answers
Answer:
a 14.3
Step-by-step explanation:
23 divided by 4 is 14.3
Solve . What is the value of x? There is no real number solution for x.
Answers
Answer:
There is no problem.
Step-by-step explanation:
Answer:
1/6 on ed
Step-by-step explanation: