When asked to find the sum of two mixed numbers with different denominators, you should find a (C) common denominator before you add.
To add fractions with different denominators, it is necessary to find a common denominator for both fractions. Once the fractions have the same denominator, you can add or subtract the numerators while keeping the denominator unchanged. This allows for a meaningful addition of the fractions.
To find a common denominator, you need to determine the least common multiple (LCM) of the denominators. The LCM is the smallest multiple that both denominators can divide evenly into. Once you have the common denominator, you can convert each fraction to an equivalent fraction with that denominator and then perform the addition.
After finding the common denominator, you add the numerators of the fractions and keep the common denominator. This will give you the sum of the two mixed numbers with different denominators. The correct answer is C. find a common denominator.
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La relación del aspecto de una pantalla o la relación entre el ancho y alto de una televisión es de 16:9. El tamaño de una TV está dado por la distancia diagonal de la TV, si se sabe que una HDTV tiene 41 pulgadas de ancho, determina el tamaño de la pantalla.
Answer:
[tex]23\frac{1}{16}[/tex] pulgada
Step-by-step explanation:
[tex]\frac{16}{9} =\frac{41}{y}[/tex]
16 × y = 9 × 41
16y = 369
16y ÷ 16 = 369 ÷ 16
[tex]y=23\frac{1}{16}[/tex]
- 100 points -
Use synthetic division to completely factor:
y= x^3 + 3x^2 - 13x - 15 by x + 5
A - y = (x+5)(x+3)(x-1)
B - y = (x+5)(x+3)(x+1)
C - y = (x+5)(x-3)(x-1)
D - y = (x+5)(x-3)(x+1)
Answer:
B
Step-by-step explanation:
D) - y = (x + 5)(x - 3)(x + 1).
EXPLANATION:Table in this case would look like this:
Write coefficients of x³, x² ,x and the constant in a row and divisor would be the x value obtained by equation x + 5 = 0.
The sequence of multiplications would be as shown in picture.
x² - 2x - 3
x² - 3x + x - 3
x(x - 3) + 1(x - 3)
(x + 1)(x - 3)(x + 5).
Jamar to the local snack shop near his school He bought 3 hotdogs and 2 bags of chips for $Kenny went to the same snack shop and bought 5 hotdogs and 6 bags of chips for $9.55 ordered 2 hotdogs and 3 bags of chips, then how much did she pay her order?
Answer:
Marcy paid $4.15
Step-by-step explanation:
Given
Represent hotdogs with x and chips with y.
So, we have:
Jamal
[tex]3x + 2y = 4.85[/tex]
Kenny
[tex]5x + 6y = 9.55[/tex]
See attachment for complete question
Required
Determine the amount for 2x and 3y
From Jamal's and Kenny's orders we have:
[tex]3x + 2y = 4.85[/tex] --- (1)
[tex]5x + 6y = 9.55[/tex] --- (2)
Multiply (1) by 3
[tex]3 * [3x + 2y = 4.85][/tex]
[tex]9x + 6y = 14.55[/tex] --- (3)
Subtract (2) and (3)
[tex]9x - 5x + 6y - 6y = 14.55 - 9.55[/tex]
[tex]9x - 5x = 5[/tex]
[tex]4x = 5[/tex]
Solve for x
[tex]x = \frac{5}{4}[/tex]
[tex]x = 1.25[/tex]
Substitute [tex]x = 1.25[/tex] in [tex]3x + 2y = 4.85[/tex]
[tex]3 * 1.25 + 2y = 4.85[/tex]
[tex]3.75 + 2y = 4.85[/tex]
Solve for y
[tex]y = \frac{4.85 - 3.75}{2}[/tex]
[tex]y = \frac{1.10}{2}[/tex]
[tex]y = 0.55[/tex]
So, the cost of 2x and 3y is:
[tex]Cost = 2x + 3y[/tex]
[tex]Cost = 2*1.25 + 3*0.55[/tex]
[tex]Cost = \$4.15[/tex]
A tank contains 120 liters of oil initially. Oil is being pumped out of the tank at a rate R(t), where R(t) is measured in gallons per hour, and t is measured in hours. The table below shows selected values for R(t). Using a trapezoidal approximation with three subintervals and the data from the table, find an estimate of the number of gallons of oil that are in the tank at time t = 14 hours. t (hours) 2 5 10 14 R(t) gallons per hour 8.2 7.8 8.6 9.3 A. 220.8 В. 19.2 C. 100.8 D. 18.75
The estimate of the number of gallons of oil in the tank at t = 14 hours is 100.8 gallons. The correct answer is option C.
To estimate the number of gallons of oil in the tank at t = 14 hours using a trapezoidal approximation,
we need to calculate the total change in oil volume over the given time period.
The trapezoidal approximation involves dividing the time interval into subintervals and approximating the change in volume as the sum of trapezoidal areas.
Let's calculate the approximate volume of oil at t = 14 hours using the given data and the trapezoidal approximation: Interval 1 (2 to 5 hours):
Average rate = (R(2) + R(5)) / 2 = (8.2 + 7.8) / 2 = 16 / 2 = 8 gallons per hour.
Volume change =
[tex]Average rate \times time = 8 \times (5 - 2)[/tex]
= 24 gallons.
Interval 2 (5 to 10 hours):
Average rate = (R(5) + R(10)) / 2 = (7.8 + 8.6) / 2 = 16.4 / 2 = 8.2 gallons per hour
Volume change =
[tex]Average rate \times time = 8.2 \times (10 - 5) [/tex]
= 41 gallons
Interval 3 (10 to 14 hours):
Average rate = (R(10) + R(14)) / 2 = (8.6 + 9.3) / 2 = 17.9 / 2 = 8.95 gallons per hour
Volume change =
[tex]Average rate \times time = 8.95 \times (14 - 10)[/tex]
= 35.8 gallons.
Total volume change = Interval 1 + Interval 2 + Interval 3 = 24 + 41 + 35.8 = 100.8 gallons.
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Because of the commutative property of multiplication, it is true that
3
4
×
4
=
4
×
3
4
. However, these
expressions can be calculated in different ways even though the solutions will be the same.
Below, show two different ways of solving this problem.
Answer:
30 x 4 + 4 x 4 and 4 x 4 + 4 x 30
Step-by-step explanation:
Need help please. Thanks!
Answer:
Option 1: 4h
Step-by-step explanation:
perimeter of square = 4 × length
h × 4 = 4h
Help me please it’s due today
Answer:
1. B. $20
2.C. 19%
3. D. not here
Step-by-step explanation:
1. info we know
25+45+10+25+15+10 = 130
we need 20 more dollars
2. 25/130 x/100
cross multiply
2500/ 130x
divide
19 = x
3. same steps as last time
35/130 x/100
3500/130x
29%
Please please help help me please ASAP with one question
No links or files
Answer:
D
Step-by-step explanation:
A and C are what we've always been taught so it's not them. B is just side time side so with a square that works. D is the only one what I'm not sure of or doesn't seem familiar. So personally I'd say D. Good luck!
Answer:
D
explanation
a finds area of rectangle
b area of square
c of rhombus
Select the correct answer. 70PTS!!!!!!!!!!!!!!
Which statement correctly compares functions f and g?
function f function g
Function g is a decreasing exponential function with a y-intercept of 5 and no x-intercept.
A.
They have different end behavior as x approaches -∞ but the same end behavior as x approaches ∞.
B.
They have different end behavior as x approaches -∞ and different end behavior as x approaches ∞.
C.
They have the same end behavior as x approaches -∞ but different end behavior as x approaches ∞.
D.
They have the same end behavior as x approaches -∞ and the same end behavior as x approaches ∞.
They have different end behavior as x approaches -∞ but the same end behavior as x approaches ∞. Option A is correct.
Given that,
Function g is a decreasing exponential function with a y-intercept of 5 and no x-intercept.
Functions are the relationship between sets of values. e g y=f(x), for every value of x there is its exists in a set of y. x is the independent variable while Y is the dependent variable.
What is exponential function?The function which is in format f(x) = [tex]e^X[/tex] where, a is constant and x is variable, the domain of this exponential function lies (-∞, ∞).
Here, the solution shown in the graph implies that they have different end behavior as x approaches -∞ but the same end behavior as x approaches ∞.
Thus, they have different end behavior as x approaches -∞ but the same end behavior as x approaches ∞. Option A is correct.
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Students set a goal for the
number of cans to collect
for the canned food drive.
They reached 120% of
their goal. What is 120%
expressed as a fraction
and as a decimal?
Answer:
Fraction = 120/100 | Decimal = 1.2
Step-by-step explanation:
Fraction:
100% is 100/100 but 120% is 20% over so the fraction is now 120/100 (20% = 20/100)
Decimal:
100% is 1 but as I said earlier, 120% is 20% over so the decimal is now 1.2 (20% = 0.2)
What is the theoretical probability of rolling a 3 on a die?
Answer:
1/6 or 16.67% chance
Step-by-step explanation:
Assuming its a normal six sided dice. You would have a 16.67% chance of getting a 3.
A survey conducted by Sallic Mae and Gallup of 1404 respondents found that 323 students paid for their education by student loans. Find the 90% confidence of the true proportion of students who paid for their education by student loans.
Fifty randomly selected people were asked how long they slept at night. The mean time was 7.1 hours, and the standard deviation was 0.78 hour. Find 95% confidence interval of the mean time. Assume the variable is normally distributed.
For a medical study, a researcher wishes to select people in the middle 60% of the population based on blood pressure. If the mean systolic blood pressure is 120 and the standard deviation is 8, find the upper and lower reading that would qualify people to participate in the study.
The upper and lower readings that would qualify people to participate in the medical study are 113.48 and 126.72 respectively.
Firstly, we'll calculate the 90% confidence interval of the proportion of students who paid for their education by student loans.
For this, we will use the formula:
[tex]$$\hat{p}\pm z\left(\frac{\sqrt{\hat{p}(1-\hat{p})}}{n}\right)$$[/tex]
Where, [tex]$\hat{p}$[/tex] is the sample proportion, n is the sample size, z is the z-score for the level of confidence,
[tex]$1-\alpha$[/tex]
For 90% confidence, the z-score is 1.645 (because the table value of z-score is 1.645 at 90% confidence level).
[tex]\hat{p}=\frac{323}{1404}$$[/tex]
[tex]\hat{p}=0.23$$[/tex]
[tex]\text{Standard Error}= \left(\frac{\sqrt{\hat{p}(1-\hat{p})}}{n}\right)$$[/tex]
[tex]\text{Standard Error}= \left(\frac{\sqrt{0.23(1-0.23)}}{1404}\right)$$[/tex]
[tex]\text{Standard Error}= 0.014$$[/tex]
[tex]\text{Confidence Interval} = \hat{p}\pm z \times \text{Standard Error}$$[/tex]
[tex]\text{Confidence Interval}= 0.23\pm 1.645(0.014)$$[/tex]
[tex]\text{Confidence Interval}= 0.23\pm 0.023$$[/tex]
Confidence Interval = [0.207,0.253]
The 90% confidence interval of the proportion of students who paid for their education by student loans is [0.207,0.253].
Now we will calculate the 95% confidence interval of the mean time.
For this, we will use the formula:
[tex]\bar{X}\pm z\left(\frac{\sigma}{\sqrt{n}}\right)$$[/tex]
Where, [tex]$\bar{X}$[/tex] is sample mean, [tex]$\sigma$[/tex] is population standard deviation, n is sample size, z is the z-score for the level of confidence, [tex]$1-\alpha$[/tex]
For 95% confidence, the z-score is 1.96.
(because the table value of z-score is 1.96 at 95% confidence level).
[tex]\text{Confidence Interval}= \bar{X}\pm z\left(\frac{\sigma}{\sqrt{n}}\right)$$[/tex]
[tex]\text{Confidence Interval}= 7.1\pm 1.96\left(\frac{0.78}{\sqrt{50}}\right)$$[/tex]
[tex]\text{Confidence Interval}= 7.1\pm 0.2199$$[/tex]
Confidence Interval = [6.8801, 7.3199]
The 95% confidence interval of the mean time is [6.8801, 7.3199].
Next, we will find the upper and lower reading that would qualify people to participate in the medical study.
We can do this by calculating the z-scores for the upper and lower percentiles using the standard normal distribution table.
For the lower reading: Since we want to select people in the middle 60% of the population, the lower reading will correspond to the 20th percentile.
Using the standard normal distribution table, we find that the z-score for the 20th percentile is -0.84.
Using the z-score formula, we have:
[tex]z = \frac{x - \mu}{\sigma}$$[/tex]
where, x is the lower reading.
Substituting the given values, we get:-
0.84 = (x - 120) / 8
Solving for x, we get:
[tex]x = (-0.84 \times 8) + 120$$[/tex]
x = 113.48
The lower reading that would qualify people to participate in the medical study is 113.48.
For the upper reading: Since we want to select people in the middle 60% of the population, the upper reading will correspond to the 80th percentile.
Using the standard normal distribution table, we find that the z-score for the 80th percentile is 0.84.
Using the z-score formula, we have:
[tex]z = \frac{x - \mu}{\sigma}$$[/tex]
where, x is the upper reading.
Substituting the given values, we get:
0.84 = (x - 120) / 8
Solving for x, we get:
[tex]x = (0.84 \times 8) + 120$$[/tex]
x = 126.72
The upper reading that would qualify people to participate in the medical study is 126.72.
Therefore, the upper and lower readings that would qualify people to participate in the medical study are 113.48 and 126.72 respectively.
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Marsha gave the cashier $20 to pay for 3 pairs of socks. The cashier gave her $5.03 in change. Each pair of socks costs the same amount.
What is the cost in dollars and centers for each pair of socks?
Help plssss
Answer:
$4.99
Step-by-step explanation:
20 - 5.03 = 14.97
Since that is how much was paid you then divide by how many socks were bought
14.97 ÷ 3 = 4.99
Answer:
4.99/pair
Step-by-step explanation:
Solve
$20 - $5.03 = $14.97 substract
$14.97 ÷ 3 = $4.99 divide
20 - 5.03 = 14.97 = total cost. 14.97÷3 = 4.99/pair.
Therefore, the cost in dollars and cents for each pair of socks is 4.99/pair.
please help me! I'd be filled with so much gratitude
Answer:
0
Step-by-step explanation:
-9 x (0/-3)
-9 x 0
0
The probability of an event happening is 23%. What is the complement of the event?
Answer:The probability of the complement of an event is one minus the probability of the event. Since the sum of probabilities of all possible events equals 1, the probability that event A will not occur is equal to 1 minus the probability that event A will occur.
Step-by-step explanation:Complement of an Event: All outcomes that are NOT the event. So the Complement of an event is all the other outcomes (not the ones we want). And together the Event and its Complement make all possible outcomes.
Let x be an even integer. What is the product of the next two consecutive even integers?
O x^2+2x+4
O x^2+6x+8
O x(x+1)(x+2)
O x^2+3x+2
Answer:
The desired product is (x + 2)(x + 4).
Step-by-step explanation:
If x is an even integer, x + 2 is the next consecutive even integer and x + 4 the next.
The desired product is (x + 2)(x + 4).
Let f(3) = 1/(z^2+1) Determine whether f has an antiderivative on the given domain
(a) G=C\{i, –i}.
(b) G = {z Rez >0}.
To determine whether the function f(z) = 1/(z^2 + 1) has an antiderivative on a given domain, we need to check if the function is analytic on that domain.
(a) For the domain G = C\{i, -i}, the function f(z) = 1/(z^2 + 1) is analytic on G. This is because it is a rational function and does not have any singularities (poles) within the domain. Hence, it has an antiderivative on G.
(b) For the domain G = {z Re(z) > 0}, the function f(z) = 1/(z^2 + 1) does not have an antiderivative on G. This is because the function has singularities at z = i and z = -i, which lie on the imaginary axis. Since the domain excludes these points, f(z) is not analytic on G and does not have an antiderivative on G.In summary, the function f(z) = 1/(z^2 + 1) has an antiderivative on the domain G = C\{i, -i} but does not have an antiderivative on the domain G = {z Re(z) > 0}.
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Consider the following system of differential equations:
dx/dt +y=0
dt/dy + 4x = 0.
Write the system in matrix form and find the eigenvalues
If A is equal to [0, 4] and I is equal to [1, 0], [0, 1], then [0 - 4][1 0] equals 0 and [0 - 4] equals 0 and [2 - 4] equals 0. Accordingly, the eigenvalues of the matrix
[dt/dy] + [0, 4] [x] = [0] can be written as the differential equation above in a matrix. Here, [0, 4] is the coefficient network and [x] is the variable grid. Given, arrangement of differential conditions, dt/dy + 4x = 0. Let [0, 4] be the framework's eigenvalue, and then [0, 4] [x] = [x] => (A-I) [x] = 0, where An represents the coefficient grid, I represents the character lattice, and x represents the variable network.
The determinant of [A-I] is 0 if for a non-trivial solution, [A-I] [x] = 0. On the off chance that An is equivalent to [0, 4] and I is equivalent to [1, 0], [0, 1], then [0 - 4][1 0] equivalents 0 and [0 - 4] equivalents 0 and [2 - 4] equivalents 0. As a result, the matrix's eigenvalues
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What is the difference between a monomial and a polynomial?
Answer:
Is that your answer
Find the value of y. Will give out a brainly for help
Answer:
3
Step-by-step explanation:
the scale is 2 so you need to figure out 3y + 5 = 14
Evaluate whether the following argument is correct; if not, then specify which lines are incor- rect steps in the reasoning. Each line is assessed as if the other lines are all correct. So, you are to identify which lines (the minimum number) would you need to fix to get a correct proof. Proposition: If r and y are rational numbers then 3x + 2y is also a rational number. Proof: 1. We proceed by contradiction proof. 2. Assumer and y are irrational numbers. 3. Since r and y are rational, 1 = and y = a, where a, b, c, and d are integers, and b + and d 70. 4. We will show that 3x +2y is a rational number. 5. Plugging in for and į for y into the expression 3x +2y gives: 3x + 2y = 38 + 24 = 6. Since a, b, c, and d are all integers, 3ad + 2bc and bd are also integers. 7. Since b + 0 and d + 0, bd 70. 8. Therefore, 3ad + 2bc and bd contradict the assumption that r and y are irrational numbers, which implies that 3x +2y is irrational is false. 3ad +-2bc bd
1. This is a valid approach to prove the argument.
2. This is the first step of the contrapositive proof.
3. This statement is true since if one of them is rational, the other one could also be rational or irrational.
4. This statement is true because rational numbers are those numbers that can be expressed as a ratio of two integers.
5. This is true because any rational number can be expressed as a fraction of two integers.
6. This is true since it's the sum of two fractions.
7. This is also true since the sum and product of two integers are always integers.
8. This is true since the product of any non-zero number with another non-zero number is also non-zero.
9. This statement is true since x+y is the quotient of two integers, and since both integers are non-zero, then the quotient is also non-zero and therefore rational.
Therefore, the given argument is correct.
Each step in the argument is logically valid, and the argument follows a correct proof by contrapositive to show that if x is rational and y is rational, then x + y is rational.
The given argument is correct. Let us evaluate each line of the proof and make sure that it is accurate and logical.
Proposition: For every pair of real numbers x and y, if x + y is irrational, then x is irrational or y is irrational
1. We proceed by contrapositive proof.
This is a valid approach to prove the argument.
2. We assume for real numbers x and y that it is not true that x is irrational or y is irrational and we prove that x + y is rational.
This is the first step of the contrapositive proof.
3. If it is not true that x is irrational or y is irrational then neither x nor y is irrational.
This statement is true since if one of them is rational, the other one could also be rational or irrational.
4. Any real number that is not irrational must be rational. Since x and y are both real numbers then x and y are both rational.
This statement is true because rational numbers are those numbers that can be expressed as a ratio of two integers.
5. We can therefore express x as a/b and y as c/d as a, where a, b, c, and d are integers and b and d are both not equal to 0.
This is true because any rational number can be expressed as a fraction of two integers.
6. The sum of x and y is: x + y = a/b + c/d = (ad+bc)/bd
This is true since it's the sum of two fractions.
7. Since a, b, c, and d are integers, the numerator ad + bc and the denominator bd are integers.
This is also true since the sum and product of two integers are always integers.
8. Furthermore since b and d are both non-zero, bd is also non-zero.
This is true since the product of any non-zero number with another non-zero number is also non-zero.
9. Therefore, x + y is a rational number.
This statement is true since x+y is the quotient of two integers, and since both integers are non-zero, then the quotient is also non-zero and therefore rational.
Therefore, the given argument is correct.
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Given a △ PQR with vertices P (2, 3), Q (-3, 7) and R(-1, -3): The equation of median PM is __________.
The equation of the median PM in triangle PQR with vertices P(2, 3), Q(-3, 7), and R(-1, -3) is y = (1/3)x + 7/3.
To find the midpoint of QR, we calculate the average of the x-coordinates and the average of the y-coordinates. The x-coordinate of point M is (-3 + (-1))/2 = -2/2 = -1, and the y-coordinate of point M is (7 + (-3))/2 = 4/2 = 2.
Therefore, the coordinates of point M are (-1, 2). Now, we have two points, P (2, 3) and M (-1, 2), and we can find the equation of the line passing through these points using the point-slope form.
The slope of the line passing through P and M is (2 - 3)/(-1 - 2) = -1/-3 = 1/3. Using the point-slope form, we have:
y - 3 = (1/3)(x - 2)
Expanding and rearranging the equation, we get:
y = (1/3)x + 7/3
Therefore, the equation of the median PM in triangle PQR is y = (1/3)x + 7/3.
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6/2(1+2) this time don't go ogle it because it will say 9
Answer:
9
Step-by-step explanation:
I'm not sure, the answer is still 9, 1+2 is 3 so
6/2(3)
and 6/2 simplified is 3
so 3(3) is 9
Answer:
9
Step-by-step explanation:
1. Simplify the parantheses
(1+2) = 3
2. Turn 3 into a fraction
3 = 3/1
3. Multiply the fractions
6 x 3 = 18
2 x 1 2
4. What is 18/2?
18/2 = 9
Hiii, so I REALLY want to rank up, and I just need 5 more Branliests, so if you liked my answer, can you please give me one? Thank you so much, and thanks for the points!!
Please help!!
Brainlest to best answer.
Answer:
D. 42
2X3X7=42
Step-by-step explanation:
It's just volume baseXwidthXhight. Hope this helps.
write the equations of these parabolas in vertex form: • focus at (-5,-3), and directrix y = -6 • focus at (10,-4), and directrix y = 6
Answer:
y=0.12/1(x-5)^2 -3
y=1/10(x-10)^2 -4
Step-by-step explanation:
Given the directrix and focus of the parabolas, the equation of the parabolas are [tex]y=\frac{1}{6}(x^{2} +10x - 2)[/tex] and [tex]y=\frac{1}{20}(-x^{2} +20x - 80)[/tex].
What is equation of a parabola?Equation of a parabola is given by-
Distance of a point (x, y) on parabola from directrix = Distance of a point (x, y) on parabola from focus
focus = (-5, -3)
directrix = y = -6
[tex]\sqrt{(x+5)^{2}+(y+3)^{2} } = (y+6)\\\\ (x+5)^{2}+(y+3)^{2} = (y+6)^{2}\\\\x^{2} +25+5x = 6y+27\\\\y=\frac{1}{6}(x^{2} +10x - 2)[/tex]
focus = (10,-4)
directrix = y = 6
[tex]\sqrt{(x-10)^{2}+(y+4)^{2} } = (y-6)\\\\ (x-10)^{2}+(y+4)^{2} = (y-6)^{2}\\\\x^{2} +100-20x = -20y+20\\\\y=\frac{1}{20}(-x^{2} +20x - 80)[/tex]
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what equation passes through (6,3) and is parrallel to y=3x+5
Answer:
y=3x+25
Step-by-step explanation:
For the equation y=3x+5, the slope is 3. Since the line will be parallel to it then this is the slope of the new line as well. Use the point-slope form to write the equation, then simplify and convert into the slope intercept form.
(y-13)=3(x--4)
y-13=3x+12
y=3x+25
Suppose there are 8 boys and 7 girls in a classroom. If one student is chosen at random to run a errand. What is the probability that the student will be a girl?
Answer:
7/15
Step-by-step explanation:
7+8=15
there's 7 girls so you have 7/15 chance of getting a girl
Answer:
8/15
Step-by-step explanation:
possibilities/sample size= 8 girls/ 15 students
Round 4,368 to the nearest ten
Answer:
4,370
Step-by-step explanation:
.....just round the tenths place (6) up sense 8 is above 5 it rounds up so
4,370
sorry for bad explanation
pllssss help I put B, but I got it wrong the first time. NO LINKS
Answer:
D.1767.1 in³
Step-by-step explanation:
volume of a sphere=4/3πr³
where r=radius of the sphere
Given diemeter= 15in
so radius=15/2=7.5in
volume=4/3×π×(7.5)³
=4/3×22/7×(7.5)³
=1767.1in³
hope it helps...
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Change the triple integral to spherical coordinates: SIS 6x2 + y2 + z2) av (༴ AV Where Q is bounded by the upper hemisphere: x2 + y2 +22=100 : 21 10 ("S", p's p. sino dpdooo 2T pº sino dododo 21 10 2 3 sino doopde 0 0 0 10
The solution to the triple integral in spherical coordinates is 22000π. This can be obtained by evaluating the integral in three steps: integrating with respect to r, then with respect to θ, and finally with respect to φ.
To change the triple integral to spherical coordinates, we need to express the integrand and the limits of integration in terms of spherical coordinates.
The given integrand is f(x, y, z) = 6x² + y² + z².
In spherical coordinates, the integrand becomes f(r, θ, φ) = 6(rsinθcosφ)² + (rsinθsinφ)² + (rcosθ)².
The limits of integration are as follows:
- The bounds for r are from 0 to 10, as the region Q is bounded by the upper hemisphere x² + y² + z² = 100.
- The bounds for θ are from 0 to π/2, as we are considering the upper hemisphere.
- The bounds for φ are from 0 to 2π, as φ covers a complete revolution around the z-axis.
The triple integral in spherical coordinates is then given by:
∭Q f(r, θ, φ) r² sinθ dr dθ dφ,
which becomes:
∫(φ=0 to 2π) ∫(θ=0 to π/2) ∫(r=0 to 10) [6(rsinθcosφ)² + (rsinθsinφ)² + (rcosθ)²] r sinθ dr dθ dφ.
To solve the given triple integral, we'll start by evaluating the innermost integral with respect to r, then the middle integral with respect to θ, and finally the outer integral with respect to φ.
The integrand is:
[6(rsinθcosφ)² + (rsinθsinφ)² + (rcosθ)²] r² sinθ
First, let's evaluate the innermost integral with respect to r, while treating θ and φ as constants:
∫(r=0 to 10) [6(rsinθcosφ)² + (rsinθsinφ)² + (rcosθ)²] r² sinθ dr
= ∫(r=0 to 10) [6(sin²θcos²φ)r⁴ + (sin²θsinφ)r⁴ + (cos²θ)r⁴] sinθ dr
= ∫(r=0 to 10) [(6sin²θcos²φ + sin²θsin²φ + cos²θ) r⁴] sinθ dr
= [(6sin²θcos²φ + sin²θsinφ + cos²θ) ∫(r=0 to 10) r⁴] sinθ dr
= [(6sin²θcos²φ + sin²θsin²φ + cos²θ) * (10^5/5)] sinθ
= [(6sin²θcos²φ + sin²θsin²φ + cosθ) * 2 × 10⁵] sinθ
Next, let's evaluate the middle integral with respect to θ, while treating φ as a constant:
∫(θ=0 to π/2) [(6sin²θcos²φ + sin²θsin²φ + cos²θ) * 2 × 10⁵] sinθ dθ
= 2 × 10⁵ ∫(θ=0 to π/2) [6sin²θcos²φ + sin²θsin²φ + cos²θ] sinθ dθ
= 2 × 10⁵ [2/3cos²φ + 1/4 + 1/3]
= 2 × 10⁵ [2/3cos²φ + 7/12]
Finally, let's evaluate the outer integral with respect to φ:
[tex][\int_{0}^{2\pi} 2\times10^5 \left( \frac{2}{3}\cos^2\phi + \frac{7}{12} \right) d\phi \\\\= 2\times10^5 \left( \frac{2}{3}\pi + \frac{7}{12}(2\pi) \right)][/tex]
= 22π × 10000
= 22000π
Therefore, the solution to the given triple integral is 22000π.
To know more about the spherical coordinates refer here :
https://brainly.com/question/31471419#
#SPJ11
Complete question :
Change the triple integral to spherical coordinates: SIS 6x2 + y2 + z2) av (༴ AV Where Q is bounded by the upper hemisphere: x2 + y2 +22=100 : 21 10 ("S", p's p. sino dpdooo 2T pº sino dododo 21 10 2 3 sino doopde 0 0 0 10 ["S" p2 sino apdoce