Which expression represents the sixth term in the binomial expansion of (2a - 3b)10?
10C5(2a)5(-3b)5
10C5(-2a)5(3b)5
10C6(2a)4(-3b)6
10C6(-2a)4(3b)6
10C6(2a)6(-3b)4
10C6(-2a)6(3b)4
Answer:
[tex]^{10}C_5(2a)^{5}(-3b)^5[/tex]
Option 1 is correct.
Step-by-step explanation:
Given: [tex](2a-3b)^{10}[/tex]
We need to find 6th term of binomial expansion.
Formula:
[tex]T_{r+1}=^nC_rx^{n-r}y^{r}[/tex]
We need to find 6th term, [tex]T_6[/tex]
[tex]T_6=T_{r+1}[/tex]
So, r=5
Put r=5 into formula
[tex]n=10, x=2a \text{ and }y=-3b[/tex]
[tex]T_6=^{10}C_5(2a)^{10-5}(-3b)^5[/tex]
Now we will simplify it to calculate 6th term
[tex]T_6=^{10}C_5(2a)^{5}(-3b)^5[/tex]
Hence, The 6th term of binomial is [tex]^{10}C_5(2a)^{5}(-3b)^5[/tex]
The expression that represents the sixth term in the binomial expansion of (2a - 3b)¹⁰ is expressed as ¹⁰C₆(2a)⁴(-3b)⁶.
To find the sixth term in the binomial expansion of (2a - 3b)¹⁰, we can use the binomial theorem,
which states that the k-th term in the expansion of (a + b)ⁿ can be calculated using the following formula:
T(x) = C(n, x) × (aⁿ⁻ˣ) × (bˣ)
Where:
T(x) is the x-th term in the expansion.
C(n, x) is the binomial coefficient, also denoted as "n choose x," and it is equal to n! / (x! × (n-x)!), where "!" denotes factorial.
a is the first term in the binomial (2a in this case).
b is the second term in the binomial (-3b in this case).
n is the exponent of the binomial (10 in this case).
x is the term we want to find (the sixth term in this case).
Now, let's calculate the sixth term:
x = 6
n = 10
a = 2a
b = -3b
First, calculate the binomial coefficient:
C(10, 6) = 10! / (6! × (10-6)!)
C(10, 6) = 210
Now, plug these values into the formula:
T(6) = 210 × (2a)¹⁰⁻⁶ × (-3b)⁶
Simplify the exponents:
T(6) = 210 × (2a)⁴ × (-3b)⁶
T(6) = 210 × 16a⁴× 729b⁶
Now, multiply the constants:
T(6) = 30,240a⁴b⁶
Therefore, the sixth term in the binomial expansion of (2a - 3b)¹⁰ is 30,240a⁴b⁶ which is written as ¹⁰C₆(2a)⁴(-3b)⁶.
learn more about binomial expansion here
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Subtract. (x2+3x−7)−(3x2−5x+3) Express the answer in standard form.
Answer is -2x^2+8-10 hope this helps!!!!
For the parallelogram, find the value of the variables. Show your work
ok trying to figure out how to solve this problem
5x + 2
3y - 6 21
17
One Question Please Help
Suppose x is any positive number. circle 1: center (5, 4) and radius 5x circle 2: center (5, 4) and radius 2x why is circle 1 similar to circle 2? circle 1 and circle 2 have the same radius. circle 1 is a dilation of circle 2 with a scale factor of 2.5. circle 1 is a dilation of circle 2 with a scale factor of 0.4. circle 1 is congruent to circle 2.
Answer:
circle 1 is a dilation of circle 2 with a scale factor of 2.5.
Step-by-step explanation:
The radius of circle 1 is 5x and the radius of circle 2 is 2x.
This means the scale factor of the dilation is 5x/2x = 2.5.
Circle 1 is a dilation of circle 2 with a scale factor of 0.4.
What are similar figures?Two figures are said to be similar if they have the same shape and the ratio of their corresponding sides are in the same proportion.
Given that circle 1 has radius of 5x and circle 2 has radius of 2x, hence:
Circle 1 scale factor = 2x / 5x = 0.4
Hence:
Circle 1 is a dilation of circle 2 with a scale factor of 0.4.
Find out more on similar figures at: https://brainly.com/question/14285697
Using 70 POINTS NEED YOUR HELP NOT SPAMMERS. To make an ice-cream cone, employees at an ice-cream shop completely fill the cone and then add one scoop of ice cream on top. The cone has a height of 6 inches and a diameter of 2 inches.
To the nearest cubic inch, how much ice cream is in the cone before the scoop is added on the top?
6 in³
19 in³
25 in³
75 in³
also
A candy machine contains 120 spherical candies. Each candy is solid and has a diameter of 0.25 inches.
What is the total volume of the candies in the machine?
Select from the drop-down menu to correctly complete the statement.
The total volume of the candies in the machine is about
Solution for Q1:
Given is the height of ice-cream cone, h = 6 inches.
Given is the diameter of ice-cream cone, d = 2 inches. Then radius of cone would be half of diameter, r = d/2 = 1 inch.
The ice-cream without scoop would be in the shape of cone only. So we need to find the volume of cone.
We know the formula for volume of cone is given as follows :-
[tex] Volume = \frac{1}{3} \pi r^{2} h \\\\Volume = \frac{1}{3} \pi (1)^{2} (6) \\\\Volume = \frac{6\pi}{3} = 2\pi = 2(3.14) = 6.28 \;cubic \;inches. [/tex]
So the volume is 6.28 ≈ 6 in³.
Hence, option A is correct i.e. 6 cubic inches.
Solution for Q2:
A candy is in shape of sphere with diameter, d = 0.25 inches.
Then radius of sphere, r = d/2 = 0.250/2 = 0.125 inches.
The formula for volume of sphere is given as follows :-
[tex] Volume = \frac{4}{3} \pi r^{3} \\\\Volume = \frac{4}{3} \pi (0.125)^{3} \\\\Volume = 0.00818123 \;cubic \;inches. [/tex]
So, volume of one candy = 0.00818123 cubic inches.
There are total 120 candies in the machine.
Total volume of all 120 candies = 120 x 0.00818123 = 0.981748 cubic inches.
Hence, the total volume of all candies in the machine is about 0.982 in³.
Answer:
This should definitely help for question 1 :)
What is the volume of a ball if its diameter is 6 inches? Leave your answer in terms of π.
A)V = 12 π cubic inches
B)V = 36 π cubic inches
C)V = 288 π cubic inches
D)V = 27.75 π cubic inches
Explanation:The volume of the ball above is V = 36 π cubic inches. Use the formula V = 4/3π 3. Just remember to cut the diameter in half to get the radius.
Step-by-step explanation:
A customer who opens a savings account at a bank, in turn, becomes a(n) _____.
A. lender
B. investor
C. insurer
D. borrower
@lulu22,
PLEASE HELP ASAP!!!!!!!!!!!!!!
19. What is the sum of the measures of the exterior angles in a nonagon? Explain.
Answer:
360°
Step-by-step explanation:
I just took the test.
The table shows the predicted growth of a particular bacterium after various numbers of hours.
Write an explicit formula for the sequence of the number of bacterium.
A) an=24n+1
B) an=24n
C) an=1/24n
D) an=n+24
A regular pentagon has an apothem measuring 3 cm and a perimeter of 21.8 cm. What is the area of the pentagon, rounded to the nearest tenth? 13.8 cm2 17.3 cm2 32.7 cm2 69.0 cm2
plz dont ask for a pic i cant put one
Answer:
[tex]\text{Hence, area of regular pentagon is }32.7 cm^2[/tex]
Step-by-step explanation:
Given that a regular pentagon has an apothem measuring 3 cm and a perimeter of 21.8 cm.
we have to find the area of pentagon.
Apothem=a=3 cm
Perimeter=s=21.8 cm
[tex]\text{Area of regular pentagon=}\frac{1}{2}\times apothem\times perimeter[/tex]
[tex]\frac{1}{2}\times s \times a[/tex]
[tex]=\frac{1}{2}\times 21.8\times 3[/tex]
[tex]=\frac{65.4}{2}=32.7 cm^2[/tex]
[tex]\text{Hence, area of regular pentagon is }32.7 cm^2[/tex]
Option 3 is correct.
While driving your rental car on your trip to Europe, you find that you are getting 12.4 kilometers per liter of gasoline. What does this correspond to in miles per gallon? PLEASE HELP ME!!!!
Claudia dumped her 200-penny coin collection on the floor and counted the number of pennies that landed heads up. Claudia repeated this process 5 times and had an average of 84 pennies landing heads up on each try. Which of the following statements is true?
If Claudia had repeated this process fewer times, the average number of pennies landing heads up would be closer to 100.
Each penny has a greater probability of landing heads up than tails up.
The theoretical probability of a penny landing heads up is
If Claudia had repeated this process more times, the average number of pennies landing heads up would be closer to 100.
In a 45-45-90 triangle, what is the length of the hypotenuse when the length of one of the legs is 5 in.? 25√ in. 52√ in. 5√ in. 55√ in.
Answer: [tex]5\sqrt{2}\ in[/tex]
Step-by-step explanation:
Given: In a 45-45-90 triangle, the length of one of the legs= 5 in.
Let H be the hypotenuse of the given right triangle.
Since the two angles are equal (45°) in the given right triangle,also in a triangle the side opposite to the equal angles are equal.
Therefore, the other leg of given right triangle = 5 in.
Now applying Pythagoras theorem, we get
[tex]H^2=5^2+5^2=25+25\\\\\Rightarrow\ H^2=50\\\\\Rightarrow H=\sqrt{50}\\\\\Rightarrow H=5\sqrt{2}\ in[/tex]
Hence, the length of the hypotenuse = [tex]5\sqrt{2}\ in[/tex]
The height in feet, h, of a model rocket t seconds after launch is given by the equation h(t) = 3+70t - 16t^2. The average rate of change in h(t) between t = 1 second and t = 3 second is 6. What does the average rate of change tell you about the rocket?
Answer:
The rocket is at a greater height when t = 3 than it is when t = 1.
Option B
Step-by-step explanation:
Just took the test
Answer:
B
Step-by-step explanation:
!
A bank offers a CD that pays a simple interest rate of 5.5%. How much must you put in this cd now in order to have $8000 for a kitchen remodeling project in two years? Could you please show work?,
Can someone help me with this one
Find the area of the following ellipse. (Round answer to the nearest tenth).
2a = 10 cm; 2b = 20 cm.
A = _____cm²,
Answer:
Approximately 157.08 sq. cm.
Step-by-step explanation:
The Area of an Ellipse is given by the formula [tex]A=\pi a b[/tex]
Here 2a = 10, so a = 5
Also, 2b = 20, b = 10
Substituting these into the area formula gives us:
[tex]A=\pi a b\\A=\pi (5)(10)\\A=50\pi\\A=157.08[/tex]
The Area is approximately 157.08 sq. cm.
A circle could be circumscribed abut the quadrilateral below
Answer:
B. False.
Step-by-step explanation:
We have been given an image of a quadrilateral. We are asked to determine whether a circle could be circumscribed about the given quadrilateral.
We know that opposite angles of a cyclic quadrilateral are supplementary.
Let us check is this true for our given quadrilateral or not.
[tex]m\angle B+m\angle D=180^{\circ}[/tex]
[tex]80^{\circ}+60^{\circ}=180^{\circ}[/tex]
[tex]140^{\circ}\neq 180^{\circ}[/tex]
Now let us check other pair of angles.
[tex]m\angle A+m\angle C=180^{\circ}[/tex]
[tex]110^{\circ}+110^{\circ}=180^{\circ}[/tex]
[tex]220^{\circ}\neq 180^{\circ}[/tex]
Since opposite angles of our given quadrilateral are not supplementary, therefore, a circle could not be circumscribed about the given quadrilateral.
Properties of Parallelograms
find x.
If b is the midpoint of ac and the length of ac is 23.8 what is the length of ab
The equation of a circle is (x + 12)2 + (y + 16)2 = (r1)2, and the circle passes through the origin. the equation of the circle then changes to (x – 30)2 + (y – 16)2 = (r2)2, and the circle still passes through the origin. what are the values of r1 and r2?
a.r1 = 10 and r2 = 17
b.r1 = 10 and r2 = 34
c.r1 = 20 and r2 = 17
d.r1 = 20 and r2 = 34
Answer:
Option D is correct
Step-by-step explanation:
Given Equations of Circles:
Circle 1 - [tex](x+12)^2+(y+16)^2=(r_1)^2[/tex]
Circle 2 - [tex](x-30)^2+(y-16)^2=(r_2)^2[/tex]
Both circles passes through origin.
To find: Values of [tex]r_1\:,\:r_2[/tex]
Coordinates of origin = ( 0 , 0 )
Circles passes through origin means x = 0 & y = 0 must satisfy the equation of circles.
So, Substituting x = 0 & y = 0 in Eqn of Circle 1
we get
[tex](0+12)^2+(0+16)^2=(r_1)^2[/tex]
[tex](r_1)^2=12^2+16^2[/tex]
[tex](r_1)^2=144+256[/tex]
[tex](r_1)^2=400[/tex]
[tex]r_1=\sqrt{400}[/tex]
[tex]r_1=20[/tex]
Now, Substituting x = 0 & y = 0 in Eqn of Circle 2
we get
[tex](0-30)^2+(0-16)^2=(r_2)^2[/tex]
[tex](r_2)^2=(-30)^2+(-16)^2[/tex]
[tex](r_2)^2=900+256[/tex]
[tex](r_2)^2=1156[/tex]
[tex]r_2=\sqrt{1156}[/tex]
[tex]r_2=34[/tex]
Therefore, Option D is correct .i.e., [tex]r_1=20\:\:,\:\:r_2=34[/tex]
which of the following statements is not always true ?
a. if a function contains the origin , then its inverse contains the origin.
b.if a function has 3 x-intercepts, then its inverse has 3 y-intercepts.
c.if a function contains no points in the fourth quadrant, then its inverse contains no points in the second quadrant.
d.if the slope of a linear function is less than 1, then the slope of its inverse is greater than 1.,
The following set of numbers is going to be graphed on a histogram.
3, 19, 11, 29, 4, 6, 10, 16, 2, 21, 15, 22, 13, 9, 1, 17, 2, 26, 18, 7
If there are going to be six intervals in the display, what is the length of each interval?
1.) 5
2). 6
3.) 3
Answer:
The answer is 5.
Step-by-step explanation:
What we know about prime and common factors help us to find the ___________ of two or more numbers.
Answer:
greatest common factor the largest factor that any given numbers have in common
Step-by-step explanation:
so the answer is greatest common factor
Final answer:
Knowledge of prime and common factors is essential for finding the greatest common divisor (GCD) or greatest common factor (GCF) of two or more numbers, a key element in simplifying fractions and understanding ratios.
Explanation:
Understanding prime and common factors is instrumental in finding the greatest common divisor (GCD) or greatest common factor (GCF) of two or more numbers. The GCD is the highest number that divides evenly into each of the numbers in question. For example, the GCD of 48 and 60 is 12 since 12 is the largest number that can divide both 48 and 60 without leaving a remainder. Finding the GCD is crucial in simplifying fractions, solving problems involving ratios, and is also a fundamental concept in more advanced mathematics including algebra and number theory.
The process often involves factoring the numbers down to their prime factors, identifying the common primes, and then multiplying these common prime factors together to find the GCD. This method not only highlights the importance of prime numbers in the mathematical hierarchy but also underlines the indelible role of factorization in simplifying and solving numerical problems.
Explain in a short paragraph how the figure below shows this is a right triangle, by the converse of the Pythagorean Theorem.
Given an exponential function for compounding interest, A(x) = P(.91)^x, what is the rate of change?
A. −0.09%
B. −9%
C. 0.91%
D. 91%
The confirmed correct answer is -9%. Trust me.
Answer:
Option B = -9%
Step-by-step explanation:
Given : Given an exponential function for compounding interest, [tex]A(x) = P(.91)^x[/tex]
To find : What is the rate of change?
Solution :
The general form of the exponential function is [tex]f(x)=a(1+r)^x[/tex]
Where,
r is the rate of change
if r> 1 then it is growth rate
if r< 1 then it is decay rate.
Comparing given function with exponential function,
[tex]A(x) = P(.91)^x[/tex]
[tex]1+r=0.91[/tex]
[tex]r=0.91-1[/tex]
[tex]r=−0.09[/tex]
It is a decay rate.
Convert into percentage, multiply with 100
[tex]-0.09\times 100=-9\%[/tex]
Therefore, Option B is correct.
The rate of change is -9%.
Which of these statements is true for f(x)....
Answer:
B. the y -intercept is (0,3)
Step-by-step explanation:
What is the length of the diagonal of a square with side lengths 7 square root of 2 ? round to the nearest tenth if necessary?
The length of the diagonal of a square with side lengths 7√2 is approximately 14.1 units.
Explanation:The length of the diagonal of a square can be found using the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the diagonal in this case) is equal to the sum of the squares of the lengths of the other two sides. Since a square has all sides equal, the length of each side can be represented as 7√2. Substituting this value into the Pythagorean theorem, we get:
Diagonal² = (7√2)² + (7√2)²
Diagonal² = 98 + 98 = 196
Taking the square root of 196, we find the length of the diagonal to be approximately 14 units. Rounding to the nearest tenth gives us 14.1 units.
Arthur borrowed $380 for six months, with $67 monthly payments. How much interest will he pay on this loan?
Given is the borrowed amount = 380 dollars.
Repayment period = 6 months.
Monthly installments = 67 dollars.
After six months, total repaid amount would be sum of all six monthly installments i.e. 6 x 67 = 402 dollars.
The interest to be paid on this loan would be difference of total repaid amount and borrowed amount.
Interest paid = Repaid amount - Borrowed amount
Interest = 402 dollars - 380 dollars
Interest = 22 dollars.
So, he would pay 22 dollars as interest on this loan.
Are the number of steps in a stairway continuous or discrete data
Answer:
The number of steps in a stairway is discrete.
Step-by-step explanation:
A discrete data is that, which can take only integer values like number of apples in a basket.
But the continuous data is that, which can take on any value like a decimal. For example, the weight of patients in a hospital.
So, the number of steps in a stairway will be a discrete data. As stairs are counted in whole numbers and not decimals.