Hello,Can you please help me with question# 11 ? it is express each sum using summation notation. Use 'i' as the index of th sum
Answers
Given:
The sum of terms
[tex]4^3+5^3+6^3..........+13^3[/tex]Required:
Find sum.
Explanation:
We know sum of cube of first n terms of natural numbers
[tex]\sum_{n\mathop{=}1}^{\infty}n^3=[\frac{n(n+1)}{2}]^2[/tex]Now,
[tex]\begin{gathered} =(1^3+2^3+....+13^3)-(1^3+2^3+3^3) \\ =[\frac{13(13+1)}{2}]^2-36 \\ =8281-36 \\ =8245 \end{gathered}[/tex]Answer:
The sum of terms is 8245.
Related Questions
help please and thankyou Write an equation that represents the graph of the line shown in the coordinate plane below.
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Slope intercept-form equation:
y=mx+b
Where:
m= slope
b= y-intercept
by looking at the graph we can see that the line crosses the origin (0,0), so it crosses the y-axis at x=0.
b=0
For the slope:
[tex]m=\frac{y2-y1}{x2-x1}[/tex]point 1= (x1,y1)= (-2,-4)
Point2= (x2,y2)= (4,8)
Replacing:
[tex]m=\frac{8-(-4)}{4-(-2)}=\frac{12}{6}=2[/tex]So the final expression:
y= 2x
50th term 64 57 50 43...
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hello
to solve this question, we need to know if this sequence is an arithmetic or geometric progression
first term (a) = 64
common difference (d) = -7
the nth term of an arithemetic progression is given as
[tex]\begin{gathered} T_n=a+(n-1)d_{} \\ n=\text{nth term} \\ a=\text{first term} \\ d=\text{common difference} \end{gathered}[/tex]now let's substitute the values into the equation above
[tex]\begin{gathered} T_n=a+(n-1)d_{} \\ a=64 \\ d=-7 \\ T_{50}=64+(50-1)\times-7 \\ T_{50}=64+(49\times-7) \\ T_{50}=64+(-343) \\ T_{50}=64-343 \\ T_{50}=-279 \end{gathered}[/tex]from the calculations above, the 50th term of the sequence is -279
Write a polynomial function of minimum degree with real coefficients whose zeros include those listed. Write the polynomial in standard form. 4)-1 and 3 + 2i A) (x)= x3 + 5x2 +7x + 13 B)(x) = x3 - 5x2 + 7x - 13 C)(x)= x3 - 5x2 + 7x + 13 D) f() = x3 - 5x2 + 7x + 14
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u ptsBirths are approximately Uniformly distributed between the 52 weeks of the year. They can be saidto follow a Uniform distribution from 1 to 53 (a spread of 52 weeks). Round answers to 4 decimalplaces when possible.a. The mean of this distribution isb. The standard deviation isC. The probability that a person will be born at the exact moment that week 18 begins isP(x = 18) =d. The probability that a person will be born between weeks 10 and 43 isP(10 < x < 43) =e. The probability that a person will be born after week 35 isP(x > 35)f. P(x > 18 x < 32) =g. Find the 47th percentile.h. Find the minimum for the upper quarter.
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Step 1
A) The mean distribution
[tex]\frac{1+53}{2}=\frac{54}{2}=27.0000[/tex]Step 2
B) The standard deviation
[tex]\begin{gathered} SD=\sqrt[]{\frac{1}{12}\times(b-a)^2} \\ SD=\sqrt[]{\frac{1}{12}(53-1)^2} \\ SD=\text{ }15.0111 \end{gathered}[/tex]Step 3
C)
[tex]P(x=18)=0[/tex]Step 4
D)
[tex]\begin{gathered} P(10Step 5E)
[tex]P(x>35)=\text{ }\frac{53-35}{52}=\frac{18}{52}=0.3462[/tex]Step 6
F)
[tex]P(x>18|x<32)=\text{ }\frac{32-18}{32-1}=\frac{14}{31}=0.4516[/tex]Step 7
G)
[tex]\begin{gathered} \text{The 47th percentile}=1\text{ + }\frac{47}{100}(53-1)_{} \\ =1+0.47(52)=25.44_{}00 \end{gathered}[/tex]Step 8
[tex]\begin{gathered} \text{The minimum for the upper percentile = 1+((}\frac{3}{4})(53^{}-1) \\ =1+0.75(52) \\ =1+\text{ 39=40}.0000 \end{gathered}[/tex]Find the area of the circle with a circumference of 62.8 inches. Use 3.14 for pi
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The area of a circumference can be calculated with this formula:
[tex]C=2\pi r[/tex]Where "r" is the radius of the circle.
The area of a circle can be found with this formula:
[tex]A=\pi r^2[/tex]Where "r" is the radius of the circle.
If you solve for "r" from the formula of a circumference, you get:
[tex]r=\frac{C}{2\pi}[/tex]Knowing that:
[tex]\begin{gathered} C=62.8in \\ \pi\approx3.14 \end{gathered}[/tex]You get:
[tex]\begin{gathered} r=\frac{62.8in}{(2)(3.14)} \\ \\ r=10in \end{gathered}[/tex]Knowing the radius, you can find the area of the circle:
[tex]\begin{gathered} A=(3.14)(10in)^2 \\ A=314in^2 \end{gathered}[/tex]The answer is:
[tex]A=314in^2[/tex]Solve: 9x 2+ 2x = –3 using the quadratic formula. step by step please to understand better
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Explanation: To solve the following equation
[tex]9x^2+2x=-3[/tex]We can use the following quadratic formula
[tex]x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}[/tex]Step 1: Let's compare our equation with a generic quadratic equation as follows
As we can see above, first we move -3 from the second term to the first term and when we do that we change its sign to +3. Now we know that a = +9, b = +2 and c = +3.
Step 2: Now all we need to do is to substitute the values of a, b and c into our quadratic formula and solve it to find the roots as follows
[tex]\begin{gathered} x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a} \\ x=\frac{-2\pm\sqrt[]{2^2-4\cdot9\cdot3}}{2\cdot9} \\ x=\frac{-2\pm\sqrt[]{4^{}-108}}{18} \\ x=\frac{-2\pm\sqrt[]{-104}}{18} \\ x=\frac{-2\pm\sqrt[]{-104}}{18} \end{gathered}[/tex]Final answer: As we can see above inside the square root there is a negative number -104 which means this quadratic equation has no real solutions.
Determine if the triangles, △YPQ and △NPD, are similar. if so, Identify criterion.
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Answer:
Yes, they are similar.
Criterion: AA Similarity
Explanation:
Looking at triangles YPQ and NPD, we can see that angles NPD and YPQ are vertically opposite angles and are congruent since vertically opposite angles are always congruent;
[tex]\angle YPQ\cong\angle NPD\text{ (vertically opposite angles)}[/tex]We can also observe that angles N and Y are congruent since they are alternate angles;
[tex]\angle N\cong\angle Y\text{ (alternate angles)}[/tex]From the AA similarity rule, we know that two triangles are said to be similar if two angles in one triangle are equal to two triangles in the other triangle.
Therefore, from the AA rule, we can say that triangles YPQ and NPD are similar.
A researcher would like to determine whether a new drug has an effect on IQ. A sample of n = 100 participants is obtained, and each person is given a standard dose of the medication one hour beforebeing given an IQ test. For the general population, scores on the IQ test are normally distributed with μ= 100 and o=15. The individuals in the sample who took the drug had an average score of M =103.a. Use a two-tailed test with a= .05. Conduct the four steps for hypothesis testing and labeleach step: Step1, Step 2, Step 3, and Step 4.b. Calculate Cohen's d.c. Are the data sufficient to conclude that there is a significant difference? Write youranswer in the form of a sentence.
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There is sufficient evidence to conclude that there is a significant difference in the hypothesis testing of the normal distribution.
a) Let us consider the hypothesis for the experiment.
Step 1:
H₀ : μ = 100
Similarly now we find H₁
H₁ : μ ≠ 100
Step 2:
Now we have to find the test statistic.
Z = (x - μ )/(σ÷√n)
Now we have x =103 , μ = 100 , σ =15 and n = 100
Z= (103-100) /15 × 10
Z=2.00
Step 3:
Now critical region is rejected if Z > Z(α/2)
Z(α/2) = 1.96
Now H₀ is rejected if z >-1.96 nor Z>1.96
Step 4:
As Z= 2>1.96 hence we will reject the null hypothesis.
b) Cohen's dc = 103 -100 / 15 = 0.2
c) Hence we can say that there is sufficient evidence to conclude that there is a significant difference.
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How to solve question 21? Area of the shaded region
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The shaded region covers an area of 86.
Given that,
In the picture,
We have to find the area of the shaded region of question 21.
We know that,
The Area of the square is side square.
The area of the circle is πr².
The radius of the circle is 10.
We know that,
The circle's diameter is the same as a square's side length.
The diameter=10+10 =20
The side is 20
The area of the square
= 20² = 400
The area of the circle = π (10)²=π×100=314
Subtract the area of the square and area of the circle.
400-314
86
Therefore, The shaded region covers an area of 86.
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If x is multiplied by 5 and then 3 is subtracted, then the function isf(x) = 5x -3.What are the steps to find the inverse to this function?
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Step:
Concept:
First, find the inverse of subtraction which is addition
x + 3
Step 2:
The multiplicative inverse is division, hence, you will divide x + 3 by 5.
Therefore, we have
[tex]\begin{gathered} y\text{ = }\frac{x\text{ + 3}}{5} \\ \end{gathered}[/tex]The inverse of the function is given below.
[tex]f^{-1}(x)\text{ = }\frac{x\text{ + 3}}{5}[/tex]Method 2
[tex]\begin{gathered} \text{If f(x) = 5x - 3} \\ \text{let y = 5x - 3} \\ \text{Make x subject of the formula} \\ \text{y + 3 = 5x} \\ x\text{ = }\frac{y\text{ + 3}}{5} \\ \text{Write the inverse of f(x) by changing y to x} \\ f^{-1}(x)\text{ = }\frac{x\text{ + 3}}{5} \end{gathered}[/tex]Answer:
Add 3, then divide by 5
Step-by-step explanation:
4. Tickets for a carnival cost $6 for adults and $4 for children. The school has abudget of $120 for a field trip to the carnival. An equation representing thebudget for the trip is 120 = 6x + 4y. Here is a graph of this equation:
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Given:
The equation is 6x + 4y = 120.
Explanation:
The points that lies on the line satifies the equation. So point (0,30) lies on the number which 0 adults and 30 children could go to school. So "if no adult chaperons were needed, 30 students could go to school is true.
For ten students and 15 adults point is (15,10). The point (15,10) does not lie on number line and not satifies the equation so second statement is false.
The cost of tickets for 4 adults is,
[tex]4\cdot6=24[/tex]and cost of tickets for six students is,
[tex]6\cdot4=24[/tex]Both costs are equal, means for six fewer students 4 additional adults can go to the zoo. Thus third statement is correct.
The cost of tickets for two children is,
[tex]4\cdot2=8[/tex]The cost of tickets for 3 adults is,
[tex]6\cdot3=18[/tex]Since cost of tickets for 3 adults is more than cost of tickets for two children which means two children can not go to the zoo for 3 fewer adults in the trip. Thus fourth statement is wrong.
For 16 adults and 6 students point is (16,6). The point (16,6) lies on the number line, which point (16,6) satifies the equation. So fifth statement is correct.
35/25 covert fraction to percent
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To convert fraction to decimal you multiply by 100%
Therefore, 35/25 to percentage
[tex]\begin{gathered} =\text{ }\frac{35}{25}\text{ x 100\%} \\ =\text{ }\frac{35\text{ x 100}}{25} \\ =\text{ }\frac{3500}{25} \\ =\text{ 140\%} \end{gathered}[/tex]A randomly generated list of integers from O to 4 is being used to simulate anevent, with the number 3 representing a success. What is the estimatedprobability of a success?
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We have that:
• A randomly generated list of, integers from 0 to 4 i,s being used to simulate an event.
• The number 3 represents a success.
And we need to find the estimated probability of success.
We can achieve that if we know that:
1. We have the following sample space for the experiment - we have a list of integers from 0 to 4:
[tex]\Omega=\lbrace0,1,2,3,4\rbrace[/tex]2. Then the probability of having a 3 is:
[tex]P(3)=\frac{1}{5}=0.2\Rightarrow20\%[/tex]We have one possibility of getting a 3 (one possibility) out of 5 possibilities (0, 1, 2, 3, 4).
Therefore, the estimated probability of success is 20% (option D.)
Here are the graphs of three equations:y = 50(1.5) ^xy = 50(2)^xY = 50(2. 5)^xWhich equation matches each graph? Explain how you know
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The graphs below are exponential function graphs, the general formular takes the form
[tex]y=ab^x[/tex]The graph of
[tex]y=50(1.5)^x[/tex]Is shown below
The graph of
[tex]y=50(2^x)[/tex]Is shown below
The graph of
[tex]y=50(2.5^x)[/tex]Is shown below
Hence,
[tex]\begin{gathered} y=50(1.5)^x\rightarrow C \\ y=50(2)^x\rightarrow B \\ y=50(2.5)^x\rightarrow A \end{gathered}[/tex]The equation of the exponential function is
[tex]\begin{gathered} y=ab^x \\ a=50\rightarrow the\text{ initial value} \\ b\rightarrow growht\text{ factor} \end{gathered}[/tex]Thus the higher the growth factor the greater the rate of attaining a higher value within a short period.
That is why you see that the function with growth factor of 2.5 grows faster than that of 2 and also 1.5.
So the at x value of 3, the function with the greatest growth factor will have the highest y-value.
This implies , growth factor of 2.5 will have the highest, that corresponds to graph with colour green. Function with growth factor 2 will be the next to that of 2.5, that is red colored graph, and the last will be blue.
Brenda Ortiz earns $18,200 per year. Find her semimonthly salary
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Semi monthly salary are paid twice a month, since the year has twelve months then the total number of semi monthly salaries she'll recieve is:
[tex]\text{n = 12}\cdot2\text{ = 24}[/tex]She'll receive 24 salaries in a year. To calculate the value of each salary we need to divide the total amount she earns in a year by 24.
[tex]\text{semimonthly = }\frac{18200}{24}\text{ = }758.34[/tex]Her semi monthly salary is $758.34.
A gardener builds a rectangular fence around a garden using at most 56 feet of fencing. The length of the fence is four feet longer than the widthWhich inequality represents the perimeter of the fence, and what is the largest measure possible for the length?
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We know that
• The gardener used at most 56 feet of fencing.
,• The length of the fence is four feet longer than the width.
Remember that the perimeter of a rectangle is defined by
[tex]P=2(w+l)[/tex]Now, let's use the given information to express as inequality.
[tex]2(w+l)\leq56[/tex]However, we have to use another expression that relates the width and length.
[tex]l=w+4[/tex]Since the length is 4 units longer than the width. We replace this last expression in the inequality.
[tex]\begin{gathered} 2(w+w+4)\leq56 \\ 2(2w+4)\leq56 \\ 2w+4\leq\frac{56}{2} \\ 2w+4\leq28 \\ 2w\leq28-4 \\ 2w\leq24 \\ w\leq\frac{24}{2} \\ w\leq12 \end{gathered}[/tex]The largest width possible is 12 feet.
Now, we look for the length.
[tex]\begin{gathered} 2(12+l)\leq56 \\ 24+2l\leq56 \\ 2l\leq56-24 \\ 2l\leq32 \\ l\leq\frac{32}{2} \\ l\leq16 \end{gathered}[/tex]Therefore, the largest measure possible for the length is 16 feet.Aubrey paints 4 square feet of her house each minute. How many square feet does she paint in 20 seconds? Round to the nearest tenth.
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We were told that Aubrey paints 4 square feet of her house each minute.
Recall that 1 minute = 60 seconds.
The statement can be written as
Aubrey paints 4 square feet of her house each 60 seconds
If x represents the number of square feets that she paints in 20 seconds, it means that
4 = 60
x = 20
By cross multiplying, it becomes
60 * x = 4 * 20
60x = 80
x = 80/60
x = 1.333
Rounding to the nearest tenth, it becomes 1.3 square feet
She paints 1.3 square feet in 20 seconds
At the sewing store, Kimi bought a bag of mixed buttons.The bag included 100 buttons, of which 10% were large.How many large buttons did kimi get?
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to find the 10% of 100 buttons, we multiply 100 by 0.1 to get the following:
[tex]100\cdot0.1=10[/tex]therefore, Kimi got 10 large buttons
(10,-3),(5,-4)i need help finding the midpoint
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The given points are (10, -3) and (5, -4).
The midpoint formula is
[tex]M=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})_{}[/tex]Where,
[tex]\begin{gathered} x_1=10 \\ x_2=5 \\ y_1=-3 \\ y_2=-4 \end{gathered}[/tex]Replacing these coordinates, we have
[tex]\begin{gathered} M=(\frac{10+5}{2},\frac{-3-4}{2}) \\ M=(\frac{15}{2},-\frac{7}{2}) \end{gathered}[/tex]Therefore, the midpoint is M(15/2, -7/2).Find anequation for the perpendicular bisector of the line segment whose endpointsare (5,-4) and (-9, -8).
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First, we need to find the slope of the line:
Let:
(5, -4) = (x1,y1)
(-9, -8) = (x2,y2)
m = (y2-y1)/(x2-x1) = (-8-(-4))/(-9-5) = -4/-14 = 2/7
Also, we need to find the midpoint:
let:
MP = (xp,yp)
xp= (x1+x2)/2 = (5-9)/2 = -2
yp = (y1+y2)/2 = (-4-8)/2 = -6
MP = (-2, -6)
Now, the slope for the perpendicular bisector is -m = -7/2
y = -mx + b
Using the midpoint
-6 = -7/2(-2) + b
-6 = 7 + b
Solving for b:
b = -13
Therefore, the equation for the perpendicular bisector is:
y = -7x/2 - 13
[tex](x-1)(x^{2}+2)[/tex]
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Answer:
x³-x²+2x-2
That's the answer
Which expression is equivalent to 8c + 6 - 3c - 2 ?A. 5c +4B. 50 + 8C.11c +4D.11c + 8
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This means that the answer is option A
A tourist at scenic Point Loma, California uses a telescope to track a boat approaching the shore. If the boat moves at a rate of5 meters per second, and the lens of the telescope is 30 meters above water level, how fast is the angle of depression of thetelescope (0) changing when the boat is 200 meters from shore? Round any intermediate calculations to no less than sixdecimal places, and round your final answer to four decimal places.
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Lest first we hte sine theorem to relate the given measures:
[tex]\frac{\sin (\theta)}{30}=\frac{\sin(90^{\circ})}{\sqrt[]{x^2+30^2}}[/tex]x represents the distance from the boat to the shore.
[tex]\frac{\sin (\theta)}{30}=\frac{1}{\sqrt[]{x^2+30^2}}[/tex][tex]\frac{\sin(\theta)}{1}=\frac{30}{\sqrt[]{x^2+30^2}}[/tex][tex]\sin (\theta)=\frac{30}{\sqrt[]{x^2+30^2}}[/tex][tex]\theta=\sin ^{-1}(\frac{30}{\sqrt[]{x^2+30^2}})[/tex]Then we must calculate the derivative in order to know the rate of change at a certain point.
[tex]\frac{d}{dx}(\sin ^{-1}(\frac{30}{\sqrt[]{x^2+30^2}}))=-\frac{30x}{\sqrt[]{\frac{x^2}{x^2+900}}\cdot(x^2+900)^{\frac{3}{2}}}[/tex]To find how fast is the angle of depression of the telescope is changing when the boat is 200 meters from shore, replace by 200 on the derivative:
[tex]-\frac{30\cdot200}{\sqrt[]{\frac{200^2}{200^2^{}+900}}\cdot(200^2+900)^{\frac{3}{2}}}=-0.0007\text{ rad/s}[/tex]find the second and third derivative of [tex]y = \sqrt{x} [/tex]
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We can use the power rule to get the second and third derivative of the function.
[tex]\begin{gathered} \text{First derivative:} \\ y^{\prime}=\mleft(\frac{1}{2}\mright)x^{\frac{1}{2}-1} \\ y^{\prime}=(\frac{1}{2})x^{-\frac{1}{2}} \\ y^{\prime}=\frac{x^{-\frac{1}{2}}}{2} \end{gathered}[/tex][tex]\begin{gathered} \text{Second derivative} \\ y^{\prime}^{\prime}=(-\frac{1}{2})\frac{x^{-\frac{1}{2}-1}}{2} \\ y^{\prime\prime}=-\frac{x^{-\frac{3}{2}}}{4}\text{ or }y^{\prime\prime}=-\frac{1}{4x^{\frac{3}{2}}} \\ \end{gathered}[/tex][tex]\begin{gathered} \text{Third derivative} \\ y^{\prime}^{\prime}^{\prime}=(-\frac{3}{2})-\frac{x^{-\frac{3}{2}-1}}{4} \\ y^{\prime\prime\prime}=\frac{3x^{-\frac{5}{2}}}{8}\text{ or }y^{\prime\prime\prime}=\frac{3}{8x^{\frac{5}{2}}} \end{gathered}[/tex]9.A piece of wood is cut into 3 pieces. The lengths are 8'15, 634 and953/8".If 1/4" is used up for each saw cut (kerf), what is the length of the original board?HINT: 2 kerfs are made in cutting the board. Reduce fraction to simplest terms.
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The Solution.
First, we shall convert the given lengths to inches.
[tex]undefined[/tex]points a,b, and are b lies between a and c. if ac=48,ab =2x+2and bc =3x+6, what is bc?
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Given that A, B, and C are collinear and B lies between A and C, then:
AB + BC = AC
Replacing with data:
(2x + 2) + (3x + 6) = 48
(2x + 3x) + (2 + 6) = 48
5x + 8 = 48
5x = 48 - 8
5x = 40
x = 40/5
x = 8
Then,
BC = 3x + 6
BC = 3(8) + 6
BC = 24 + 6
BC = 30
I need help with fractions and word problem can you help me
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Grace wants to bring a small wedge of cheese for the next 12 days, but there are three small rounds of cheddar cheese in Grace's refrigerator, That is to say, that Grace needs to buy cheese to fulfill this purpose, the small wedge of cheese that Grace needs are:
[tex]12-3=9[/tex]but we must represent this infraction, for this, we will take the 12 small wedges of cheese as a unit, which is formed by having the 12 portions.
That is, out of 12/12 portions, we only have 3/12, To do this we do subtraction and see how many we need in fractional form.
[tex]\begin{gathered} \frac{12}{12}=1 \\ \frac{12}{12}-\frac{3}{12}=\frac{9}{12} \end{gathered}[/tex]In conclusion, Grace need 9 portions i.e. 9/12
Now, that we know this fraction, this is a correct answer, however, we can simplify this fraction.
[tex]\frac{9}{12}=\frac{3}{4}[/tex]In conclusion, Grace needs 3/4 small wedge of cheese.
Note: the answer can be 9/12, but also its simplified form which corresponds to the same, this simplified form is 3/4.
What is the slope for this equation y = -2.5x + 92
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EXPLANATION
Given the equation y = -2.5x + 92
The slope is equal to -2.5
For the interval expressed in the number line, write it using set-builder notation and interval notation.
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Answer:
Writing the number line in set builder notation we have;
[tex]\mleft\lbrace x\mright|x>0\}[/tex]Writing in interval notation.
[tex]x=(0,\infty)[/tex]Explanation:
Given the number line in the attached image.
x starts on 0, with a non shaded circle and pointed to the right/positive direction.
So;
[tex]x>0[/tex]Writing the number line in set builder notation we have;
[tex]\mleft\lbrace x\mright|x>0\}[/tex]Writing in interval notation.
[tex]x=(0,\infty)[/tex]Since the upper boundary of x is not stated then we will represent it with infinity in the interval notation.
[tex]\begin{gathered} (\text{ }\rightarrow\text{ greater than} \\ \lbrack\text{ }\rightarrow\text{ greater than or equal to } \\ so,\text{ } \\ 0Linda must choose a number between 55 and 101 that is a multiple of 3,5, and 9. Write all the numbers that she could choose.
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We will have the following:
*The LCM of the numbers given (3, 5 & 9) is 72. [This is the value to chosse]. We will have that 90 is also a multiple of 3, 5 & 9 [This is other value that can be chosen].