The number of animals are given as;
[tex]\begin{gathered} \text{Turtles}=8 \\ \text{Fish}=15 \\ \text{Pelicans}=17 \\ \text{Total animals}=40 \end{gathered}[/tex]The percentage that are pelicans would be calculated as follows;
[tex]\begin{gathered} \text{Percentage Pelicans}=\frac{17}{40}\times\frac{100}{1} \\ \text{Percentage Pelicans}=\frac{1700}{40} \\ P\text{ercentage Pelicans}=42.5\text{ } \end{gathered}[/tex]ANSWER:
The percentage of the animals that are Pelicans is 42.5%
Question 2-22
A cake is cut into 12 equal slices. After 3 days Jake has eaten 5 slices. What is his weekly rate of eatin
5
36
35
36
O
●
cakes/week
1
01 cakes/week
35
1
1 cakes/week
4
< Previous
cakes/week
2022-2023 T-Math-Gr7Reg-T2-CBT: Section 2-...
Jake's weekly rate of eating of cake slices is 11.6
Given,
Number of equal slices of cake = 12
Number of slices Jake eaten after 3 days = 5
We have to find the weekly rate of eating;
Here,
Number of days in a week = 7
Jake eaten 5 slices in 3 days so, 7 - 3 = 4
Again after 3 days 5 slices.
Then,
4 - 3 = 1
That is, Jake eaten 10 slices of cake in 6 days.
Number of cake slices eaten in 1 day = 5/3 = 1.6
Therefore,
Weekly rate of eating is 10 + 1.6 = 11.6
That is,
Jake's weekly rate of eating of cake slices is 11.6
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Trying to find the length of the robot arm given the xy point along with angle theta and angle theta r
8a. The length of the robot arm, and the angle in radian and degrees are as follows
length of the robot arm = 30 unitsangle in radian = 7π/6 radian (anticlockwise)angle in degrees = 210 degrees (anticlockwise)8b. The length of the robot arm, and the angle in radian and degrees are as follows
length of the robot arm = 60 unitsangle in radian = π/6 radian (clockwise)angle in degrees = 210 degrees (clockwise)How to find the length of the robot arm and the angles8a
i. let the length of the robot arm be r
The length of the lever arm using the points is done by
r^2 = ( -26 )^2 + ( -15 )^2
r^2 = 676 + 225
r^2 = 900
r = √ 900
r = 30
ii. theta in radian
tan θ = opposite / adjacent
tan θ = y direction / x direction
tan θ = -15 / -26
tan θ = 15/26
θ = Arc tan ( 15/26 )
θ = 29.9816 degrees
θ ≈ 30 degrees
θ ≈ 30 degrees + 180 = 210 degrees
In radian
π = 180 degrees
? = 210 degrees
we cross multiply to get
? * 180 degrees = π * 210 degrees
? = π * 210 degrees / 180 degrees
= 7π/6 radian
8b
i. let the length of the lever arm be r
The length of the lever arm using the points is done by
r^2 = ( -30 )^2 + ( 52 )^2
r^2 = 900 + 2704
r^2 = 3604
r = √ 3604
r = 60.033
r ≈ 60
ii. theta in radian
tan θ = opposite / adjacent
tan θ = y direction / x direction
tan θ = -30 / 52
tan θ = -30/52
θ = Arc tan ( -30/52 )
θ = -29.9816 degrees
θ ≈ -30 degrees (clockwise direction)
In radian
π = 180 degrees
? = 30 degrees
we cross multiply to get
? * 180 degrees = π * 30 degrees
? = π * 30 degrees / 180 degrees
= π/6 radian
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Graph the y = 2x ^ 2 - 12x + 15 Plot five points on the parabolathe vertextwo points to the left of the vertex, and two points to the right of the vertexThen click on the graph-a-function button
Explanation:
Given the function:
[tex]y=2x^2-12x+15[/tex]First, we find the vertex of the parabola.
Vertex
The equation of the axis of symmetry is calculated using the formula:
[tex]x=-\frac{b}{2a}[/tex]From the function: a=2, b=-12
[tex]\implies x=-\frac{-12}{2(2)}=\frac{12}{4}=3[/tex]Substitute x=3 into y to find the y-coordinate at the vertex.
[tex]\begin{gathered} y=2x^2-12x+15 \\ =2(3)^2-12(3)+15 \\ =18-36+15 \\ =-3 \end{gathered}[/tex]The vertex is at (3, -3).
Two points to the left of the vertex
When x=2
[tex]\begin{gathered} y=2x^2-12x+15 \\ =2(2)^2-12(2)+15=8-24+15=-1 \\ \implies(2,-1) \end{gathered}[/tex]When x=1
[tex]\begin{gathered} y=2x^2-12x+15 \\ =2(1)^2-12(1)+15=2-12+15=5 \\ \implies(1,5) \end{gathered}[/tex]Two points to the right of the vertex
When x=4
[tex]\begin{gathered} y=2x^2-12x+15 \\ =2(4)^2-12(4)+15=32-48+15=-1 \\ \implies(4,-1) \end{gathered}[/tex]When x=5
[tex]\begin{gathered} y=2x^2-12x+15 \\ =2(5)^2-12(5)+15=50-60+15=5 \\ \implies(5,5) \end{gathered}[/tex]Answer:
Plot these points on the graph: (3, -3), (2,-1), (1,5), (4,-1), and (5,5).
-3-23x=-14(x-21)-15(x+33)
Find (x)
Please I keep messing up somewhere so please show step by step
The value of (x) that satisfy the equation → - 3 - 23x = - 14(x - 21) - 15(x + 33) is x = - 33.
What is Equation?
An equation is a mathematical statement with an 'equal to' symbol between two expressions that have equal values.
Given is the following equation -
- 3 - 23x = - 14(x - 21) - 15(x + 33)
The given equation is -
- 3 - 23x = - 14(x - 21) - 15(x + 33)
Simplifying for (x), we get -
- 3 - 23x = - 14x + 294 - 15x - 495
- 23x + 14x + 15x = 3 + 294 - 495
6x = - 198
x = - 33
Therefore, the value of (x) that satisfy the equation → - 3 - 23x = - 14(x - 21) - 15(x + 33) is x = - 33.
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Solve the system of inequalities by graphing.
y ≥ 2
y < 4
Select a line to change it between solid and dotted. Select a region to shade it.
The area between the solid line (y=x+4) and the dotted line (y=-2x-2) represents the system of inequalities solution set.
Two linear inequalities system on a coordinate plane. The first has a solid line graphed with a negative slope of one, a negative y-intercept, and a shaded origin area. The area encompassing the origin is shaded, and the second is a dashed vertical line 3 units to the left of the origin.
x ≥ –3; y ≥ x – 2
x > –3; 5y ≥ –4x – 10
x > –3; y ≥ –x + 1
x > –2; y ≥ –x – 1
The solution set for the system of inequalities is represented by the region between the solid line (y=x+4) and the dotted line (y=-2x-2).
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Find the area of a triangular window with the given base and height base = 10 ft height =7 ft
The area of a triangular shape is given by the following formula
[tex]A=\frac{bh}{2}[/tex]where,
base, b = 10 ft
height, h = 7 ft
therefore,
[tex]A=\frac{10\cdot7}{2}=\frac{70}{2}=35[/tex]thus, the answer is 35 ft^2
Use y = 3x2 + 18x - 2 to answer the following question(1, 19) is a point on the graph. What point is the reflection of (1, 19) across the axis of symmetry of the parabola?
Since y is a parabolla, there will be two values for y = 19. We already know that x = 1 is one value, to find the other, we can substitute y = 19 on the equation and solve for x to get the following:
[tex]\begin{gathered} 19=3x^2+18x-2 \\ \Rightarrow3x^2+18x-2-19=0 \\ \Rightarrow3x^2+18x-21=0 \\ \Rightarrow3(x^2+6x-7)=0 \\ \Rightarrow3(x-1)(x+7)=0 \end{gathered}[/tex]the solutions of the equation are x = 1 and x = -7. Since we already have that (1,19) is a point on the graph, then we have that the other point is (-7,19)
HELP!!
Triangle ABC is shown with exterior ∠z.
triangle ABC with angle A labeled 58 degrees, angle B labeled 44 degrees, and side AC extended with angle z labeled as exterior angle to angle C
Determine m∠z.
136°
102°
78°
58
The sum of all interior angles of a triangle is 180° thus the measure of the exterior angle m∠K is 102° so option (B) is correct.
What is a triangle?A triangle is a 3-sided shape that is occasionally referred to as a triangle. There are 3 sides and three angles in every triangle, some of which may be the same.
Triangle is a very common figure to deal with in our daily life.
It is known that the sum of all three angles inside a triangle will be 180°.
So, m∠A + m∠B + m∠C = 180°
m∠C = 180° - 58° - 44°
m∠C = 78°.
The exterior angle of C = 180- 78 = 102°.
Hence "The sum of all interior angles of a triangle is 180° thus the measure of the exterior angle to m∠K is 102°".
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Answer:
136
Using the exterior angle theorem,
Step-by-step explanation:
Determine if the line passing through A(7,5) and B(-14, -9) is parallel, or perpendicular to the line passing through C(0,1) and D(4, -5).
To solve this problem, we will use the two pair of points to find the slope of the equation of each line. Then, by comparing these slopes, we can determine either if they are perpendicular or parallel.
Slope calculations
To calcula the slopes, given the pairs of points, we are going to use the following formula: Given points (a,b) and (c,d) the slope of the line that passes through them is given by the formula
[tex]m=\frac{d\text{ - b}}{c\text{ -a}}=\frac{b\text{ - d}}{a\text{ -c}}[/tex]Let us calculate first the slope of the line that passes through the points (7,5) and (-14,-9). In this case, we have a=7,b=5,c=-14 and d=-9. So we get
[tex]m=\frac{5\text{ - (-9)}}{7\text{ - (-14)}}=\frac{14}{21}=\frac{2\cdot7}{3\cdot7}=\frac{2}{3}[/tex]Now, let us calculate the slope of the line that passes through the points (0,1) and (4,-5). In this case, we have a=0,b=1,c=4 and d=-5. So we get
[tex]m=\frac{1\text{ -(-5)}}{0\text{ - 4}}=\frac{6}{\text{ -4}}=\text{ -}\frac{3\cdot2}{2\cdot2}=\text{ -}\frac{3}{2}[/tex]Slope comparison
Now, we compare the slopes to determine if the lines are perpendicular or parallel. Recall that two lines are parallel if they have the same slope and they are perpendicular if the product of their slopes is -1. From our calculations, we can see that the slopes are not equal. Let us confirm that they are perpendicular. To do so, we multiply both slopes. So we get
[tex]\frac{2}{3}\cdot(\text{ -}\frac{3}{2})=\text{ -1}[/tex]Since their product is -1, this confirms that both lines are perpendicular.
Simplify the expression by first transforming the radical to exponential form. Leave the answer in exact form as a radical or a power, not as a decimal approximation.
Answer:
[tex]\textsf{Radical form}: \quad \sqrt[4]{2}\\\\\textsf{Exponent form}: \quad 2^{\frac{1}{4}}[/tex]
Step-by-step explanation:
Given expression:
[tex]\sqrt{8} \div \sqrt[4]{32}[/tex]
[tex]\textsf{Apply the exponent rule} \quad \sqrt[n]{a}=a^{\frac{1}{n}}:[/tex]
[tex]\implies 8^{\frac{1}{2}} \div 32^{\frac{1}{4}}[/tex]
Rewrite 8 as 2³ and 32 as 2⁵:
[tex]\implies (2^3)^{\frac{1}{2}} \div (2^5)^{\frac{1}{4}}[/tex]
[tex]\textsf{Apply the exponent rule} \quad (a^b)^c=a^{bc}:[/tex]
[tex]\implies 2^{\frac{3}{2}} \div 2^{\frac{5}{4}}[/tex]
[tex]\textsf{Apply the exponent rule} \quad a^b \div a^c=a^{b-c}:[/tex]
[tex]\implies 2^{\frac{3}{2}-\frac{5}{4}}[/tex]
[tex]\implies 2^{\frac{6}{4}-\frac{5}{4}}[/tex]
[tex]\implies 2^{\frac{1}{4}}[/tex]
[tex]\textsf{Apply the exponent rule} \quad \sqrt[n]{a}=a^{\frac{1}{n}}:[/tex]
[tex]\implies \sqrt[4]{2}[/tex]
When you mix two colors of paint in equivalent ratios, the resulting color is always the same. Complete the table as you answer the questions.
How many cups of yellow paint should you mix with 1 cup of blue paint to make the same shade of green? Explain or show your reasoning.
Make up a new pair of numbers that would make the same shade of green. Explain how you know they would make the same shade of green.
row 1
cups of blue paint
cups of yellow paint
row 2
2
10
row 3
1
5
row 4
3
15
What is the proportional relationship represented by this table?
What is the constant of proportionality? What does it represent?
By Calculating the Constant of Proportionality, we get,
1. 5 cups of yellow paint is required for 1 cup of blue paint.
2. For 4 cups of blue paint, 20 cups of yellow paint is required.
For 5 cups of blue paint, 25 cups of yellow paint are required.
3. Cups of Yellow Paints = 5* Cups of Blue Paints
4. The constant of Proportionality is 5.
Let the cups of blue paint = x
and cups of yellow paint = y
From the table, we can infer that to make the same shade of green, we need to mix 5 cups of yellow color with 1 cup of blue.
We have, To make the same shade of green,
we need to mix 5 cups of yellow color with 1 cup of blue.
we need to mix 10 cups of yellow color with 2 cups of blue.
we need to mix 15 cups of yellow color with 3 cups of blue.
So, here we can see a relationship between the two colors, blue (x) and green(y)
Let k be the constant of proportionality, then, we have :
10 = k *2
k =[tex]\frac{10}{2} = 5[/tex]
Hence, For one cup of blue paint, we need 5 cups of yellow paint to make the same shade of green.
And the equation of the same is y =kx, that is y =5x....equation(1)
To make new pair of numbers that would make the same shade of green.
We can use the equation 1,
for x = 4, we need y = 20
foe x =5, we need y = 25
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What is the slope of the line passing through (-2, 4)
and (3, -4)?
Answer:
[tex]\frac{-8}{5}[/tex]
Step-by-step explanation:
Slope is the change in y over the change in x
[tex]\frac{y_{2 - y_{1} } }{x_{2 - y_{1} } }[/tex]
(-2,4) is ([tex]x_{1}[/tex], [tex]y_{1}[/tex])
(3. -4) is ([tex]x_{2}[/tex],[tex]y_{2}[/tex])
[tex]\frac{-4 - 4}{3 - -2}[/tex] = [tex]\frac{-8}{3+ 2}[/tex] = [tex]\frac{-8}{5}[/tex]
Answer:
-8 / 5
Step-by-step explanation:
We can use the slope formula
m = ( y2-y1)/(x2-x1)
= ( -4 -4)/( 3 - -2)
= -8/ ( 3+2)
= -8 / 5
equati 1. v = 3.1 + 14 y = -1
y= 3x +14 (a)
y=-4x (b)
Replace the value of y (b) on equation (a), and solve for x
-4x = 3x+14
-4x-3x = 14
-7x = 14
x= 14/-7
x = -2
Replace the value of x on any initial equation and solve for y:
y= -4 (-2) = 8
y=8
Answer:
1*v=3,0.1+14y=-1 : v=3,y=-0.07857
[1.v=3}
[0.1+14y=-1]
v=3,y=-0.07857
Step-by-step explanation:
I need help with part b.
log ( √2.86668684 ) is value of logarithm 18 .
What is logarithm used for in math?
In order to get another number, a number must be raised to a certain power, which is known as a logarithm (see Section 3 of this Math Review for more about exponents). For instance, the base ten logarithm of 100 is 2, since ten multiplied by two yields the number 100: log 100 = 2.The answer is found using a logarithm (or log).[tex]log_{b} \sqrt{18}[/tex] = [tex]log_{b} = log_{b} \sqrt{15} + log_{b} \sqrt{3}[/tex]
= log( √15 + √3 )
= log ( √1.0986 * 1.6094 + √1.0986 )
= log ( √1.76808684 + √1.0986 )
= log ( √2.86668684 )
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please help with this practice question thank you
Answer:
Formula for slope of line is given as y2-y1 ÷ x2-x1 or y1-y2 ÷ x1-x2, where x and y are the coordinates of the points.
First, identify the coordinates of the two points shown on the graph.
First coordinate is (0,-1) and second coordinate is (3,1).
After that, find the slope of the line using the formula.
Slope = (1-(-1))÷(3-0)
= 2/3
Suppose you buy two sweaters. Each sweater cost the same amount. The amount of money you spend varies directly with the number of sweaters you buy. If you spent $37.50 for two sweaters, what is the constant of variation? A. 2.00 B. 37.50 C. 0.053 D. 18.75
Answer:
D
Step-by-step explanation:
37.50 ÷ 2 = 18.75. jdjdjfbffcv
4^5 x 4^3 x 4^2 x 4^3 / 4^2 x 4 x 4^2 simplify help
The simplification form of the given mathematical equation 4^5 x 4^3 x 4^2 x 4^3 / 4^2 x 4 x 4^2 is [tex]4^{8}[/tex]
In the given question, it is given
We know the simplification property of the division of numbers as,
Division in exponential form
[tex]\frac{a^{x} }{a^{y} }[/tex] = [tex]a^{x-y}[/tex] , and
Multiplication in exponential form
[tex]a^{x} . a^{y}[/tex] = [tex]a^{x+y}[/tex]
Similarly, we'll apply the same property to solve this question,
[tex]4^{5+3+2+3 + ( -2 -1 -2)}[/tex]
[tex]4^{5+3+2+3 - ( 2 + 1 + 2)}[/tex]
[tex]4^{13 - 5}[/tex]
[tex]4^{8}[/tex]
Hence, the simplification form of the given mathematical equation 4^5 x 4^3 x 4^2 x 4^3 / 4^2 x 4 x 4^2 is [tex]4^{8}[/tex]
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a box contains 3 white balls and 4 black balls. a ball is drawn at random the color is recorded and then the ball is put back in the box. Then a second ball is drawn at random from the same box. find the probability of the event that at least one of the balls is white
The box has 3 white balls and 4 black balls.
Total number of balls = 3 + 4 = 7
First draw:
The probability of getting a white ball is given by
[tex]\begin{gathered} P(white)=\frac{\text{number of white balls}}{total\text{ number of balls}} \\ P(white)=\frac{3}{7} \end{gathered}[/tex]Second draw:
Notice that after the first draw the ball is put back in the box.
The probability of getting a white ball is given by
[tex]P(white)=\frac{3}{7}[/tex]At least one of the balls is white means that one white ball or two white balls.
[tex]P(x\ge1)\; =P(x=1)+P(x=2)_{}[/tex]We have already found the probability of getting one white ball that is P(x=1) = 3/7
The probability of getting two white balls is
[tex]\begin{gathered} P(two\; white)=P(white)\times P(white) \\ P(two\; white)=\frac{3}{7}\times\frac{3}{7}=\frac{9}{49} \end{gathered}[/tex]Finally, the probability of at least one white ball is
[tex]\begin{gathered} P(x\ge1)\; =P(x=1)+P(x=2)_{} \\ P(x\ge1)\; =\frac{3}{7}+\frac{9}{49} \\ P(x\ge1)\; =\frac{30}{49} \end{gathered}[/tex]Therefore, the probability of the event that at least one of the balls is white is 30/49
Write a two-column proof.
4. Given: AB EF
AC DF
Prove: ABC ~ FED
Please help
Explanation:
The following is a proof that ∆ABC ~ ∆FED.
Statement . . . . Reason1. AB║EF, AC║FD . . . . given
2. ∠BCA ≅ ∠EDF . . . . alternate exterior angles theorem
3. ∠ABC ≅ ∠FED . . . . alternate interior angles theorem
4. ∆ABC ~ ∆FED . . . . AA similarity postulate
In reference to 3x + 2 ≥-4Is 0 a solution?Yes or No?
ANSWER :
yes
EXPLANATION :
From the problem, we have the inequality :
[tex]3x+2\ge-4[/tex]To check if a number is a solution, substitute it to the inequality and check the truthfulness.
Check if 0 is a solution :
[tex]\begin{gathered} 3(0)+2\ge-4 \\ 2\ge-4 \end{gathered}[/tex]Since 2 is greater than -4, therefore 0 is a solution
The graph shows the reciprocal parent function.Which statement best describes the function?O A. The function is negative when x < 0.OB. The function is never negative.C. The function is negative when x > 0.O D. The function is negative when x < 0 and also when x > 0.PREVIOUS
Let's begin by identifying key information given to us in the graph:
The reciprocal parent function is given as 1/f(x). This is better written as shown below
[tex]\begin{gathered} f(x)=\frac{a}{(x-h)}+k \\ when\colon h=0,k=0,a=1 \\ f(x)=\frac{1}{x-0}+0 \\ f(x)=\frac{1}{x} \\ f(x)=y \\ \Rightarrow y=\frac{1}{x} \end{gathered}[/tex]When the value for x is greater than zero, the function is positive as shown below:
[tex]\begin{gathered} x=2 \\ y=\frac{1}{2}=\frac{1}{2} \\ y=\frac{1}{2} \end{gathered}[/tex]When the value of x is lesser than zero, the function is negative as shown below:
[tex]\begin{gathered} x=-1 \\ y=\frac{1}{-1}=-1 \\ y=-1 \end{gathered}[/tex]Therefore, the correct answer is option A (The function is negative when x < 0)
jamals lawn is shaped like a square with an area of 224.9 ft2. which measurement is closest to the side length of his lawn in feet?
Given:
Area of the square shaped lawn = 224.9 ft²
A square has all four side lengths equal.
To find the side length of this lawn, use the formula for area of a square below:
[tex]\text{Area = }L^2[/tex]Take the square root of both sides to find the sile length L:
[tex]\begin{gathered} \sqrt[]{Area\text{ }}=\sqrt[]{L^2} \\ \\ \sqrt[]{Area}\text{ = L} \\ \\ \sqrt[]{224.9}\text{ = L} \\ \\ 14.99\text{ ft = L} \end{gathered}[/tex]Therefore, the measurement that is closest to the side length in feet is 15 ft
ANSWER:
15 ft
Hi I need help with this! I completed most of it, but I need help with the last portion. Thank you!
The equation of the axis of symmetry is x = 2
Explanation:The equation of the axis of symmetry is the value of x that divides the parabola into equal halves (mirror images).
The value of x which gives the mirror images is the value of x coordinate of the vertex. The vertex is at (2, 0). The x coordinate = 2
From the graph, we can also see it occurred at x = 2
The axis of symmetry: x = x-coordinate of the vertex
The equation of the axis of symmetry is x = 2
I BEG YOU FOR HELP!!! Determine the relationship between the two triangles and whether or not they can be proven to be congruent.
Answer:
Step-by-step explanation:
The triangles are congruent because they follow the SSS triangle congruence postulate. As the SSS postulate says that all 3 sides of one triangle are congruent to another triangle's sides, these triangles shown have all three of their sides congruent to each other.
Answer: I am no sure but the only way to decide whether a pair of triangles are congruent would be to measure all of the sides and angles, and these triangles do not look the same so I would say that these tringles are not congruent.
Step-by-step explanation:
Answer the questions below.
(a) Amount of data entered
The amount of data entered affects his pay at the end of the week.(b) R
The independent variable is the input.(c) Average temperature
The average temperature depends on the city's distance from the equator.Which of the expressions are equivalent to the one below? Check all that apply.
( 15 • 3) - 20
Answer:
I don't know your answer choices but...
Step-by-step explanation:
( 15 • 3) - 20 is equal to:
45-20
25
(5)(3)(3)-20
You plan to work for 40 years and then retire using a 25-year annuity. You want to arrange a retirement income of $4500 per month. You have access to an account that pays an APR of 8.4% compounded monthly.
The desired monthly yield at the retirement time will be equal to $565,714.28.
Compound Interest may be defined as the interest earned by the bank on the basis of principle and also accumulated interest which increases exponentially and not linearly with respect to time. In calculating compound interest, the amount earned at the end of first year becomes principle for the next year and so on. Compound interest can be calculated, annually, half-yearly or quarterly etc.
Time for which work is planned = 40 years, Principle = $4500 and APR = 8.4% = 0.084/12 = 0.007.
The value of n = 12 × 25 = 300
The amount can be calculated by the formula A = P/r [1 - (1 + r) ⁻ⁿ]
A = (4500/0.007) [1 - (1 + 0.007) ⁻³⁰⁰]
A = 642,857.14 [1 - 0.12]
A = 642,857.14 × 0.88
A = $565,714.28 which is required amount.
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Complete Question:
You plan to work for 40 years and then retire using a 25-year annuity. You want to arrange a retirement income of $4500 per month. You have access to an account that pays an APR of 8.4% compounded monthly. What monthly deposits are required to achieve the desired monthly yield at retirement?
Please help me with my calculus homework, only question 3****
I would start by stating the Fundamental Theorem of Calculus which states that;
If a function f is continuous on a closed interval [a,b] and F is an antiderivative of f on the interval [a,b], then
[tex]\int ^b_af(x)dx=F(b)-F(a)\text{ }[/tex]Let
[tex]\begin{gathered} f(x)=x^3-6x^{} \\ F(x)=\int f(x)dx=\int (x^3-6x)dx \end{gathered}[/tex]Recall that;
[tex]\int x^n=\frac{x^{n+1}}{n+1},n\ne-1[/tex]That implies that,
[tex]F(x)=\int (x^3-6x)dx=\int x^3dx-\int 6xdx=\frac{x^4}{4}-6(\frac{x^2}{2})=\frac{x^4}{4}-3x^2+C[/tex]Applying the Fundamental Theorem of Calculus, where a=0, b=3
[tex]\begin{gathered} \int ^3_0(x^3-6x)dx=F(3)-F(0) \\ F(3)=\frac{3^4}{4}-3(3)^2+C=\frac{81}{4}-27+C=-\frac{27}{4}+C \\ F(0)=\frac{0^4}{4}-3(0)^2+C=C \\ \Rightarrow\int ^3_0(x^3-6x)dx=-\frac{27}{4}+C-C=-\frac{27}{4} \end{gathered}[/tex]So the answer is -27/4
10a-4a simplify
PLEASEE I NEED HELLPPP
they answer is 6a because you mines 10-4 and you get 6 then just put the loke 6a
i need help with this asap please
Answer:
Given that,
[tex]2+(-8)+32+(-128)+.\ldots_{}[/tex]To find the sum of the first 5 terms.
First, to find the first 5 terms of the given sequence.
The given sequence is 2,-8,32,-128,...
It follows geometric series with initial term 2, and common ratio as -4
The explicit formula of the given sequence is,
[tex]t_n=2(-4)^{n-1}_{}_{}[/tex]To find the 5th term of the sequence,
Put n=5 in the above equation we get,
[tex]t_5=2(-4)^{5-1}[/tex][tex]t_5=2(-4)^4[/tex][tex]t_5=2(256)[/tex][tex]t_5=512[/tex]Since common ratio is less than 1, we get the sum of the series formula as,
[tex]S_n=\frac{a(1-r^n)}{1-r}[/tex]Substituting the values we get,
[tex]S_5=\frac{2(1-(-4)^5)}{1+4}[/tex][tex]=\frac{2(1+1024)}{5}[/tex][tex]=\frac{2(1025)}{5}[/tex][tex]=2(205)[/tex][tex]=410[/tex]The sum of the first 5 terms of the given series is 410.
Answer is: option B: 410