Answer and Step-by-step explanation: There are a number of ways of calculating an area under a curve. The more precise way is to use Definite Integral:
The function is [tex]2x\sqrt{x^{2}+1}[/tex], then the area under, with interval between 0 and 6 is:
[tex]\int\limits^6_0 {(2x\sqrt{x^{2}+1}) } \, dx[/tex]
To solve this integration, use substitution method, in which:
[tex]u=x^{2}+1[/tex]
[tex]\frac{du}{dx}=2x[/tex]
du = 2xdx
Replacing into the integral:
[tex]\int\limits^a_b {\sqrt{u} } \, du[/tex]
Solving:
[tex]\int\limits^a_b {\sqrt{u} } \, du=\frac{2}{3} \sqrt{u^{3}}[/tex]
Replacing it back to x:
[tex]\int\limits^6_0 {2x\sqrt{x^{2}+1} } \, dx =\frac{2}{3}\sqrt{(x^{2}+1)}[/tex]
Substituing limits between 0 and 6:
[tex]= \frac{2}{3}[\sqrt{(6^{2}+1)^{3}}-\sqrt{(0^{2}+1)^{3}} ][/tex]
= 149.37
Area under the curve using Integration is 149.37 square units
Another way of calculating area under the curve is dividing the area into a number of small rectangles and then adding the area of each one. This method is called Riemann Sums and it is an approximation of the area.
The method is done by the following relation:
in which
i is the n, the number of subintervals the area is dividing into
Δx is width of each subintervals.
For the function f(x) = [tex]2x\sqrt{x^{2}+1}[/tex], interval between 0 and 6:
subinterval n1 = 12:[tex]\Delta x=\frac{6-0}{12}[/tex]
[tex]\Delta x=[/tex] 0.5
[tex]A=\Sigma f(x_{i}).\Delta x[/tex]
[tex]A = f(0)*0.5+f(0.5)*0.5+f(1)*0.5+f(1.5)*0.5+f(2)*0.5+f(2.5)*0.5+f(3)*0.5+f(3.5)*0.5+f(4)*0.5+f(4.5)*0.5+f(5)*0.5+f(5.5)*0.5[/tex]
[tex]A=0+1.12*0.5+2.83*0.5+...+50.99*0.5+61.49*0.5[/tex]
A = 131.575 square units
subinterval n2 = 5:[tex]\Delta x=\frac{6-0}{5}[/tex]
Δx = 1.2
[tex]A=f(0)*1.2+f(1.2)*1.2+f(2.4)*1.2+f(3.6)*1.2+f(4.8)*1.2[/tex]
[tex]A=0*1.2+3.75*1.2+12.48*1.2+26.90*1.2+47.07*1.2[/tex]
A = 108.24 square units
Comparing results, notice that with less subintervals, the area is far from the exact measure. It occurs because Riemann Sums is an approximation method. So, if there are more subintervals, more approximate is the area, therefore, more precise it will be.
PLEASE HELP
A car dealer offers a 15% discount off the list price x for any car on the lot. At the same time, the manufacturer offers a $1500 rebate for each purchase of a car.
a. Write a function f(x) to represent the price after the discount is applied.
b. Write a function g(x) to represent the price after the rebate is applied.
Suppose the list price of a car is $19,000. Use a composite function to find the price
of the car:
C. if the discount is applied before the rebate;
D. if the rebate is applied before the discount
Answer:
A) f(x) = 0.85x
B) g(x) = x - 1500
C) g(f(x)) = $14650
D) f(g(x)) = $14875
Step-by-step explanation:
A) The list price is x and a 15% discount is applied.
Thus;
f(x) = x - 15%x
f(x) = 0.85x
B) We are told that the manufacturer offers a $1500 rebate for each purchase of a car.
Thus, the function g(x) to represent the price after the rebate is applied is;
g(x) = x - 1500
C)if the discount is applied before the rebate, the function is;
g(f(x))
Now,
f(x) = 0.85(19000)
f(x) = 16150
g(x) = x - 1500
Thus;
g(x) = 16150 - 1500
g(x) = $14650
D) If discount after rebate, then we have; f(g(x))
g(x) = 19000 - 1500
g(x) = 17500
f(g(x)) = 0.85(17500)
f(g(x)) = $14875
A meat inspector has randomly selected 30 packs of 95% lean beef. The sample resulted in a mean of 96.2% with a sample standard deviation of 0.8%. Calculate an upper prediction bound for the leanness of a new pack using a prediction level of 99%. Assume normality. The contents of seven similar containers of sulfuric acid are 9.8, 10.2, 10.4, 9.8,10.0, 10.2, and 9.6 liters. Find a 95% confidence interval for the mean contents of all such containers, assuming an approximately normal distribution.
Answer:
a
The upper bound of the 99% prediction level is [tex] 98.2 [/tex]
b
The 95% confidence interval is [tex]9.7383 < \mu < 10.2617 [/tex]
Step-by-step explanation:
Considering first question
From the question we are told that
The sample size is n = 30
The sample mean is [tex]\= x = 96.2\%[/tex]
The standard deviation is [tex]s = 0.8\%[/tex]
Generally the degree of freedom is mathematically represented as
[tex]df = n - 1[/tex]
=> [tex]df = 30 - 1[/tex]
=> [tex]df = 29[/tex]
From the question we are told the confidence level is 99% , hence the level of significance is
[tex]\alpha = (100 - 99 ) \%[/tex]
=> [tex]\alpha = 0.01[/tex]
Generally from the t distribution table the critical value of at a degree of freedom of is
[tex]t_{\alpha , 29} = 2.462[/tex]
Generally the 99% prediction level is mathematically represented as
[tex]\= x \pm [(t_{\alpha , df }) * s * (\sqrt{1 + \frac{1}{ n} } )}] [/tex]
Generally the upper bound of the 99% prediction level is mathematically represented as
[tex]\= x + [(t_{\alpha , df }) * s * (\sqrt{1 + \frac{1}{ n} } )}] [/tex]
=> [tex] 96.2 + (2.462 ) * 0.8 * (\sqrt{1 + \frac{1}{ 30} } )}] [/tex]
=> [tex] 98.2 [/tex]
Considering second question
Generally the sample is mathematically represented as
[tex]\= x = \frac{\sum x_i}{n}[/tex]
=> [tex]\= x = \frac{ 9.8 + 10.2 + \cdots +9.6 }{7}[/tex]
=> [tex]\= x = 10[/tex]
Generally the standard deviation is mathematically represented as
[tex]\sigma = \sqrt{ \frac{ \sum ( x_ i - \= x)}{n-1} }[/tex]
=> [tex]\sigma = \sqrt{ \frac{ ( 9.8 -10)^2 + ( 10.2 -10)^2 + \cdots + ( 9.6 -10)^2 }{7-1} }[/tex]
=> [tex]\sigma = 0.283[/tex]
Generally the degree of freedom is mathematically represented as
[tex] df = n- 1 [/tex]
=> [tex] df = 7- 1 [/tex]
=> [tex] df = 6 [/tex]
From the question we are told the confidence level is 95% , hence the level of significance is
[tex]\alpha = (100 - 95 ) \%[/tex]
=> [tex]\alpha = 0.05[/tex]
Generally from the t distribution table the critical value of at a degree of freedom of is
[tex]t_{\frac{\alpha }{2} , 6 } = 2.447[/tex]
Generally the margin of error is mathematically represented as
[tex]E = t_{\frac{\alpha }{2} , 6 } * \frac{\sigma }{\sqrt{n} }[/tex]
=> [tex]E =2.447* \frac{0.283 }{\sqrt{7} }[/tex]
=> [tex]E =0.2617[/tex]
Generally 95% confidence interval is mathematically represented as
[tex]\= x -E < \mu < \=x +E[/tex]
=> [tex]10 -0.2617 < \mu < 10 + 0.2617[/tex]
=> [tex]9.7383 < \mu < 10.2617 [/tex]
Please help!!!
If h(x)= -3x-10, find h(-3)
Please give an explanation too. Thank you.
Answer:
h(-3)=-1
Step-by-step explanation:
Replace all x in h(x)= -3x-10 by -3:
h(-3)=-3(-3)-10
Now multiply -3 and -3:
h(-3)=9-10
Simplify 9 and -10 (aka add 9 and -10 together):
h(-3)=-1
Answer: h(-3)=-1
Hope this helps!
Answer:
-1
Step-by-step explanation:
If h(x)= -3x-10 and we want to solve for h(-3) then we have to plug in -3 for all x variables. h(-3)= -3(-3)-10, then solve using PEMDAS.
First multiple h(-3)= 9-10, multiple -3*-3 to get 9.
Then subtract, h(-3)=-1
I feel it could be AAS since both angles could be alternate interior but it could also be SAS so could someone help please?
Answer: D) Side-Angle-Side (SAS)
Check out the diagram below. Note the color coding to see how the angles and sides correspond to the given information. We aren't given RT = RT, but we use the reflexive property in this case. The angles are between the sides so we use SAS.
We don't have enough information to determine anything about the other angles, so we cannot use AAS or ASA. Also, recall that SSA is not a valid congruence theorem. Luckily the angle is between the sides so we don't have to worry about SSA.
Answer This math question and ill give brainliest
Answer:
16 miles
Step-by-step explanation:
there are 8 units between the points each unit is 2 miles so 8*2= 16
Azrael and Simon are training for track. Azrael trains at a rate of 6 miles every half hour. Simon trains at a rate of 2 miles every 15 minutes. What is the unit rate in mile per hour for each runner? *
A. Azrael = 12 miles per hour ; Simon = 8 miles per hour
B. Azrael = 8 miles per hour ; Simon = 12 miles per hour
C. Azrael = 6 miles per hour ; Simon = 2 miles per hour
D. Azrael =2 miles per hour ; Simon = 6 miles per hour
Answer:
A The first one
Answer:
A
Step-by-step explanation: 30 minutes is half an hour so 6 miles is half the miles he runs an hour. Azrael runs 12 miles in an hour. 15 minutes is 1/4 of an hour. 2 is 1/4 of the miles Simon runs in an hour. 2 times 4 is 8. I'm really bad at explaining, sorry. the answer is A.
Can you do this for me?
Answer:
10, 20, 30, 40
2.5, 5, 7.5, 10
120.8, 60.4, 30.2, 15.1
3624, 1812, 906, 453
hope this helps!
2. Determine whether quadrilateral EFGH can
be classified as a parallelogram.
Justify your reasoning.
Now consider quadrilateral EFGH shown
Answer:
3.4
Step-by-step explanation:
Answer:
yes
Step-by-step explanation:
there are two sets of parrallel lines.
I need help with this
The distance an object falls is directly proportional to the square of the time it falls. A ball falls 144 feet in 3 seconds. How far will an object fall in 4 seconds?
Answer:
d = 256 ft.
Step-by-step explanation:
The distance an object falls is directly proportional to the square of the time it falls.
First, we name the variables.
Let d= the distance.
t= time
Write the formula for direct variation, where y varies directly with the square of x.
y=kx2
We will use d in place of y and t in place of x.
d=kt2
Substitute the given values for the variables.
d=144whent=3144=k⋅32
Solve for the constant of variation.
144916=k⋅99=k
Write the equation that relates d and t.
d=kt2
Substitute in the constant of variation.
d=16t2
Find d when t=4.
Write the equation that relates d and t.
d=16t2
Substitute the given value for t.
d=16⋅42
Simplify.
d=256
An object will fall 256 feet in 4 seconds.
The required distance fall by fall in 4 seconds is 192 feet.
Given that,
The distance an object falls is directly proportional to the square of the time it falls. A ball falls 144 feet in 3 seconds, how far will an object fall in 4 seconds is to be determined.
In mathematics, it deals with numbers of operations according to the statements. There are four major arithmetic operators, addition, subtraction, multiplication, and division.
here,
The rate of fall of the ball = 144 / 3
= 48 feet second
Now distance fall by the ball in 4 seconds = 48 * 4 = 192 feets.
Thus, the required distance fall by fall in 4 seconds is 192 feet.
Learn more about arithmetic here:
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I will put brainliest
The bowling club at your school had a car wash last week.
They washed 72 cars in 6.5 hours.
About how many cars did they wash per hour?
Answer:
11.077 cars / 11 cars (nearest whole number)
Step-by-step explanation:
6.5 hours → 72 cars
1 hour → 72/6.5 cars = 11.077 cars (5 s.f.)
On Monday, Deborah ran 3 2− 5miles and on Tuesday she ran 4 1− 5miles. How many miles did she run on these two days together?
Mixed fraction:
3 2/5 = 17/5
4 4/5 = 24/5
Miles she ran:
17/5 + 24/5
= 41/5 miles or 8.2 miles
Find the missing measure and write your solution on the blank provided
Step-by-step explanation:
sum of internal angles of any triangle = 180 degrees
sum of an exterior angle and an interior angle = 180 degrees
Use spherical coordinates. Find the volume of the solid that lies within the sphere x^2 + y^2 + z^2 = 81, above the xy-plane, and below the cone z = x^2 + y^2.
Answer:
The volume of the solid is 243[tex]\sqrt{2} \ \pi[/tex]
Step-by-step explanation:
From the information given:
BY applying sphere coordinates:
0 ≤ x² + y² + z² ≤ 81
0 ≤ ρ² ≤ 81
0 ≤ ρ ≤ 9
The intersection that takes place in the sphere and the cone is:
[tex]x^2 +y^2 ( \sqrt{x^2 +y^2 })^2 = 81[/tex]
[tex]2(x^2 + y^2) =81[/tex]
[tex]x^2 +y^2 = \dfrac{81}{2}[/tex]
Thus; the region bounded is: 0 ≤ θ ≤ 2π
This implies that:
[tex]z = \sqrt{x^2+y^2}[/tex]
ρcosФ = ρsinФ
tanФ = 1
Ф = π/4
Similarly; in the X-Y plane;
z = 0
ρcosФ = 0
cosФ = 0
Ф = π/2
So here; [tex]\dfrac{\pi}{4} \leq \phi \le \dfrac{\pi}{2}[/tex]
Thus, volume: [tex]V = \iiint_E \ d V = \int \limits^{\pi/2}_{\pi/4} \int \limits ^{2\pi}_{0} \int \limits^9_0 \rho ^2 \ sin \phi \ d\rho \ d \theta \ d \phi[/tex]
[tex]V = \int \limits^{\pi/2}_{\pi/4} \ sin \phi \ d \phi \int \limits ^{2\pi}_{0} d \theta \int \limits^9_0 \rho ^2 d\rho[/tex]
[tex]V = \bigg [-cos \phi \bigg]^{\pi/2}_{\pi/4} \bigg [\theta \bigg]^{2 \pi}_{0} \bigg [\dfrac{\rho^3}{3} \bigg ]^{9}_{0}[/tex]
[tex]V = [ -0+ \dfrac{1}{\sqrt{2}}][2 \pi -0] [\dfrac{9^3}{3}- 0 ][/tex]
V = 243[tex]\sqrt{2} \ \pi[/tex]
if my dog was born 4 weeks before August 27th how old will my dog be
Answer:
the dog was born on july 30th so the dog would be 3 months and a week old
Step-by-step explanation:
What is the solution set for 24+ 2 = 6 given the replacement set {1,2,3,4} 1=2 1-4
x = 2
x = 4
x = 1
x = 3
Answer:
x is 2
Step-by-step explanation:
Brainliest Answer
(has to be right for brainliest)
Answer:
it is 0.7x0.4=0.28
Step-by-step explanation:
Answer:
3627514
Step-by-step explanation:
Which number is divisible by 3?
A) 1,794
B) 1,912
C) 1,270
D) 473
Answer:
Step-by-step explanation:
should be A.
Help ASAP Find the measure of 2.
21
69
111
Answer:
69°
Step-by-step explanation:
m∠1 + 111° = 180° ⇒ m∠1 = 69°
m∠1 = m∠2 = 69°
b. Multiply (3 x - 1) by (x + 1)
Answer:
(3x - 1)(x + 1) = 3x² + 2x - 1
Step-by-step explanation:
[tex] \rm \longrightarrow (3x - 1)(x + 1) \\ \\ \rm \longrightarrow 3x(x + 1) - 1(x + 1) \\ \\ \rm \longrightarrow (3x)(x) + (3x)(1) + ( - 1)(x) + ( - 1)(1) \\ \\ \rm \longrightarrow 3 {x}^{2} + 3x + ( - x ) + ( - 1) \\ \\ \rm \longrightarrow 3 {x}^{2} + 3x - x - 1 \\ \\ \rm \longrightarrow 3 {x}^{2} + 2x - 1[/tex]
Answer:
(3x - 1)(x + 1) = 3x² + 2x - 1
Step-by-step explanation:
got it right on edge
Problem
What is the constant of proportionality in the equation y=3xy
Answer:
-3
Step-by-step explanation:
y=3xy and subtract y on both sides that gives u -1 and then you have -1=3x and then divide 3 on both sides you are left with -3 =x
The number 0.003 is 1/10 of which decimal?
Answer:
I'd say it's 0.03
Step-by-step explanation:
0.03 divided by 10 is 0.003 so it's 1/10
0.003 is 1/10th of 0.03.
What is a decimal ?A decimal is represented by a point or a dot it separates the decimal part and the fractional part for example 24.75 here the whole part is is 24 and the fractional part is 75.
if we divide 0.03 by 10 = [tex]\frac{0.03}{10}[/tex] = 0.003. (dividing a number by 10
moves the decimal point
to the left by 1 because
10 has 1 no. of zeroes)
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The length of one side of a square pond is 15 feet. The pond is surrounded by a 3 foot wide walkway. What is the total area of the pond and the walkway, in square feet
Answer:
Area = 324 ft²
Step-by-step explanation:
Given
Length = 15ft
Walkway = 3ft
Required
Determine the area of the pond + the walkway
To calculate the required area, we need to get the total length of a side .
This is calculated by adding the length of a side of the pond and the side of the walkway
Total Length = Pond Length + Walkway
Total Length = 15ft + 3ft
Total Length = 18ft
Now, the required area can be calculated as thus:
Area = Total Length * Total Length
Area = 18ft * 18ft
Area = 324 ft²
An equilateral triangle and a regular hexagon have the same area. The ratio of the side length of the regular hexagon to the side length of the equilateral triangle can be written as m/√n. What is m+n?
Help me plz!
Answer:
7
Step-by-step explanation:
Let a and b be the sides of the equilateral triangle and the regular hexagon respectively as shown in figures (1) and (2).
The area of the equilateral having side a is
[tex]A_T= \frac {\sqrt3}{2}a^2\cdots(i)[/tex]
Now, join all the 6 corner points of the regular hexagon to the center of the hexagon as shown in figure (3).
As there are 6 equal sides, so angle subtended by each side at the center is 360/6=60 degrees.
Considering the triangle POR:
[tex]\angle POR= 60[/tex] degree [angle at center]
As sides PO=RO, so [tex]\angle OPR = \angle ORP = 60[/tex] degree.
Hence, triangle POR is an equilateral triangle.
Similarly, all the remaining 5 triangles are equilateral triangles.
So, the area of hexagon = 6 x Area of triangle POR
As the length of sides of the triangle POR is b,
so the area of the triangle POR [tex]= \frac {\sqrt3}{2}b^2[/tex].
Hence, the area of the regular hexagon,
[tex]A_H = 6 \times \frac {\sqrt3}{2}b^2\cdots(ii)[/tex]
As the equilateral triangle and a regular hexagon have the same area, so from the equations (i)and (ii), we have
[tex]A_T=A_H\\\\\frac {\sqrt3}{2}a^2=6 \times \frac {\sqrt3}{2}b^2\\\\\Rightarrow a^2=6b^2\\\\\Rightarrow \frac {b^2}{a^2}=\frac 1 6\\\\\Rightarrow \frac {b}{a}=\frac {1}{\sqrt 6}\cdots(iii)[/tex]
Given that the ratio of the side length of the regular hexagon to the side length of the equilateral triangle, b/a= m/√n.
On comparing with the equation (iii), we have
m=1 and n=6
Hence, m+n=1+6=7.
The derivative of the function f is given by f′(x)=−3x+4 for all x, and f(−1)=6. Which of the following is an equation of the line tangent to the graph of f at x=−1 ?
Answer:
The equation of the line tangent to the graph of f at x = -1 is [tex]y = 7\cdot x +13[/tex].
Step-by-step explanation:
From Analytical Geometry we know that the tangent line is a first order polynomial, whose form is defined by:
[tex]y = m\cdot x + b[/tex] (1)
Where:
[tex]x[/tex] - Independent variable, dimensionless.
[tex]y[/tex] - Dependent variable, dimensionless.
[tex]m[/tex] - Slope, dimensionless.
[tex]b[/tex] - Intercept, dimensionless.
The slope of the tangent line at [tex]x = -1[/tex] is:
[tex]f'(x) = -3\cdot x +4[/tex] (2)
[tex]f'(-1) = -3\cdot (-1) +4[/tex]
[tex]f'(-1) = 7[/tex]
If we know that [tex]m = 7[/tex], [tex]x = -1[/tex] and [tex]y = 6[/tex], then the intercept of the equation of the line is:
[tex]b = y-m\cdot x[/tex]
[tex]b = 6-(7)\cdot (-1)[/tex]
[tex]b = 13[/tex]
The equation of the line tangent to the graph of f at x = -1 is [tex]y = 7\cdot x +13[/tex].
please help, the question says what statement best describes the placement of the lines Carson Drew
Answer:
Only line A is a well-placed line of best fit.
Step-by-step explanation:
I did this on Plato and got it correct.
HELP!!!!! I dont understand this!
Answer:
-14
Step-by-step explanation:
I might be wrong, but this is my interpretation of the problem.
To solve this problem, find the endpoints of the line in the interval. Then draw a line between those two points. Finally, find the slope of the line that passes between the two points.
The line is f(x) = -2x^(2) + 2x + 1
The interval (in standard notation) is (3 [tex]\leq[/tex] x [tex]\leq[/tex] 5)
So when x = 3,
-2x^(2) + 2x + 1
-2(3)^(2) + 2*3 + 1
-2 * 9 + 6 + 1
-18 + 7
- 11
One endpoint is: (3, -11).
When x = 5
-2x^(2) + 2x + 1
-2(5)^2 + 2*5 + 1
-2 * 25 + 10 + 1
-50 + 11
-39
The other endpoint is; (5, -39).
Find the slope of a line passing through these two points.
The formula to find the slope of a line is:
[tex]\frac{y_{2} - y_{1} }{x_{2} - x_{1} }[/tex]
[tex]x_{1}[/tex] = 3
[tex]y_{1}[/tex] = -11
[tex]x_{2}[/tex] = 5
[tex]y_{2}[/tex] = -39
((-39) - ( -11))/((5) - (3))
Simplify
-28/2
-14
what is 19+1 pls help me
Math problem pls help I will give brainiest
Answer:
y=5
Step-by-step explanation:
125 +125+13y+9y=360
punch it into math-way.
250+22y=360
22y= 110
y=5
How many pounds of chocolates that cost $7 per pound must be mixed with 3 pounds of caramels that cost $3.50 per pound to make a candy mix that
cost $5.25 per pound?
Answer:
You know somw things are changed i can't even answer that so please answer it yourself because hahhahaahahaahahhahahahha hello
Step-by-step explanation:
Because i dont know how to answer that ok?! Tell that to ur mom haha