2ax+1=ax+5 solve for x
How would you use the Fundamental Theorem of Calculus to determine the value(s) of b if the area under the graph g(x)=4x between x=1 and x=b is equal to 240?
Answer:
[tex]\displaystyle b = 11[/tex]
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDAS
Brackets Parenthesis Exponents Multiplication Division Addition Subtraction Left to RightEquality Properties
Multiplication Property of Equality Division Property of Equality Addition Property of Equality Subtraction Property of EqualityAlgebra I
Functions
Function NotationCalculus
Integration
IntegralsDefinite IntegralsIntegration Constant CIntegration Rule [Reverse Power Rule]: [tex]\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C[/tex]
Integration Rule [Fundamental Theorem of Calculus 1]: [tex]\displaystyle \int\limits^b_a {f(x)} \, dx = F(b) - F(a)[/tex]
Integration Property [Multiplied Constant]: [tex]\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx[/tex]
Area of a Region Formula: [tex]\displaystyle A = \int\limits^b_a {[f(x) - g(x)]} \, dx[/tex]
Step-by-step explanation:
Step 1: Define
Identify
g(x) = 4x
Interval [1, b]
A = 240
Step 2: Solve for b
Substitute in variables [Area of a Region Formula]: [tex]\displaystyle \int\limits^b_1 {4x} \, dx = 240[/tex][Integral] Rewrite [Integration Property - Multiplied Constant]: [tex]\displaystyle 4\int\limits^b_1 {x} \, dx = 240[/tex][Integral] Integrate [Integration Rule - Reverse Power Rule]: [tex]\displaystyle 4(\frac{x^2}{2}) \bigg| \limits^b_1 = 240[/tex]Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]: [tex]\displaystyle 4(\frac{b^2}{2} - \frac{1}{2}) = 240[/tex][Distributive Property] Distribute 4: [tex]\displaystyle 2b^2 - 2 = 240[/tex][Addition Property of Equality] Add 2 on both sides: [tex]\displaystyle 2b^2 = 242[/tex][Division Property of Equality] Divide 2 on both sides: [tex]\displaystyle b^2 = 121[/tex][Equality Property] Square root both sides: [tex]\displaystyle b = \pm 11[/tex]Choose: [tex]\displaystyle b = 11[/tex]Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Integration
Book: College Calculus 10e
A trucker had a load of grain containing 2 tons. She unloaded 1 ton 1,200 pounds at the warehouse. How much grain does she still have left on the truck?
The average yearly temperature T in degrees for Newport, Oregon is given by: |T−55|≤10 Find the average temperature range in degrees.
The average temperature range in Newport, Oregon is between 45 and 65 degrees.
The average yearly temperature T in degrees for Newport, Oregon is given by: |T−55|≤10. This means that the temperature T is within 10 degrees of 55 degrees. To find the average temperature range, we consider the range between 55+10 = 65 degrees and 55-10 = 45 degrees.
Therefore, the average temperature range in Newport, Oregon is 45 to 65 degrees.
Mrs white wants to crochet beach hits and baby afghans for a church fund raising bazaar. she needs 7 hours to make a hat and 4 hours to make an Afghan and she has 68 hours available. she wants to make no more than 14 items and no more than 11 afghans. the bazaar will sell hats for $21 each and the afphans for $9 each. How many of each should she make to maximize the income for the bazaar?
a full-time employee who works 40 hours per week earns 29.85 per hour estimate the person's annual income
Jonathan's Antiques purchased an old wooden sled at an auction for $210. Jonathan wants to mark up the sled 60% of the selling price. What would be the selling price of the sled?
The senate in a certain state is compromised of 58 republicans, 39 democrats, and 3 independents. How many committees can be formed if each committee must have 3Republicans and 2 Democrats?
p=m/1+rt solve for t
If the following system of equations was written as a matrix equation in the form AX = C, and matrix A was expressed in the form: A= {A C} {B D}, find the value of a-b +c+d. 2x+8y=7 4x-2y=9 Please help, i dont know which number would be which letters
Answer: a-b+c+d =4
Step-by-step explanation:
The given system of equation is
[tex]2x+8y=7\\4x-2y=9[/tex]
from this we have the following matrices
[tex]A_1 =\begin{bmatrix}\\2 &8 \\ \\4&2 \\\end{bmatrix}\ ,X=\begin{bmatrix}\\x\\ \\y\\\end{bmatrix}\text{and}\ C=\begin{bmatrix}\\7\\ \\9\\\end{bmatrix}[/tex]
the given matrix A =[tex]\begin{bmatrix}\\a &c \\ \\b &d \\\end{bmatrix}[/tex]
On comparing Matrix [tex]A_1[/tex] with Matrix A
[tex]\begin{bmatrix}\\a &c \\ \\b &d \\\end{bmatrix}=\begin{bmatrix}\\2&8 \\ \\4 &-2 \\\end{bmatrix}[/tex]
we have the following values
a=2 ,b=4,c=8,d=-2
Thus a-b+c+d =2-4+8+(-2)=4
The sum of two #'s is 29. The sum of the smaller and 2 times the larger is 45. Find the #'s.
x+y=29
y=29-x
x+2y=45
x+2(29-x)=45
x+58-2x=45
-1x=-13
x=13
y=29-13=16
x+y = 13+16 = 29
x+2y= 13 + 2(16) = 13+32 = 45
so the numbers are 13 & 16
Find the Taylor series for
f(x), centered at the given value of a.
f(x) = sin(x), a = π
Written as a summation?
Radius of convergence?
The Taylor series for [tex]\( \sin(x) \)[/tex] centered at [tex]\( a = \pi \)[/tex] is:[tex]\[ \sin(x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n + 1)!}(x - \pi)^{2n + 1} \][/tex]
The radius of convergence of this Taylor series is infinite, meaning it converges for all values of ( x ).
To find the Taylor series for [tex]\( f(x) = \sin(x) \)[/tex] centered at [tex]\( a = \pi \),[/tex] we first need to find the derivatives of \( \sin(x) \) and evaluate them at \( x = \pi \). Then, we'll write out the Taylor series using the formula:
[tex]\[ f(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \frac{f'''(a)}{3!}(x - a)^3 + \cdots \][/tex]
Let's find the derivatives of [tex]\( \sin(x) \)[/tex] and evaluate them at [tex]\( x = \pi \):[/tex]
1. [tex]\( f(x) = \sin(x) \)[/tex]
2. [tex]\( f' (x) = \cos(x) \)[/tex]
3.[tex]\( f''(x) = -\sin(x) \)[/tex]
4.[tex]\( f'''(x) = -\cos(x) \)[/tex]
Now, evaluate these derivatives at \( x = \pi \):
1. [tex]\( f(\pi) = \sin(\pi) = 0 \)[/tex]
2. [tex]\( f'(\pi) = \cos(\pi) = -1 \)[/tex]
3.[tex]\( f''(\pi) = -\sin(\pi) = 0 \)[/tex]
4. [tex]\( f'''(\pi) = -\cos(\pi) = 1 \)[/tex]
Now, plug these values into the Taylor series formula:
[tex]\[ \sin(x) = 0 - 1(x - \pi) + \frac{0}{2!}(x - \pi)^2 + \frac{1}{3!}(x - \pi)^3 + \cdots \][/tex]
Simplify:
[tex]\[ \sin(x) = -\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n + 1)!}(x - \pi)^{2n + 1} \][/tex]
So, the Taylor series for [tex]\( \sin(x) \)[/tex] centered at [tex]\( a = \pi \)[/tex] is:
[tex]\[ \sin(x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n + 1)!}(x - \pi)^{2n + 1} \][/tex]
The radius of convergence of this Taylor series is infinite, meaning it converges for all values of ( x ).
After 5 years of earning interest at an annual rate of 3% an investment has earned $950 in interest. To the nearest whole dollar, determine the amount of the initial investment
A drawer contains 12 identical white socks, 18 identical black socks and 14 identical brown socks. What is the least number of socks you must choose, without looking, to be certain that you have chosen two brown socks?
The only time you can be so certain that you have picked 2 brown socks is when there are no more white socks and black socks. This means that to be 100% sure that you have picked 2 brown socks, you must pick all 12 white socks and all 18 black socks and only then you can pick 2 brown socks without looking. Therefore the total number of socks that should be picked is:
Total number of socks = 12 white socks + 18 black socks + 2 brown socks
Total number of socks = 32 socks
A total picking of 32 socks is required to be certain without looking that 2 brown have already been chosen.
If you have 18 out of 20 homework sections completed what percentage do you have
What is 40 kilometers per hour in meters per hour
By what factor will the rate of the reaction change if the ph decreases from 5.50 to 2.00? express your answer numerically using two significant figures.
a. Rate law expression: [tex]\(\text{rate} = kxyz\)[/tex]
b. [tex]\[ \text{Rate factor} \approx 3.16 \times 10^3 \][/tex]
**Part A: Rate Law Expression**
Given that the reaction is first order in [tex]\( (IO_3)^- \)[/tex], first order in [tex]\( (SO_3)^{2-} \)[/tex], and first order in [tex]\( H^+ \)[/tex], the rate law for the reaction can be expressed as:
[tex]\[ \text{rate} = k \cdot [(IO_3)^-]^1 \cdot [(SO_3)^{2-}]^1 \cdot [H^+]^1 \][/tex]
Simplifying this, we get:
[tex]\[ \text{rate} = kxyz \][/tex]
Here, [tex]\( x \), \( y \), and \( z \)[/tex] are the concentrations of [tex]\( (IO_3)^- \), \( (SO_3)^{2-} \), and \( H^+ \)[/tex], respectively. [tex]\( k \)[/tex] is the rate constant.
**Part B: Rate Change with pH**
The pH of a solution is a measure of its hydrogen ion concentration [tex](\( [H^+] \))[/tex]. As the reaction is first order in [tex]\( H^+ \)[/tex], a change in pH will directly impact the rate. The relationship between pH and [tex]\( [H^+] \)[/tex] is logarithmic:
[tex]\[ \text{pH} = -\log[H^+] \][/tex]
To calculate the rate change factor when the pH decreases from 5.50 to 2.00, we need to consider the relationship between pH and [tex]\( [H^+] \)[/tex]:
[tex]\[ \text{pH} = -\log[H^+] \][/tex]
1. **At pH 5.50:**
[tex]\[ [H^+] = 10^{-\text{pH}} = 10^{-5.50} \][/tex]
2. **At pH 2.00:**
[tex]\[ [H^+] = 10^{-\text{pH}} = 10^{-2.00} \][/tex]
Now, calculate the rate factor:
[tex]\[ \text{Rate factor} = \frac{\text{rate at pH 2.00}}{\text{rate at pH 5.50}} = \frac{[H^+]_{\text{pH 2.00}}}{[H^+]_{\text{pH 5.50}}}\][/tex]
[tex]\[ \text{Rate factor} = \frac{10^{-2.00}}{10^{-5.50}} \][/tex]
[tex]\[ \text{Rate factor} \approx \frac{0.01}{3.16 \times 10^{-6}} \][/tex]
[tex]\[ \text{Rate factor} \approx 3.16 \times 10^3 \][/tex]
Expressing the answer numerically using two significant figures, the rate change factor is approximately [tex]\(3.2 \times 10^3\).[/tex]
The question probable maybe:
Part A:
The reaction is found to be first order in [tex](IO_3)^-[/tex], first order in [tex](SO_3)^2^-[/tex], and first order in H^+.
If [[tex](IO_3)^-[/tex]] = x, [[tex](SO_3)^2^-[/tex]] = y and [[tex]H^+[/tex]]=z, what is the rate law for the reaction in terms of x, y, and z and the rate constant k?
Express the rate in terms of k, x, y, and (e.g., kxy^3z^2).
▸ View Available Hint(s)
rate= kx^1y^1z
Part B:
By what factor will the rate of the reaction change if the ph decreases from 5.50 to 2.00? express your answer numerically using two significant figures.
Find the dimensions of the rectangular box with largest volume if the total surface area is given as 64cm^2
pie =????????????????????.??
9.8 is 2% of what number
Triangle xyz has sidesxy equals 3 yz equals 4 and xz equals 5.if angle y is a right angle and yz is opposite angle x what is the tan of angle x
tan 2θ; cos θ = 8 17 , θ in Quadrant I
We first found sin θ using the Pythagorean identity, then found tan θ, and finally used the double-angle formula for tan to find tan 2θ.
Explanation:To find the value of tan 2θ when cos θ = 8/17 and θ is in Quadrant I, you need first to find the value of sin θ. Since we are in Quadrant I, both cos and sin are positive. You can use the Pythagorean Identity for sin, cos, and tan, which states sin² θ + cos² θ = 1, to find sin θ. Substituting the given value of cos θ in this identity, we can find that sin θ = sqrt(1 - (8/17)²) = 15/17.
With sin and cos known, we can now find tan θ using the formula tan θ = sin θ/cos θ which gives tan θ = (15/17)/(8/17) = 15/8.
Finally, to find tan 2θ, use the Double-Angle formula for the tangent, which states tan 2θ = 2 tan θ / (1 - tan² θ). Substituting tan θ = 15/8 into this formula, we get tan 2θ = 2 * (15/8) / (1 - (15/8)²).
Learn more about Trigonometric values here:https://brainly.com/question/29069676
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To find tan 2θ, given cos θ is 8/17 and θ is in the first quadrant, first find sin θ using the Pythagorean identity, then use the double-angle formula where tan 2θ equals 2 tan θ divided by 1 minus tan squared θ, yielding -2.
To find tan 2θ given that cos θ is 8/17 and θ is in the first quadrant, we first need to find the value of sin θ. Since θ is in the first quadrant, all trigonometric functions are positive. Using the Pythagorean identity, we have:
sin θ = √(1 - cos² θ)
Substituting cos θ = 8/17:
sin θ = √(1 - (8/17)²) = √(1 - 64/289) = √(225/289) = 15/17.
Next, we use the double-angle formula for tangent:
tan 2θ = (2 tan θ) / (1 - tan² θ)
To find tan θ, we use:
tan θ = sin θ / cos θ = (15/17) / (8/17) = 15/8
Now, substitute tan θ into the double-angle formula:
tan 2θ = (2 × 15/8) / (1 - (15/8)²)
= (30/8) / (1 - 225/64)
= (30/8) / (-161/64) = -1920/1288 = -15/7.5 = -2
The complete question is :
Given that [tex]\(\cos \theta = \frac{8}{17}\)[/tex] with [tex]\(\theta\)[/tex] in the first quadrant, determine [tex]\(\tan 2\theta\).[/tex]
How much times does 7 go into 48??
How much do you need to invest in an account earning an annual interest rate of 2.938% compounded weekly, so that your money will grow to $7,880.00 in 50 weeks?
Tom predicted that the Giants would score three 3-point field goals, score two 6-point touchdowns, and score 1 extra point. If Tom were correct, what would the Giants' score be at the end of the game?
3x3 = 9
2x6 = 12
12+9 = 21+1 =22
score would be 22 points
Answer:
The Giants' score will be 22 points.
Step-by-step explanation:
Consider the provided information.
It is given that tom predicted that the Giants would score three 3-point field goals. Which can be written as:
3 × 3 = 9
Total score by field goals is 9 points.
If he score two 6-point touchdowns, this can be written as:
2 × 6 = 12
Total score by touchdown is 6 points.
And 1 extra point.
Now add all the points as shown below:
Total score is: 9 + 12 + 1 = 22 points
Hence, the Giants' score will be 22 points.
The total cost to rent a row boat is $14 times the number of hours the boat is used. Write an equation to model this situation if c = total cost and h = number of hours.
c = 14h
this would give you the total cost by multiplying 14 by the number of hours
Find the decimal notation for 84.3%
to turn a percent into a decimal, move the decimal point 2 places to the left
84.3% = 0.843
Four students did a survey to find the soda flavor sixth-grade students perefer. The table below shows the method each student used to conduct the survey:
Student. Method
Trey- asked 100 students at random from his seventh- grade class what their favorite soda flavor is
Jesse- asked 100 sixth-grade students at random what their favorite soda flavor is
Nita-asked 100 eighth-grade students at random what their favorite soda flavor is
Ruben- asked 100 third- grade students at random what their favorite soda flavor is
Which students survey is most likely not biased?
Trey
Jesse
Nita
Ruben
What is 46 2/3% of 28
Answer:
46 2/3% of 28 is 13.06
Step-by-step explanation:
Hello,
thank you very much for asking this here in brainly, I think I can help you with this one
Let's remember
[tex]a\frac{b}{c} =\frac{(a*c)+b}{c}\\[/tex]
Step 1
[tex]46\frac{2}{3} =\frac{(46*3)+2}{3} =\frac{140}{3} =46.67[/tex]
so 46 2/3% =46.67%
Step 2
using a rule of three
define the relationships
if
28⇒ 100%
x? ⇒46.67%
[tex]\frac{28}{100}=\frac{x}{46.67} \\[/tex]
isolating x
[tex]\frac{28}{100}=\frac{x}{46.67} \\x=\frac{28*46.67}{100}\\x=\frac{1306.67}{100} \\x=13.06[/tex]
46 2/3% of 28 is 13.06
I hope it helps, have a great day
Ethel is arranging rides so that 27 members go bowling. Some people can ride in a van that belonges to the center where they meet the rest must ride in cars if 12 can go in the van and 5 can go in each car how many cars will they need?
Answer:
Number of cars needed=3
Step-by-step explanation:
Ethel is arranging rides for 27 members.
If 12 can go in the van and 5 can go in each car.
Let c cars are needed
Then, 12+5c=27
5c=27-12
5c= 15
c= 15/5
c=3
Hence, number of cars needed equals:
3