Note that √((1 - x)/x) is defined only as long as 0 < x ≤ 1.
Consider a right triangle with reference angle θ such that
[tex]\cot(\theta) = \sqrt{\dfrac{1-x}x}[/tex]
In other words, on an appropriate domain,
[tex]\theta = \cot^{-1}\left(\sqrt{\dfrac{1-x}x}\right)[/tex]
In such a triangle, you would find that
[tex]\sin(\theta) = \sqrt x[/tex]
so f(x) reduces a bit to
[tex]f(x) = \cos\left(2 \tan^{-1}(\sqrt x)\right)[/tex]
Now consider another triangle with reference angle ɸ such that
[tex]\tan(\phi) = \sqrt x \implies \phi = \tan^{-1}(\sqrt x)[/tex]
In this triangle, you would find
[tex]\cos(\phi) = \dfrac1{\sqrt{1+x}}[/tex]
Recalling the double angle identity for cosine, it follows that
[tex]f(x) = \cos(2\phi) = 2 \cos^2(\phi) - 1 = \dfrac{1-x}{1+x}[/tex]
Differentiating with respect to x yields
[tex]f'(x) = -\dfrac2{(1+x)^2}[/tex]
while
[tex]f(x)^2 = \dfrac{(1-x)^2}{(1+x)^2}[/tex]
It follows that
[tex]f'(x)(x-1)^2 - 2f(x)^2 = 0[/tex]
so B is the correct choice.
Note: Enter your answer and show all the steps that you use to solve this problem in the space provided.
Write [tex](8a^{-3})^{-2/3}[/tex] in simplest form.
Answer:
[tex] \frac{1}{64 {a}^{2} } [/tex]
Step-by-step explanation:
[tex](8 {a}^{ - 3} {)}^{ \frac{ - 2}{3} } [/tex]
[tex](8 a {)}^{ - 2} [/tex]
[tex] \frac{1}{ {8}^{2} {a}^{2} } [/tex]
[tex] \frac{1}{64 {a}^{2} } [/tex]
You are mowing your neighbor's lawn. He is going to pay you $20.00 for the first two hours and $5.00 for anything after that. Which algebraic expression represents how much you will be paid for working h hours?
I NEED THIS ANSWER ASAP!!
Answer:
20+5(x-2) im pretty sure
Step-by-step explanation:
Convert the binary number 1101001 to decimal.
Answer:
105
Step-by-step explanation:
Answer:
105
Step-by-step explanation:
Decimal calculation steps
(1101001)₂ = (1 × 2⁶) + (1 × 2⁵) + (0 × 2⁴) + (1 × 2³) + (0 × 2²) + (0 × 2¹) + (1 × 2⁰) = (105)₁₀
Amy, Bob, and Carl were using flour from a 7 cup container. Amy used 0.15 of the total container , Bob used 30% of the total container, and Carl used .5 cups of the container. Determine the amount that each person uses and list them from least to greatest.
Answer:
Amy, Carl, bob
Step-by-step explanation:
Amy = 7 - 0.15 = 6.45
Bob = 7 - 0.30 = 6.70
Carl = 7 - 0.5 = 6.5
Order from least to greatest is:
Amy (6.45)
Carl (6.5)
Bob (6.7)
The radius of a circle is 5 meters. What is the length of a 180° arc?
Answer:
10 meters
Step-by-step explanation:
180° arc makes semi circle then it's diameter equal to its arc .
A line has a slope of 2 and a y-intercept of -3/4 which of the following is an equation of the line
Answer:
y = 2x - 0.75
Step-by-step explanation:
Slope-intercept form is:
⇒ y = mx + b
m being the slope and b being the y-intercept.
The given slope and y-intercept:
⇒ Slope ⇒ 2
⇒ Y-intercept ⇒ -3/4
Plug in:
⇒ y = 2x - 0.75 (3/4)
Total price of all shares algebraically when g shares of stock for w dollars per share is purchased, when shares sold when they earned a 40% capital gain?
Question:
Zeke bought g shares of stock for w dollars per share. His broker told him to sell when they earn 40% capital gain. Express the total price of all shares algebraically when g shares of stock for w dollars per share is purchased, when shares sold when they earned a 40% capital gain?
Answer:
1.40w
Step-by-step explanation:
If f(x) = x^3 - 2x^2 , which expression is equivalent to f(i)?
O -2 + i
O -2 - i
O 2 + i
O 2 - i
[tex]\text{Given that,}~f(x)=x^3-2x^2\\\\f(i)=i^3-2i^2\\\\~~~~~~=i^2 \cdot i -2(-1)~~~~~;[i^2=-1]\\\\~~~~~~=-i+2\\\\~~~~~~=2-i[/tex]
The expression equivalent to f(i) is 2-i.
What is expression?The expression is mathematical statement which consist of variables, numbers and connected by at least one operation.
Given that, a function f(x) = x³ - 2x², we need to find the expression equivalent to f(i),
To find the same we will put x = i,
So,
f(i) = i³ - 2i² [i = √-1]
= (√-1)³ - 2(√-1)²
= -i - 2(-1)
= -i+2
= 2-i
Hence, the expression equivalent to f(i) is 2-i.
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Solve the simultaneous equations
5x + y = -5
5x – 3y = 11
The solution to the simultaneous equations, 5x + y = -5 and 5x – 3y = 11, is: (-1/5, -4)
What is the Solution of Simultaneous Equations?The solution is the pair of values of x and y that is true for both equations.
Given the simultaneous equations,
5x + y = -5
5x – 3y = 11
Subtract
4y = -16
y = -4
Substitute y = -4 into equation 1.
5x + (-4) = -5
5x - 4 = -5
5x = -5 + 4
5x = -1
x = -1/5
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There are 318 calories in six ounces of a certain ice cream. How many calories are there in one pound?
1 pound = 16 ounces
Answer:
795
Step-by-step explanation:
318 x 2 (to get 12 ounces)
636 + (1/2 x318) (to get 3 ounces)
636 + 159 (to get 16 ounce)
795
The number of calories in one pound is 848 calories.
Given that, there are 318 calories in six ounces of a certain ice cream.
What are proportions?Proportion, in general, is referred to as a part, share, or number considered in comparative relation to a whole. Proportion definition says that when two ratios are equivalent, they are in proportion. It is an equation or statement used to depict that two ratios or fractions are equal.
Let x be number of calories in 1 pound = 16 ounces
Here, 318:6::x:16
⇒ 6x=318×16
⇒ 6x=5088
⇒ x=5088/6
⇒ x=848
Therefore, the number of calories in one pound is 848 calories.
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Find the slope of the line passing through (3,-4) and (3,8)
To find the slope ⇒ we must know the formula
Formula = [tex]\frac{y2-y1}{x2-x1}[/tex]
(x1, y1) ⇒ (3, -4)(x2, y2) ⇒ (3,8)Now let's solve and plug the known variables
Slope = [tex]\frac{8-(-4)}{3-3} =\frac{12}{0}[/tex]
Nothing can be divided by zero
⇒ in this case, the slope is undefined
*interesting fact: the line is vertical
Hope that helps!
HELP ME OUTT PPL (BRAINIEST)
Answer:
(0, 6)
Step-by-step explanation:
Point T is at (3, 4), and three units to the left would be (0, 4). They then tell us to go two points up, which would change the ordered pair to (0, 6).
I hope this was helpful! Have a lovely day! :D
In a school theater performance, 150 tickets were sold, some for $30 and others for $20, how many tickets were sold of each price if the total sale was $3,500?
Answer:
85 tickets for 20 dollars and 60 for 30
Step-by-step explanation:
my smart friend in AP calc said so
He decided to bike from his home to the library to return some books. Ell biked 110 miles when he remembered that he had left a book at home, so he biked back home to get it. After getting the book from home, he biked to the library. What was the total distance, in miles, en had biked when he finally reached the library Show your work. miles Answer
Answer:
2 2/10 miles
but on the part where he realized he forgot his books was he at the library already when he realized cuz if not them it's more than 2 2/10
What is the area of the trapezoid?
7 cm
10 cm
6 cm
7 cm
1
13 cm
O A. 43 cm?
O B. 60 cm?
O C. 70 cm?
Answer:
B. 60 cm²
Step-by-step explanation:
The formula for the area of a trapezoid is: [tex]\frac{1}{2}(b_{1} + b_{2} )h[/tex]
Plugging in the given values, we can solve:
[tex]\frac{1}{2} (13+7)(6)\\\\\frac{1}{2}(20)(6)\\\\10(6)\\\\= 60[/tex]
hope this helps!
This graph shows a proportional relational between the distance traveled by Tim on a road trip and the number
of hours for which he traveled.
Distance Traveled by Tim
250
200
150
Distance (mi)
100
50
0
10
6 8
Number of Hours
Which statement identifies the correct slope and the correct interpretation of the slope for this situation?
50
A The slope of the line is
1
so the distance traveled by Tim is miles every hour.
50
B The slope of the line is so the distance traveled by Tim is miles every hours.
1
1
The slope of the line is
50
so the distance traveled by Tim is miles every hour.
Using the proportional relationship, it is found that the correct statement regarding the slope of the relation is:
The slope of the line is of 25, so the distance traveled by Tim is 25 miles every hour.
What is a proportional relationship?A proportional relationship is a function in which the output variable is given by the input variable multiplied by a constant of proportionality, that is:
y = kx
In which k is the constant of proportionality, which is also called the slope.
Researching the problem on the internet, it is found that in 2 hours, Tim traveled 50 miles, hence:
k = 50/2 = 25.
The slope of the line is of 25, so the distance traveled by Tim is 25 miles every hour.
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. Carlee surveyed customers as they left a convenience store. She asked them, “How many items did you
purchase?” The results are shown in the line plot.
(a) How many customers were surveyed?
(b) What was the greatest number of items purchased by a customer?
(c) How many customers purchased 9 items?
(d) How many customers purchased at least 5 items?
(e) What was the median number of items purchased?
Answer:
We need the line plot in order to solve the questions.
Step-by-step explanation:
A spherical balloon has a circumference of 25 cm.
a.) What is the approximate surface area of the balloon to the nearest square centimeter?
b.) What is the approximate volume of the balloon to the nearest cubic centimeter?
Answer:
See below ↓↓
Step-by-step explanation:
Finding the radius
C = 2πr25 = 2 x 3.14 x rr = 25 / 6.28r = 3.98 cm ≅ 4 cmSurface Area
4πr²4 x 3.14 x 4 x 43.14 x 64200.96201 cm²Volume
4/3πr³ = 4πr² x r/3201 x 4/367 x 4268 cm³okay okay what is emoji aaaaa
okay okay what is emoji aaaaa
Rearrange the equation and solve for x.
2 = x2 + 6x + 11
Answer:
-3
Step-by-step explanation:
It's a lot of work, I'm not typing that into here
Ruby is painting her house. She uses 2 2/7 gallons of paint in the living room. To paint the kitchen, Ruby uses 5/8 of the amount of paint she uses in the living room. Which expression can be used to find the number of gallons of paint Ruby uses to paint the kitchen?
Answer:
ur momma
Step-by-step explanation:
yea so you take the 5/8 and the 2 and the 2/7 and you dd then subtract then divide
please help solve this math question
Answer:
4# | 267
Step-by-step explanation:
There are 3 numbers in the 40's in the data:
42, 46 and 47.
Find the volume of the prism please help and hurry
Answer:
volume is 360 i hope i uelped u
A flower garden is shaped like a circle. Its diameter is 40yd. A ring-shaped path goes around the garden. Its outer edge is a circle with diameter 48yd.
The gardener is going to cover the path with sand. If one bag of sand can cover 6yd^2, how many bags of sand does the gardener need? Note that sand comes only by the bag, so the number of bags must be a whole number. (Use the value 3.14 for pi number .)
Answer:
About 5 bags of sand
Step-by-step explanation:
1. Create Equation
[(48x3.14)-(40x3.14)]/6
2. Solve
[(48x3.14)-(40x3.14)]/6
[150.72-125.6]/6
25.12/6
Now if you use a calculater you will get
4.1866666667
So lets just round the number to 5 instead of
4 beacuse if we round it to 4 it won't be enough
to cover the whole path.
so your answer is 5
Of 11 possible books you plan to take 4 with you on vacation how many different collections of 6 books can you take
The number of different collections of 6 books which can be taken on vacation of 11 possible books is 462.
What is combination?Combination is the way of arrangement or the collection of items in the particular order. The order of this group of items does not matter in the combination type of arrangement.
Here, in the original problem, the number of book selection is6.
Of 11 possible books, one plans to take 6 with on vacation. The number of different collections of 6 books can be taken has to be found out.
The total number of books=11.The total number of selection of book=6.Thus, the number of combination to select 6 books collection from 11 is,
[tex]^{11}C_6=\dfrac{11!}{(11-6)!6!}\\^{11}C_6=\dfrac{11\times10\times9\times8\times7\times6!}{5\times4\times3\times2\times1\times6!}\\^{11}C_6=462[/tex]
Thus, the number of different collections of 6 books which can be taken on vacation of 11 possible books is 462.
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What is the smallest angle of a right triangle with sides 9, 40, and 41? Round your answer to the nearest degree, and enter the number only.
Answer: 13°
Explanation: x = tan^-1 ( 9/40) ≈ 13°
Find two numbers whose product is 110, and whose sum is 21.
Step-by-step explanation:
The answer is 10 and 11
10 + 11 = 21
10 × 11 = 110
What is the equation of the line that passes through the point (3, 5) and has a slope
of -1/3
Linear equations are typically organized in slope-intercept form:
[tex]y=mx+b[/tex]
m = slopeb = y-intercept (the value of y when the line crosses the y-axis)To determine the equation of a line:
Find the slopePlug the slope into y=mx+bPlug in a point that falls on the line as (x,y)Solve for bSolving the Question
We're given:
[tex]m=-\dfrac{1}{3}[/tex]The lines passes through the point (3,5)[tex]y=mx+b[/tex]
⇒ Plug the slope into the equation:
[tex]y=-\dfrac{1}{3}x+b[/tex]
⇒ Plug in the given point (3,5) and determine the y-intercept:
[tex]5=-\dfrac{1}{3}(3)+b\\\\5=-1+b\\\\b=6[/tex]
⇒ Plug this back into the equation along with the slope:
[tex]y=-\dfrac{1}{3}x+6[/tex]
Answer[tex]y=-\dfrac{1}{3}x+6[/tex]
Number two please asap
J
Step-by-step explanation:
both Rue and Zoe they earned that same amount after working for 4 hours as you see that both line both hit the point (4,24)
13tX 9
7. Dareen's family now wants to Install 1-foot-square terra-cotta tiles in the entryway and kitchen, and 4-inch-square blue tiles on each bathroom floor. The
terra-cotta tiles cost $ each, and the bathroom tiles cost 45€ cach. How many of each kind will they need? What will all the tiles cost?
The total number of terra-cotta tiles needed is 220 tiles.
The total number of blue tiles needed is 4428 tiles.
The cost of all the tiles would be $3,092.60.
What is the total number of tiles needed?The first step is to determine the area of the entryway and the kitchen.
Area of the entryway = length x width
(22 - 18) x 10 = 40 ft²
Area of the kitchen = 10 x 18 = 180ft²
Sum of the areas of the entryway and the kitchen = 180 + 40 = 220 ft² = 220 tiles
Total cost = 220 x $5 = $1100
Areas of the bathroom floors = (10 x 6) + (7 x 9) = 123ft²
Convert ft² to in²
1 ft² = 144in²
123 x 144 = 17,712 in²
Number of tiles needed = 17,712 / 4 = 4428 tiles
Total cost = 4428 x 0.45 = $1992.60
Total cost = $1992.60 + $1100 = $3,092.60
Here is the complete question: Dareen's family now wants to Install 1-foot-square terra-cotta tiles in the entryway and kitchen, and 4-inch-square blue tiles on each bathroom floor. The
terra-cotta tiles cost $5 each, and the bathroom tiles cost 45€ cach. How many of each kind will they need? What will all the tiles cost?
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