What is the probability of getting exactly 2 heads, given that the first toss is a head?
there is a 1/2 probability of getting heads on any one flip
since the first one landed on heads you have a 1/2 probability f getting a 2nd one
Probability = 1/2
If x2 + xy + y3 = 1, find the value of y''' at the point where x = 1.
The third derivative of the function is [tex]y^{'''}=\dfrac{\left(3y^2+12xy-x\right)y''+y'\left(6yy'+12xy'+12y-1\right)-\left(6y-1\right)y'-6yy'}{\left(3y^2+x\right)^2}-\dfrac{2\left(6yy'+1\right)\left(\left(3y^2+12xy-x\right)y'-y\cdot\left(6y-1\right)\right)}{\left(3y^2+x\right)^3}[/tex]and the value at the point x = 1 is 42
How to determine the third derivative at the point x = 1
From the question, we have the following parameters that can be used in our computation:
[tex]x^2 + xy + y^3 = 1[/tex]
Differentiate implicitly
So, we have
[tex]3y^2y' + xy'+y+2x=0[/tex]
Make y' the subject of formula
So, we get
[tex]y'=-\dfrac{y+2x}{3y^2+x}[/tex]
Differentiate the second time
Using a graphing tool, we have
[tex]y''=\dfrac{\left(3y^2+12xy-x\right)y'-6y^2+y}{\left(3y^2+x\right)^2}[/tex]
Differentiate the third time to get the third derivative
Using a graphing tool, we have
[tex]y^{'''}=\dfrac{\left(3y^2+12xy-x\right)y''+y'\left(6yy'+12xy'+12y-1\right)-\left(6y-1\right)y'-6yy'}{\left(3y^2+x\right)^2}-\dfrac{2\left(6yy'+1\right)\left(\left(3y^2+12xy-x\right)y'-y\cdot\left(6y-1\right)\right)}{\left(3y^2+x\right)^3}[/tex]
Recall that
x = 1
Calculating y, we have
[tex]1^2 + (1)y + y^3 = 1[/tex]
[tex]1 + y + y^3 = 1[/tex]
[tex]y^3 + y = 0[/tex]
Factorize
[tex]y(y^2 + 1) = 0[/tex]
So, we have
y = 0 or [tex]y^2 + 1 = 0[/tex]
The equation [tex]y^2 + 1 = 0[/tex] will give a complex solution
So, we have
x = 1 and y = 0
Calculating y', we have
[tex]y'=-\dfrac{0+2(1)}{3 * 0^2+1}[/tex]
[tex]y'=-\dfrac{2}{1}[/tex]
y' = -2
Calculating y", we have
[tex]y''=\dfrac{\left(3y^2+12y-1\right)y'-6y^2+y}{\left(3y^2+1\right)^2}[/tex]
[tex]y''=\dfrac{\left(3(0)^2+12(1)(0)-1\right)(-2)-6(0)^2+0}{\left(3(0)^2+1\right)^2}[/tex]
[tex]y''=\dfrac{\left2}{1}[/tex]
y" = 2
Calculating y", we have
[tex]y^{'''}=\dfrac{\left(3y^2+12xy-x\right)y''+y'\left(6yy'+12xy'+12y-1\right)-\left(6y-1\right)y'-6yy'}{\left(3y^2+x\right)^2}-\dfrac{2\left(6yy'+1\right)\left(\left(3y^2+12xy-x\right)y'-y\cdot\left(6y-1\right)\right)}{\left(3y^2+x\right)^3}[/tex]
Simplifying the denominators, we have
[tex](3y^2 + x)^2 = (3(0)^2 + 1)^2 = 1[/tex]
[tex](3y^2 + x)^3 = (3(0)^2 + 1)^3 = 1[/tex]
So, we have
[tex]y^{'''}=\dfrac{\left(3y^2+12xy-x\right)y''+y'\left(6yy'+12xy'+12y-1\right)-\left(6y-1\right)y'-6yy'}{1}-\dfrac{2\left(6yy'+1\right)\left(\left(3y^2+12xy-x\right)y'-y\cdot\left(6y-1\right)\right)}{1}[/tex]
Divide
[tex]y^{'''}=[\left(3y^2+12xy-x\right)y''+y'\left(6yy'+12xy'+12y-1\right)-\left(6y-1\right)y'-6yy']-[2\left(6yy'+1\right)\left(\left(3y^2+12xy-x\right)y'-y\cdot\left(6y-1\right)\right)][/tex]
Simplifying each term:
[tex](3y^2+12xy-x)y''+y'(6yy'+12xy'+12y-1)-(6y-1)y'-6yy' = (3(0)^2+12(1)(0)-(1))(2) + (-2)(6(0)(-2) +12(1)(-2) + 12(0) - 1) - (6(0) - 1)(-2) - 6(0)(-2)[/tex]
[tex](3y^2+12xy-x)y''+y'(6yy'+12xy'+12y-1)-(6y-1)y'-6yy' = 46[/tex]
Also, we have
[tex]2(6yy'+1)((3y^2+12xy-x)y'-y(6y-1)) = 2(6(0)(-2) + 1)((3(0)^2 + 12(1)(0) - 1)(-2) - 0(6(0)-1))[/tex]
[tex]2(6yy'+1)((3y^2+12xy-x)y'-y(6y-1)) = 4[/tex]
So, the expression becomes (by substitution)
[tex]y^{'''}= 46 -4[/tex]
This gives
[tex]y^{'''}= 42[/tex]
Hence, the third derivative at the point x = 1 is 42
the base of an exponential function cannot be a negative number true or false
Answer:
True
Step-by-step explanation:
Given statement : The base of an exponential function cannot be a negative number.
We need to check whether the given statement is true or not.
The general form of an exponential function is
[tex]f(x)=ab^x[/tex]
where, a is the initial value and b is the base of exponential function.
The value of base b is always greater than 0 because
1. The terms like [tex](-3)^\frac{1}{2}[/tex] is an imaginary number and it make no sense. So, the base of an exponential function cannot be a negative number.
2. It b=0, then [tex](0)^x=0[/tex], which is a constant function. So, the base of an exponential function cannot be 0.
It means b>0.
Therefore, the given the given statement is true.
Final answer:
The base of an exponential function must be positive, as a negative base can lead to undefined values when raised to non-integer exponents. The rate of growth of an exponential function with a positive base near one can be approximated by 1+x for small x. Bases for exponential functions can often be connected to Euler's number e (approximately 2.71818).
Explanation:
The statement that the base of an exponential function cannot be a negative number is true. The base of an exponential function needs to be positive because a negative base can lead to undefined or complex values when raised to a fractional or irrational exponent. For example, the exponential function with base e (where e is the Euler's number, approximately equal to 2.71818) demonstrates natural growth, and its rate of growth for small x values approximates to 1+x. This approximation indicates how for a small change in x, the change in the value of the function is nearly proportional when the base is a positive number close to one.
Considering other bases, they can be related to base e using the identity ax = eln(a)x, where a is a positive real number, and ln(a) is the natural logarithm of a. Thus, whether for common or natural bases, the exponential function is well-defined only for positive bases.
Write the standard form of the equation of the line passing through the point (2,5) and perpendicular to the line 4x - y = 2. The answer key says that the answer is x + 4y = 22, but I'm confused on how to get there
To find the perpendicular line's equation, first find the negative reciprocal of the original line's slope. Next, use the point-slope form with the given point. Lastly, rearrange the equation into standard form, resulting in x + 4y = 22.
To find the equation of a line that is perpendicular to another line and passes through a given point, you need to perform a series of steps. The first line's equation is given as 4x - y = 2. Firstly, solve for y to put it in slope-intercept form, y = mx + b. Here, the equation becomes y = 4x - 2, so the slope (m) is 4. The slope of the perpendicular line will be the negative reciprocal of this, which is -1/4.
The next step is to use the point-slope form of the line, which is y - y1 = m(x - x1), where (x1, y1) is the point through which the line passes. For the point (2,5), the equation of the line is y - 5 = -1/4(x - 2). Multiplying both sides by 4 to clear the fraction gives 4y - 20 = -x + 2.
Finally, rearrange the equation to get it into standard form, Ax + By = C, giving us x + 4y = 22. This is the standard form of the equation we were seeking.
Candis took out a payday loan with an effective interest rate of 15,400%. if she had 220 to invest for a year at this interest rate, how much would make in interest?
A. 3,388,000
B 338,800
C.. 3388
D 33,880
To find the interest Candis would make from a 15,400% interest rate on a $220 investment for one year, we calculate using the simple interest formula, resulting in $33,880.
Explanation:The question asks us to determine how much interest Candis would make from a payday loan with an effective interest rate of 15,400% if she invested $220 for a year. To calculate the interest earned, we can use the formula for simple interest which is I = Prt, where I is interest, P is principal amount (the initial amount of money), r is the annual interest rate (in decimal form), and t is the time in years.
Converting 15,400% to a decimal, we get 154. Then, apply the formula:
I = $220 × 154 × 1
This gives us:
I = $33,880
Therefore, Candis would make $33,880 in interest after one year, which corresponds to option D.
A local supermarket sells chicken fir $2.49/lb and pork for $3.19/lb.Todd buys "c" pounds of chicken and "p" pounds ofpork which of the following inequalities represents that Todd only has $40 to spend?
What is the length of the hypotenuse, x, if (12, 35, x) is a Pythagorean triple?
Answer:37
Step-by-step explanation:
12•12=144
35•35=1225
1225+144=1369
Square root 1369=37
Find the derivative of f(x) = 8 divided by x at x = -1.
A. 4
B. 0
C. 8
D. -8
The derivative of f(x) at x = -1 is -8.
What is a derivative?Derivative is the rate of change of a function with respect to a variable. Derivatives are fundamental to the solution of problems in calculus and differential equations.
According to the given problem,
f(x) = [tex]\frac{8}{x}[/tex]
Derivative of f(x),
⇒ f'(x) = [tex]-\frac{8}{x^{2} }[/tex]
At x = -1,
⇒ f'(-1) = [tex]-\frac{8}{(-1)^{2} }[/tex]
= -8
Hence, we can conclude, the derivative of f(x) at x = -1 is -8.
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HELP! Will give Brainliest! Using dimensional analysis, convert 293 cm into m. (1 m= 100 cm)
(and this is also a Chemistry Question)
I get how to work out the other question, but I'm confused on this one
A rectangle is placed around a semicircle as shown below. the width of the rectangle is 6ft . find the area of the shaded region. use the value 3.14 for π , and do not round your answer. be sure to include the correct unit in your answer.
The shaded region is by assumption the region which is not covered by the semicircle in in given rectangles.
The area of the shaded region is given by 15.48 sq. ft.
What is a semicircle?
A semicircle is a circle cut in half. Thus, one circle produces two semicircle.
Firstly we will find the area of the rectangle and then subtract the area of the semicircle to find the are of the shaded region.
Since the radius of the semicircle is equal to width of the rectangle(6 ft), thus the length of the diameter of the circle( twice the radius which is 12 ft) serves as length of the considered rectangle.
Thus, we have:
[tex]\text{Area of the given rectangle\:} = 6 \times 12 = 72 \: \rm ft^2[/tex]
Since the semicircle is having radius of 6 ft, thus:
[tex]\text{Area of semicircle} = \dfrac{\pi r^2}{2} = \dfrac{3.14 \times 6^2}{2} = 56.52 \: \rm ft^2[/tex]
Thus, area of the shaded region will be equal to area of rectangle - area of semicircle = 72 - 56.52 = 15.48 sq. ft.
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Give the degree and classify the polynomial by the number of terms- 3
A)degree 1, monomial
B)degree 1, binomial
C)degree 0, monomial
D)degree 0, binomial
Answer:
Step-by-step explanation:
the answer is a
solve the following equation -2x + 4 = 2 (4x - 3) -3 (-8 + 4x)
a.7
b.2
c.-7
d.3
Answer:
[tex]A) 7[/tex]
Step-by-step explanation:
[tex]-2x+4=2\left(4x-3\right)-3\left(-8+4x\right)[/tex]
[tex]2\left(4x-3\right)-3\left(-8+4x\right)[/tex]
[tex]2 (4x - 3)[/tex]
[tex]\longrightarrow8x-6[/tex]
[tex]\longrightarrow 8x-6-3\left(-8+4x\right)[/tex]
[tex]-3 (-8 + 4x)[/tex]
[tex]\longrightarrow 24-12x[/tex]
[tex]8x-6+24-12x[/tex]
Combine like terms:
[tex]8x-12x-6+24[/tex]
Add: [tex]8x-12x=-4x[/tex]
Add: [tex]-6+24=18[/tex]
[tex]\longrightarrow-2x+4=-4x+18[/tex]
Subtract 4 from both sides:
[tex]\longrightarrow-2x+4-4=-4x+18-4[/tex]
[tex]\longrightarrow-2x=-4x+14[/tex]
Add 4x to both sides:
[tex]\longrightarrow-2x+4x=-4x+14+4x[/tex]
[tex]\longrightarrow2x=14[/tex]
Divide both sides by 2:
[tex]\longrightarrow\frac{2x}{2}=\frac{14}{2}[/tex]
[tex]\longrightarrow x=7[/tex]
____________________________
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Does arkansas lie south of 40 degrees latitude
Which of the following are vertical asymptotes of the function y = 2cot(3x) + 4? Check all that apply. A.x = pi/3 B.x = +/- pi/2 C.x = 2pi D.x = 0
The vertical asymptotes of the function y = 2cot(3x) + 4 are A.x = π/3 C. x = 2π D.x = 0
How to determine the vertical asymptote?The function is given as:
y = 2cot(3x) + 4
The above function is a cotangent function, represented as:
y = Acot(Bx +C) + D
By comparison, we have:
B = 3
The vertical asymptotes are then calculated using:
[tex]x = \frac{\pi}{B}n[/tex], where n are integers
Substitute 3 for B
[tex]x = \frac{\pi}{3}n[/tex]
Using the above format, the vertical asymptotes in the options are A.x = π/3 C. x = 2π D.x = 0
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A high-altitude spherical weather balloon expands as it rises, due to the drop in atmospheric pressure. Suppose that the radius r increases at the rate of 0.07 inches per second, and that r = 36 inches at time t = 0. Determine the equation that models the volume V of the balloon at time t, and find the volume when t = 400 seconds.
Simplify 6 to the fifth power over 7 cubed all raised to the second power. 6 to the seventh power over 7 to the tenth power 6 to the tenth power over 7 to the sixth power 6 cubed over 7 12 to the fifth power over 14 cubed
Answer:
6 to the tenth power over 7 to the sixth power
Step-by-step explanation:
Given phrase,
6 to the fifth power over 7 cubed all raised to the second power,
[tex]\implies (\frac{6^5}{7^3})^2[/tex]
By using [tex](a^m)^n=a^{mn}[/tex]
[tex]=\frac{6^{5\times 2}}{7^{3\times 2}}[/tex]
[tex]=\frac{6^{10}}{7^6}[/tex]
= 6 to the tenth power over 7 to the sixth power
Simplify the expressions
(6⁵/7³)² = 2143588816/117649
(6⁷/7¹⁰) = 279936/282475249
(6¹⁰/7⁶) = 60466176/117649
(6³/7) = 216/7
(12⁵/14³) = 90855/1001
To simplify the given expressions, we can calculate the numerical values and perform the necessary operations. Let's evaluate each expression:
(6⁵/7³)²:
First, calculate the numerator and denominator:
Numerator: 6⁵ = 6 × 6 × 6 × 6 × 6 = 7776
Denominator: 7³ = 7 × 7 × 7 = 343
Now, substitute the values into the expression and square the result:
(7776/343)² = (7776/343) × (7776/343) = 2143588816/117649
The simplified form is 2143588816/117649.
(6⁷/7¹⁰):
Calculate the numerator and denominator:
Numerator: 6⁷ = 6 × 6 × 6 × 6 × 6 × 6 × 6 = 279936
Denominator: 7¹⁰ = 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 = 282475249
Substitute the values into the expression:
279936/282475249
This expression cannot be simplified further.
(6¹⁰/7⁶):
Calculate the numerator and denominator:
Numerator: 6¹⁰ = 6 × 6 × 6 × 6 × 6 × 6 × 6 × 6 × 6 × 6 = 60466176
Denominator: 7⁶ = 7 × 7 × 7 × 7 × 7 × 7 = 117649
Substitute the values into the expression:
60466176/117649
This expression cannot be simplified further.
(6³/7):
Calculate the numerator and denominator:
Numerator: 6³ = 6 × 6 × 6 = 216
Denominator: 7
Substitute the values into the expression:
216/7
This expression cannot be simplified further.
(12⁵/14³):
Calculate the numerator and denominator:
Numerator: 12⁵ = 12 × 12 × 12 × 12 × 12 = 248832
Denominator: 14³ = 14 × 14 × 14 = 2744
Substitute the values into the expression:
248832/2744 = 90855/1001
The simplified form is 90855/1001.
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Simplify 6 to the fifth power over 7 cubed all raised to the second power. 6 to the seventh power over 7 to the tenth power 6 to the tenth power over 7 to the sixth power 6 cubed over 7 12 to the fifth power over 14 cubed
(6⁵/7³)²
(6⁷/7¹⁰)
(6¹⁰/7⁶)
(6³/7)
(12⁵/14³)
A security fence encloses a rectangular area on one side of a park in a city. three sides of fencing are? used, since the fourth side of the area is formed by a building. the enclosed area measures 392392 square feet. exactly 5656 feet of fencing is used to fence in three sides of this rectangle. what are the possible dimensions that could have been used to construct this? area?
please i need help....the question is.........
area = H/2*(b1+b2)
8.1 = 1.5/2*(6.7 +b2)
8.1=0.75*(6.7+b2)
10.8=6.7+b2
b2=10.8-6.7
b2=4.1m
factors of 3x^2y^2+6x^2+12y^2+24
how many cups of grape punch containing 10% fruit juice and berry punch containing 20% fruit juice must be added together to create 12 cups of punch with 18% fruit juice?
2x − y = 3 4x = 6 + 2y
Adult male heights are normally distributed with a mean of 70 inches and a standard deviation of 3 inches. The average basketball player is 79 inches tall. Approximately what percent of the adult male population is taller than the average basketball player? 0.135% 0.875% 49.875% 99.875%
Answer:
A. 0.135%
Step-by-step explanation:
We have been given that adult male heights are normally distributed with a mean of 70 inches and a standard deviation of 3 inches. The average basketball player is 79 inches tall.
We need to find the area of normal curve above the raw score 79.
First of all let us find the z-score corresponding to our given raw score.
[tex]z=\frac{x-\mu}{\sigma}[/tex], where,
[tex]z=\text{z-score}[/tex],
[tex]x=\text{Raw-score}[/tex],
[tex]\mu=\text{Mean}[/tex],
[tex]\sigma=\text{Standard deviation}[/tex].
Upon substituting our given values in z-score formula we will get,
[tex]z=\frac{79-70}{3}[/tex]
[tex]z=\frac{9}{3}[/tex]
[tex]z=3[/tex]
Now we will find the P(z>3) using formula:
[tex]P(z>a)=1-P(z<a)[/tex]
[tex]P(z>3)=1-P(z<3)[/tex]
Using normal distribution table we will get,
[tex]P(z>3)=1-0.99865 [/tex]
[tex]P(z>3)=0.00135[/tex]
Let us convert our answer into percentage by multiplying 0.00135 by 100.
[tex]0.00135\times 100=0.135%[/tex]
Therefore, approximately 0.135% of the adult male population is taller than the average basketball player and option A is the correct choice.
How much would $125 invested at 8% interest compounded continuously be worth after 16 years? Round your answer to the nearest cent. Do not include units in your answer.
Part A: Solve -vp + 30 < 45 for v .. show your work.
Part B: Solve 3w - 6r = 30 for r .. show your work.
Oil is poured on a flat surface and it spreads out forming a circle. the area of this circle is increasing at a constant rate of 10cm2/s. determine the rate of change of the radius of the circle when the radius is 10cm
The rate of change of radius of the circle when the radius is 10 cm is
dr / dt = 1/2π cm / sec
What is implicit differentiation?
In mathematics, differentiation is the process of determining the derivative, or rate of change, of a function. Differentiation is a technique for determining a function's derivative. Differentiation is a mathematical procedure that determines the instantaneous rate of change of a function based on one of its variables. The process of finding the derivative of dependent variable in an implicit function by differentiating each term separately
Given data ,
Oil is poured on a flat surface and it spreads out forming a circle
The area of the circle is increasing at a rate of 10 cm²/sec
Let the area of the circle be = A
So , the equation is dA² / dt² = 10 cm²/sec
The area of the circle = πr²
So , dA² / dt² = 2π dr / dt
2πr dr / dt = 10 cm²/sec
Substituting the value of r in the equation , we get
2π x 10 dr / dt = 10 cm²/sec
20π dr / dt = 10 cm²/sec
Divide both sides of the equation be 20π
dr / dt = ( 1/2π ) cm / sec
Therefore , the rate of change of radius of the circle is ( 1/2π ) cm / sec
Hence , The rate of change of radius of the circle when the radius is 10 cm is dr / dt = 1/2π cm / sec
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Nancy is a telemarketer who calls 96 people each day. How many people will she call in a 5-day workweek?
A. 430
B. 460
C. 480
D. 380
-2(4g-3)= 30 how do i solve this
are all semi circles simular
When constructing a circle circumscribed about a triangle, what is the purpose of constructing perpendicular bisectors?
The first term of a geometric sequence is –2 and the common -1/4. What are the next three terms of the sequence?