c. Reflection over either the x-axis or y-axis will change the graph
Step-by-step explanation:a. Reflection over the y-axis will not change the graph since sine is an even function.This is false because [tex]y=sin(x)[/tex] is an odd function, not an even one. This means that [tex]sin(-x)=-sin(x)[/tex], and a reflection over the y-axis will change the graph.
b. Sin(x)=sin(-x)This is false because we said that [tex]sin(-x)=-sin(x)[/tex]
c. Reflection over either the x-axis or y-axis will change the graphThis is true. Since [tex]sin(x)[/tex] is an odd function, then reflection over either the x-axis or y-axis will change the graph as we said in a. So, for [tex]f(x)[/tex]:
REFLEXION IN THE X-AXIS:
[tex]h(x)=-f(x)[/tex]
REFLEXION IN THE Y-AXIS:
[tex]h(x)=f(-x)[/tex]
d. Sin(x)=-sin(x)False by the same explanation as b.
The correct statement about the function y=sin(x) is that Reflection over either the x-axis or y-axis will change the graph. Therefore, option C is the correct answer.
The statement regarding the function y=sin(x) which is true is that reflection over either the x-axis or y-axis will change the graph.
This is because the sine function is an odd function, meaning that it has rotational symmetry about the origin. A characteristic of odd functions is that they satisfy the identity y(-x) = -y(x), not y(-x) = y(x), which describes an even function.
Therefore, the assumption Sin(x)=Sin(-x) would be incorrect, as it does not reflect the odd nature of the sine function. Thus, the correct answer is c. Reflection over either the x-axis or y-axis will change the graph.
a) Estimate the volume of the solid that lies below the surface z = 7x + 5y2 and above the rectangle R = [0, 2]⨯[0, 4]. Use a Riemann sum with m = n = 2 and choose the sample points to be lower right corners.
In the [tex]x[/tex] direction we consider the [tex]m=2[/tex] subintervals [0, 1] and [1, 2] (each with length 1), while in the [tex]y[/tex] direction we consider the [tex]n=2[/tex] subintervals [0, 2] and [2, 4] (with length 2). Then the lower right corners of the cells in the partition of [tex]R[/tex] are (1, 0), (2, 0), (1, 2), (2, 2).
Let [tex]f(x,y)=7x+5y^2[/tex]. The volume of the solid is approximately
[tex]\displaystyle\iint_Rf(x,y)\,\mathrm dx\,\mathrm dy\approx f(1,0)\cdot1\cdot2+f(2,0)\cdot1\cdot2+f(1,2)\cdot1\cdot2+f(2,2)\cdot1\cdot2=\boxed{164}[/tex]
###
More generally, the lower-right-corner Riemann sum over [tex]m=\mu[/tex] and [tex]n=\nu[/tex] subintervals would be
[tex]\displaystyle\sum_{m=1}^\mu\sum_{n=1}^\nu\left(7\frac{2m}\mu+5\left(\frac{4n-4}\nu\right)^2\right)\frac{2-0}\mu\frac{4-0}\nu=\frac83\left(101+\frac{21}\mu+\frac{40}{\nu^2}-\frac{120}\nu\right)[/tex]
Then taking the limits as [tex]\mu\to\infty[/tex] and [tex]\nu\to\infty[/tex] leaves us with an exact volume of [tex]\dfrac{808}3[/tex].
The Riemann sum is used to provide an estimate of the volume of a solid under the function surface z = 7x + 5y² and above the rectangle R = [0, 2] × [0, 4]. The rectangle is divided into four equal parts, and the function's value at specific points, multiplied by the area of the base, provides the estimate.
Explanation:This Mathematics problem requires estimating the volume of a solid under the surface z = 7x + 5y² and above the rectangle R = [0, 2]⨯[0, 4] using a Riemann sum. This method is commonly used in calculus to approximate the definite integral of a function.
Applying a Riemann sum with m=n=2 implies the rectangle is split into 4 equal rectangles for the estimation. Our sample delta x and delta y = rectangle's length/2, for this example, Δx = 2/2 = 1 and Δy = 4/2 = 2. The lower right corners points will be (1,2), (1,4), (2,2) and (2,4).
We then find volume estimates by taking the function's value at these sample points and multiplying it by the area of the base. This gives us: ((7*1 + 5*2²) + (7*1 + 5*4²) + (7*2 + 5*2²) + (7*2 + 5*4²))* (Δx*Δy). Simplifying the expression gives us the estimated volume using the Riemann sum.
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A box contains 12 balls numbered 1 through 12. Two balls are drawn in succession without replacement. If the second ball has the number 4 on it, what is the probability that the first ball had a smaller number on it? An even number on it? The probability that the first ball had a smaller number is nothing.
Final answer:
To determine the probability that the first ball has a smaller number than 4, we count the three balls with numbers 1, 2, and 3, resulting in a 3 out of 11 probability. The probability that the first ball was even is calculated by counting the even-numbered balls (2, 6, 8, 10, 12), which gives a probability of 5 out of 11.
Explanation:
The question asks about the probability that the first ball drawn from a box of 12 numbered balls is smaller or even, given that the second ball drawn has the number 4 on it and the drawing occurs without replacement. We know that once the second ball has been confirmed as the number 4, there are 11 remaining possibilities for the first ball.
To find the probability that the first ball had a smaller number than 4, we count the balls that are numbered less than 4. There are 3 such balls: 1, 2, and 3. Therefore, the probability is 3 out of 11 that the first ball had a smaller number.
To calculate the probability that the first ball had an even number, we consider only the even-numbered balls among the 11 remaining. These are 2, 6, 8, 10, and 12. So, there are 5 even-numbered balls, making the probability 5 out of 11 that the first ball was even.
Simplify 16m^2/m^2+5/4m/3m^2+15
Answer:
12m
Step-by-step explanation:
We are given the following expression where a fraction is divided by another fraction:
[tex]\frac{\frac{16m^2}{m^2+5} }{\frac{4m}{3m^2+15} }[/tex]
To change this division into multiplication, we will take reciprocal of the fraction in the denominator and then solve:
[tex] \frac { 1 6 m ^ 2 } { m^2+5} } \times \frac{3m^2+15}{4m}[/tex]
Factorizing the terms to simplify:
[tex] \frac { 4 m ( 4m ) } { m ^ 2 + 5 } \times \frac { 3 ( m ^ 2 + 5 ) } { 4 m } [/tex]
Cancelling the like terms to get:
12m
Answer: [tex]12m[/tex]
Step-by-step explanation:
Given the expression [tex]\frac{\frac{16m^2}{m+5}}{\frac{4m}{3m^2+15}}[/tex], we can rewrite it in this form:
[tex](\frac{16m^2}{m+5})(\frac{3m^2+15}{4m})[/tex]
Now we must multiply the numerator of the first fraction by the numerator of the second fraction and the denominator of the first fraction by the denominator of the second fraction:
[tex]=\frac{(16m^2)(3m^2+15)}{(m^2+5)(4m)}}[/tex]
According to the Quotient of powers property:
[tex]\frac{a^m}{a^n}=a^{(m-n)}[/tex]
And the Product of powers property states that:
[tex](a^m)(a^n)=a^{(m+n)}[/tex]
Then, simplifying, we get:
[tex]=\frac{3(m^2+5)(4m)(4m)}{(m^2+5)(4m)}}\\\\=3(4m)\\\\=12m[/tex]
If F(x,y) = x^2sin(xy), find Fyx.
Answer:
[tex]F_{yx}=3x^{2} cos(xy)- yx^{3} sin(xy)[/tex]
Step-by-step explanation:
We need to find out the partial differential [tex]F_{yx}[/tex] of [tex]F(x,y)=x^{2}sin(xy)[/tex]
First, differentiate [tex]F(x,y)=x^{2}sin(xy)[/tex] both the sides with respect to 'y'
[tex]\frac{d}{dy}F(x,y)=\frac{d}{dy}x^{2}sin(xy)[/tex]
Since, [tex]\frac{d}{dt}\sin t =\cos t[/tex]
[tex]\frac{d}{dy}F(x,y)=x^{2}cos(xy)\times \frac{d}{dy}(xy)[/tex]
[tex]\frac{d}{dy}F(x,y)=x^{2}cos(xy)\times x[/tex]
[tex]\frac{d}{dy}F(x,y)=x^{3}cos(xy)[/tex]
so, [tex]F_y=x^{3}cos(xy)[/tex]
Now, differentiate above both the sides with respect to 'x'
[tex]F_{yx}=\frac{d}{dx}x^{3}cos(xy)[/tex]
Chain rule of differentiation: [tex]D(fg)=f'g + fg'[/tex]
[tex]F_{yx}=cos(xy) \frac{d}{dx}x^{3} + x^{3} \frac{d}{dx}cos(xy)[/tex]
Since, [tex] \frac{d}{dx}x^{m} =mx^{m-1}[/tex] and [tex] \frac{d}{dt} cost =-\sin t[/tex]
[tex]F_{yx}=cos(xy)\times 3x^{2} - x^{3} sin(xy)\times \frac{d}{dx}(xy)[/tex]
[tex]F_{yx}=cos(xy)\times 3x^{2} - x^{3} sin(xy)\times y[/tex]
[tex]F_{yx}=3x^{2} cos(xy)- yx^{3} sin(xy)[/tex]
hence, [tex]F_{yx}=3x^{2} cos(xy)- yx^{3} sin(xy)[/tex]
Solve the following system of equations.
0.12x - 0.07y = -1.35
0.4x + 0.8y = 4.8
Answer:
x = -6 and y = 9
Step-by-step explanation:
It is given that,
0.12x - 0.07y = -1.35 -----(1)
0.4x + 0.8y = 4.8 -----(2)
To find the value of x and y
eq(1) * 100 ⇒
12x + 7y = -135 -----(3)
eq(2) /0.4 ⇒
x + 2y = 12 -----(4)
eq(4) * 12 ⇒
12x + 24y = 144 ---(5)
eq(5) - eq(3) ⇒
12x + 24y = 144 ---(5)
12x - 7y = -135 -----(3)
0 + 31y = 279
y = 279/31 = 9
Substitute the value of y in eq(4)
x + 2y = 12 -----(4)
x + 2*9 = 12
x = 12 - 18 = -6
Therefore x = -6 and y = 9
Find the probability of the given event. A bag contains 7 red marbles, 2 blue marbles, and 3 green marbles. A randomly drawn marble is blue.
Answer: [tex]\dfrac{1}{6}[/tex]
Step-by-step explanation:
The given event : A randomly drawn marble is blue.
The number of blue marbles in the bag = 2
The total number of marbles in the bag = [tex]2+7+3=12[/tex]
Now, the probability of drawing a blue marble is given by :-
[tex]\text{P(Blue)}=\dfrac{\text{Number of blue marbles}}{\text{Total number of marbles}}\\\\\Rightarrow\text{P(Blue)}=\dfrac{2}{12}=\dfrac{1}{6}[/tex]
Hence, the probability of the given event event = [tex]\dfrac{1}{6}[/tex]
For f(x) = 2|x+3| – 5, name the type of function and describe each of the three transformations from the parent function f(x) = |x|.
Answer:
Type of function: Absolute Value
Transformations: 1) elongated by a stretch factor of 2; 2) shifted left 3; 3) shifted down 5
Answer:
Shifted 5 units downShifted 3 units to the leftVertically streched by a scale of 2.Step-by-step explanation:
The parent function is
[tex]f(x)=|x|[/tex]
The transformed function is
[tex]g(x)=2|x+3|-5[/tex]
You can deduct by comparison, that the function was shifted 5 units down, 3 units to the left, and vertically streched by a scale of 2.
We deduct this transfromations based on the following rules.
[tex]f(x)-u[/tex] indicates a movement downside [tex]u[/tex] units.
[tex]f(x+u)[/tex] indicates a movement leftside [tex]u[/tex] units.
[tex]uf(x)[/tex] indicates a vertical stretch for [tex]u>1[/tex].
A value meal package at Ron's Subs consists of a drink, a sandwich, and a bag of chips. There are 4 types of drinks to choose from, 3 types of sandwiches, and 3 types of chips. How many different value meal packages are possible?
Final answer:
To find the total number of different value meal packages possible at Ron's Subs, multiply the number of choices for drinks (4), sandwiches (3), and chips (3), resulting in 4 × 3 × 3 = 36 possible combinations.
Explanation:
To find the total number of different value meal packages possible, we calculate the product of the number of choices for each category. In this case, there are 4 types of drinks, 3 types of sandwiches, and 3 types of chips. Thus, the calculation is as follows:
Drink choices: 4Sandwich choices: 3Chip choices: 3To find the total number of combinations, we multiply the number of choices for each category:
4 (drinks) × 3 (sandwiches) × 3 (chips) = 36 different value meal packages.
Therefore, at Ron's Subs, there are 36 possible different value meal packages a customer can choose from.
please help!!! What is the decimal equivalent of this fraction?
Answer:
[tex]\bullet\ \ 0.\overline{15}[/tex]
Step-by-step explanation:
5/33 = (5·3)/(33·3) = 15/99 = 0.151515151515...
_____
You may recall that 1/9 = 0.11111...(repeating indefinitely). That is, a multiple of 1/9 is a single-digit repeating decimal.
Likewise, 1/99 = 0.01010101...(repeating indefinitely). This means when a 2-digit numerator has 99 as the denominator, the decimal equivalent is that number repeated indefinitely. Any fraction with 999 as the denominator is a 3-digit repeat in decimal; 9999 as the denominator gives a 4-digit repeat, and so on.
....Help Please.......
Answer:
linear
Step-by-step explanation:
The x-values all differ by 1, which is to say they are equally-spaced. The corresponding y-values all differ by -3. When (first) differences of equally-spaced values of y are constant, the function is of first degree, which is to say it is linear.
___
If second differences are non-zero and constant, the function is of second degree, quadratic.
Answer:
line
Step-by-step explanation:
The graphing option sounds nice...
But lines have the same slope no matter what two points you choose.
You can see that x is going up by the same number (plus 1) each time and the y's are going down by the same number each time (minus 3) so this says no matter what two points you choose you will have the same slope which means it is a line.
The sun has a radius of about 695,000 km. What is the volume of the sun (in scientific notation, using 3 decimal places in the mantissa)?
Answer:
1.406×[tex]10^{[tex]10^{18}km cubed
Step-by-step explanation:
The volume of a sphere is
[tex]V=\frac{4}{3}\pi r^3[/tex]
Filling in our formula:
[tex]V=\frac{4}{3}\pi (695,000)^3[/tex]
Cubing first gives us:
[tex]V=\frac{4}{3}\pi (3.35702[/tex]×[tex]10^{17}[/tex]
Do the multiplication and division of those numbers, multiply in the value of pi on your calculator, and you'll get 1.406×[tex]10^{18}[/tex]
To determine the volume of the Sun with a radius of about 695,000 km, we first convert the radius to centimeters and then apply the formula V = (4/3)πr³. After performing the calculations, the volume of the Sun is approximately 1.401 x 10³³ cm³in scientific notation with three decimal places in the mantissa.
Explanation:The student has asked what the volume of the Sun is, given its radius of about 695,000 km. To find the volume of a sphere, the formula to use is V = (4/3)πr³, where V represents the volume and r is the radius.
First, we need to convert the radius from kilometers to centimeters because the standard unit for volume in scientific notation often involves cubic centimeters. There are 100,000 centimeters in a kilometer, so the radius in centimeters is 695,000 km × 100,000 cm/km = 6.95 x 10¹⁰cm.
Now, we can calculate the volume using the formula:
V = (4/3)π(6.95 x 10¹⁰ cm)³
V = (4/3)π(6.95^3 x 10³⁰) cm³
V = (4/3)π(334.14 x 10³⁰) cm³
V = (4/3)π(3.3414 x 10³²) cm³
V ≈ 4.1888 x 3.3414 x 10³² cm³
V ≈ 1.401 x 10^33 cm³
Therefore, the volume of the Sun in scientific notation, using three decimal places in the mantissa, is approximately 1.401 x 10³³ cm³.
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A test score of 48.4 on a test having a mean of 66 and a standard deviation of 11. Find the z-score corresponding to the given value and use the z-score to determine whether the value is significant. Consider a score to be significant if its z-score is less than -2.00 or greater than 2.00. Round the z-score to the nearest tenth if necessary. A. -1.6; not significant B.-17.6; significant C. -1.6, significant D. 1.6; not significant
Answer:
A. -1.6; not significant
Step-by-step explanation:
The z-score of a data set that is normally distributed with a mean of [tex]\bar x[/tex] and a standard deviation of [tex]\sigma[/tex], is given by:
[tex]z=\frac{x-\bar x}{\sigma}[/tex].
From the question, the test score is: [tex]x=48.4[/tex], the mean is [tex]\bar x=66[/tex], and the standard deviation is [tex]\sigma =11[/tex].
We just have to plug these values into the above formula to obtain:
[tex]z=\frac{48.4-66}{11}[/tex].
This simplifies to: [tex]z=\frac{-17.6}{11}[/tex].
[tex]z=-1.6[/tex].
We can see that the z-score falls within two standard deviations of the mean.
Since [tex]-2\le-1.6\le2[/tex] the value is not significant.
The correct answer is A. -1.6; not significant
Winning the jackpot in a particular lottery requires that you select the correct three numbers between 1 and 53 and, in a separate drawing, you must also select the correct single number between 1 and 45. Find the probability of winning the jackpot.
[tex]|\Omega|={_{53}C_3}\cdot 45=\dfrac{53!}{3!50!}\cdot45=\dfrac{51\cdot52\cdot53}{2\cdot3}\cdot45=1054170\\|A|=1\\\\P(A)=\dfrac{1}{1054170}\approx0.00000095\%[/tex]
Complete parts a through f below to find nonnegative numbers x and y that satisfy the given requirements. Give the optimum value of P. x plus y equals 81 and Pequalsx squared y is maximized a. Solve x plus y equals 81 for y. yequals 81 minus x b. Substitute the result from part a into the equation Pequalsx squared y for the variable that is to be maximized. Pequals x squared left parenthesis 81 minus x right parenthesis c. Find the domain of the function P found in part b. left bracket 0 comma 81 right bracket (Simplify your answer. Type your answer in interval notation.) d. Find StartFraction dP Over dx EndFraction . Solve the equation StartFraction dP Over dx EndFraction equals0. StartFraction dP Over dx EndFraction equals nothing
Answer:
y = 81-xthe domain of P(x) is [0, 81]P is maximized at (x, y) = (54, 27)Step-by-step explanation:
Given
x plus y equals 81x and y are non-negativeFind
P equals x squared y is maximizedSolution
a. Solve x plus y equals 81 for y.
y equals 81 minus x
__
b. Substitute the result from part a into the equation P equals x squared y for the variable that is to be maximized.
P equals x squared left parenthesis 81 minus x right parenthesis
__
c. Find the domain of the function P found in part b.
left bracket 0 comma 81 right bracket
__
d. Find dP/dx. Solve the equation dP/dx = 0.
P = 81x² -x³
dP/dx = 162x -3x² = 3x(54 -x) = 0
The zero product rule tells us the solutions to this equation are x=0 and x=54, the values of x that make the factors be zero. x=0 is an extraneous solution for this problem so ...
P is maximized at (x, y) = (54, 27).
The problem is about using the equations x + y = 81 and P=x^2y to find the maximum possible value of P. This involves solving for y, substituting the result into the P equation, determining the domain of P, and finding the derivative of P to solve for x.
Explanation:We are given the system of equations: x + y = 81 and P = x^2*y and we're tasked with finding the optimum value for P given these constraints.
a. We solve for y in the equation x + y = 81, so y = 81 - x
b. We substitute the result from part a into the equation for P, P = x^2(81 - x)
c. The domain of function P is the set of all possible x values or [0, 81]. This is because x and y are non-negative and y is equal to 81 minus x, meaning the highest x can be is 81
d. With respect to maximization, step d usually involves calculating the derivative of P with respect to x, setting it equal to zero, and solving for x. If you apply the product rule and the chain rule, you would get dP/dx = 0 then solve for x
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Suppose that 3 cards from a standard deck of 52 playing cards are successively drawn at random without replacement (a) Find the probability that all 3 are queens (b) Find the probability that all 3 are spades (a) The probability that all 3 are queens is (Type an integer a simplified fraction) or (b) The probability that all 3 are spades is (Type integer simplified fraction) an or a
[tex]|\Omega|=52\cdot51\cdot50=132600[/tex]
a)
[tex]|A|=4\cdot3\cdot2=24\\P(A)=\dfrac{24}{132600}=\dfrac{1}{5525}[/tex]
b)
[tex]|A|=13\cdot12\cdot11=1716\\P(A)=\dfrac{1716}{132600}=\dfrac{11}{850}[/tex]
a. Probability of all 3 cards being queens:
Number of ways to choose 3 queens from 4: 4C3 = 4.Number of ways to choose 3 cards from 52: 52C3 = 22100.Probability = 4/22100 = 1/5525.b. Probability of all 3 cards being spades:
Number of ways to choose 3 spades from 13: 13C3 = 286.Number of ways to choose 3 cards from 52: 52C3 = 22100.Probability = 286/22100 = 13/1001.if f(x)=2x-1+3 and g(x)=5x-9, what is (f-g)(x)?
Answer:
[tex]\large\boxed{(f-g)(x)=-3x+11}[/tex]
Step-by-step explanation:
[tex](f-g)(x)=f(x)-g(x)\\\\f(x)=2x-1+3=2x+2\\g(x)=5x-9\\\\\text{Substitute:}\\\\(f-g)(x)=(2x+2)-(5x-9)\\\\=2x+2-5x-(-9)\\\\=2x+2-5x+9\qquad\text{combine like terms}\\\\=(2x-5x)+(2+9)\\\\=-3x+11[/tex]
Find the distance from the point to the line. (-1,-2,1);x=4+4t, y=3+t, z=6-t .The distance is ____ Typn exact answer, using radicals as needed.)
Answer:
The distance is 4.726
Step-by-step explanation:
we need to find the distance from the point to the line
Given:- point (-1,-2,1) and line ; x=4+4t, y=3+t, z=6-t .
used formula [tex]d=\frac{|a\times b|}{|a|}[/tex]
Let point P be (-1,-2,1)
using value t=0 and t=1
The point Q (4 , 3, 6) and R ( 8, 4, 5)
Let a be the vector from Q to R : a = < 8 - 4, 4 - 3, 5 - 6 > = < 4, 1, -1 >
Let b be the vector from Q to P: b = < -1 - 4, -2 - 3, 1 - 6> = < -5, -5, -5 >
The cross product of a and b is:
[tex]a \times b= \begin{vmatrix} i & j & k\\ 4 &1&-1\\-5 &-5&-5\\ \end{vmatrix}[/tex]
= -6i+15j-15k
The distance is : [tex]d=\frac{\sqrt{(-6)^{2}+(15)^{2}+(-15)^{2}}}{\sqrt{(4)^{2}+(1)^{2}+(-1)^{2}}}[/tex]
[tex]=\frac{\sqrt{36+225+225}}{\sqrt{16+1+1}}[/tex]
[tex]=\frac{\sqrt{36+225+225}}{\sqrt{16+1+1}}[/tex]
[tex]d=\frac{\sqrt{486}}{\sqrt{18}}[/tex]
≈4.726
Therefore, the distance is 4.726
Which zero pair could be added to the function fon) = x2 + 12x + 6 so that the function can be written in vertex form?
03.-3
0 6.6
03-3
O 36,-36
ANSWER
36,-36
EXPLANATION
The given function is:
[tex]f(x) = {x}^{2} + 12x + 6[/tex]
To write this function in vertex form;
We need to add and subtract the square of half the coefficient of x.
The coefficient of x is 12.
Half of it is 6.
The square of 6 is 36.
Therefore we add and subtract 36.
Hence the zero pair is:
36, -36.
The correct answer is D.
Answer:
Last option: 36,-36
Step-by-step explanation:
The vertex form of the function of a parabola is:
[tex]y=a(x-h)^2+k[/tex]
Where (h,k) is the vertex.
To write the given function in vertex form, we need to Complete the square.
Given the Standard form:
[tex]y=ax^2+bx+c[/tex]
We need to add and subtract [tex](\frac{b}{2})^2[/tex] on one side in order to complete the square.
Then, given [tex]y=x^2+12x+6[/tex], we know that:
[tex](\frac{12}{2})^2=6^2=36[/tex]
Then, completing the square, we get:
[tex]y=x^2+12x+(36)+6-(36)[/tex]
[tex]y=(x+6)^2-30[/tex] (Vertex form)
Therefore, the answer is: 36,-36
The automatic opening device of a military cargo parachute has been designed to open when the parachute is 200 m above the ground. Suppose opening altitude actually has a normal distribution with mean value 200 m and standard deviation 30 m. Equipment damage will occur if the parachute opens at an altitude of less than 100 m. What is the probability that there is equipment damage to the payload of at least one of five independently dropped parachutes?
Step-by-step answer:
Given:
mean, mu = 200 m
standard deviation, sigma = 30 m
sample size, N = 5
Maximum deviation for no damage, D = 100 m
Solution:
Z-score for maximum deviation
= (D-mu)/sigma
= (100-200)/30
= -10/3
From normal distribution tables, the probability of right tail with
Z= - 10/3
is 0.9995709, which represents the probability that the parachute will open at 100m or more.
Thus, by the multiplication rule, the probability that all five parachutes will ALL open at 100m or more is the product of the individual probabilities, i.e.
P(all five safe) = 0.9995709^5 = 0.9978565
So there is an approximately 1-0.9978565 = 0.214% probability that at least one of the five parachutes will open below 100m
The probability that at least one out of five parachutes causes equipment damage, given that the parachute opening altitude is normally distributed with a mean of 200m and standard deviation of 30m, is approximately 0.2%.
Explanation:The situation described is a question of probability related to the normal distribution. In this case, we are asked to find the probability of a parachute opening at less than 100m, which will cause damage. First, we need to standardize the value to a z-score. The z-score is calculated by subtracting the mean from the value of interest and dividing by the standard deviation. In this case, it will be (100-200)/30, which equals to about -3.33.
By looking at a z-table or using a statistical calculator we find that the probability of single parachute causing damage is approximately 0.0004. However, the question is interested in the probability of at least one out of five parachutes causing damage. This can be approached as 1 minus the probability of none of the five causing damage, which will be 1 - (1-0.0004)^5. Thus, the resulting probability of equipment damage to the payload of at least one of five independently dropped parachutes is approximately 0.002 or 0.2%.
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Type the correct answer in the box. Jason builds doghouses for a pet store. Each doghouse is a wooden structure with a rectangular base that has an area of 21 square feet and a length that is 4 feet more than its width. If x represents the width of the doghouse, write an equation in the given form that can be used to determine the possible dimensions of the base of the doghouse.
Answer:
Width = 3 feet
Length = 7 feet
Step-by-step explanation:
x represents the width of doghouse so,
Width = x
Length is 4 times more than its width
Length = x+4
Area of doghouse = 21 square feet
We know
Area of rectangle = Length * Width
21 = (x+4)*x
21 = x^2+4x
=> x^2+4x-21=0
Solving the above quadratic equation by factorization to find the value of x
x^2+7x-3x-21=0
x(x+7)-3(x+7)=0
(x+7)(x-3)=0
x+7 =0 and x-3=0
x= -7 and x =3
Since the width of rectangle can never be negative so, x=3
Width =x = 3 feet
Length = x+4 = 3+4 = 7 feet
Answer: (x + 7)(x - 3) = 0
width (x) = 3
length (x+4) = 7
Step-by-step explanation:
Area (A) = length (l) × width (x)
21 = (x + 4) × (x)
21 = x² + 4x distributed (x) into (x + 4)
0 = x² + 4x - 21 subtracted 21 from both sides
0 = (x + 7)(x - 3) factored quadratic equation
0 = x + 7 or 0 = x - 3 applied Zero Product Property
x = -7 or x = 3 solved each equation
x = -7 is not valid because lengths cannot be negative
so x = 3
and length ... x + 4 = (3) + 4 = 7
5x=k-14 solve for x (literal equation)
Answer:
[tex]\large\boxed{x=\dfrac{k-14}{5}}[/tex]
Step-by-step explanation:
[tex]5x=k-14\qquad\text{divide both sides by 5}\\\\\dfrac{\not5x}{\not5}=\dfrac{k-14}{5}\\\\x=\dfrac{k-14}{5}[/tex]
Answer:
[tex]5x = k - 14 \\ \frac{5x}{5} = \frac{k - 14}{5} \\ x = \frac{k - 14}{5} [/tex]
A chemical company makes two brands of antifreeze. The first brand is 20% pure antifreeze, and the second brand is 70% pure antifreeze. In order to obtain 30 gallons of a mixture that contains 35% pure antifreeze, how many gallons of each brand of antifreeze must be used?
Answer:
First brand of antifreeze: 21 gallons
Second brand of antifreeze: 9 gallons
Step-by-step explanation:
Let's call A the amount of first brand of antifreeze. 20% pure antifreeze
Let's call B the amount of second brand of antifreeze. 70% pure antifreeze
The resulting mixture should have 35% pure antifreeze, and 30 gallons.
Then we know that the total amount of mixture will be:
[tex]A + B = 30[/tex]
Then the total amount of pure antifreeze in the mixture will be:
[tex]0.2A + 0.7B = 0.35 * 30[/tex]
[tex]0.2A + 0.7B = 10.5[/tex]
Then we have two equations and two unknowns so we solve the system of equations. Multiply the first equation by -0.7 and add it to the second equation:
[tex]-0.7A -0.7B = -0.7*30[/tex]
[tex]-0.7A -0.7B = -21[/tex]
[tex]-0.7A -0.7B = -21[/tex]
+
[tex]0.2A + 0.7B = 10.5[/tex]
--------------------------------------
[tex]-0.5A = -10.5[/tex]
[tex]A = \frac{-10.5}{-0.5}[/tex]
[tex]A = 21\ gallons[/tex]
We substitute the value of A into one of the two equations and solve for B.
[tex]21 + B = 30[/tex]
[tex]B = 9\ gallons[/tex]
Use Stokes' Theorem to evaluate C F · dr where C is oriented counterclockwise as viewed from above. F(x, y, z) = xyi + 5zj + 7yk, C is the curve of intersection of the plane x + z = 8 and the cylinder x2 + y2 = 81.
By Stokes' theorem,
[tex]\displaystyle\int_C\vec F\cdot\mathrm d\vec r=\iint_S(\nabla\times\vec F)\cdot\mathrm d\vec S[/tex]
where [tex]S[/tex] is the surface with [tex]C[/tex] as its boundary. The curl is
[tex]\nabla\times\vec F(x,y,z)=2\,\vec\imath-x\,\vec k[/tex]
Parameterize [tex]S[/tex] by
[tex]\vec\sigma(u,v)=u\cos v\,\vec\imath+u\sin v\,\vec\jmath+(8-u\cos v)\,\vec k[/tex]
with [tex]0\le u\le9[/tex] and [tex]0\le v\le2\pi[/tex]. Then take the normal vector to [tex]S[/tex] to be
[tex]\vec\sigma_u\times\vec\sigma_v=u\,\vec\imath+u\,\vec k[/tex]
Then the line integral is equal to the surface integral,
[tex]\displaystyle\iint_S(\nabla\times\vec F)\cdot\mathrm d\vec S=\int_0^{2\pi}\int_0^9(2\,\vec\imath-u\cos v\,\vec k)\cdot(u\,\vec\imath+u\,\vec k)\,\mathrm du\,\mathrm dv[/tex]
[tex]\displaystyle=\int_0^{2\pi}\int_0^9(2u-u^2\cos v)\,\mathrm du\,\mathrm dv=\boxed{162\pi}[/tex]
Please need help in these 3 algebra questions !!!!
7. Add: (3s2 + 7s + 2) + (5s2 + 9s – 1)
A. 8s2 + 16s + 3
B. 8s4 + 16s + 1
C. 8s2 + 16s + 1
D. –2s2 – 2s + 1
8. (–3t2u3)(5t7u8) = _______.
A. –15t14u24
B. 2t9u11
C. –15t–5u–5
D. –15t9u11
11. The square of a number is equal to 6 more than the number. Find all such numbers.
A. –4; –3
B. –2
C. 3; –2
D. –3
Answer:
[tex]\large\boxed{7.\ B.\ 8s^2+16s+1}\\\\\boxed{8.\ D.\ -15t^9u^{11}}\\\\\boxed{11.\ C.\ 3,\ -2}[/tex]
Step-by-step explanation:
[tex]7.\\(3s^2+7s+2)+(5s^2+9s-1)=3s^2+7s+2+5s^2+9s-1\\\\\text{combine like terms}\\\\=(3s^2+5s^2)+(7s+9s)+(2-1)\\\\=8s^2+16s+1[/tex]
[tex]8.\\(-3t^2u^3)(5t^7u^8)=(-3\cdot5)(t^2t^7)(u^3u^8)\qquad\text{use}\ a^na^m=a^{n+m}\\\\=-15t^{2+7}u^{3+8}=-15t^9u^{11}[/tex]
[tex]11.\\n-the\ number\\\\n^2=n+6\qquad\text{subtract}\ n\ \text{and}\ 6\ \text{from both sides}\\\\n^2-n-6=0\\\\n^2+2n-3n-6=0\\\\n(n+2)-3(n+2)=0\\\\(n+2)(n-3)=0\iff n+2=0\ \vee\ n-3=0\\\\n+2=0\qquad\text{subtract 2 from both sides}\\n=-2\\\\n-3=0\qquad\text{add 3 to both sides}\\n=3[/tex]
Write the standard equation of a circle that passes through (-5 5) with center (-10 -5) brainly
Answer:
The equation of the circle is (x + 10)² + (y + 5)² = 125 in standard form
Step-by-step explanation:
* lets study the standard form of the equation of a circle
- If the coordinates of the center of the circle are(h , k) and its radius
is r, then the standard equation of the circle is:
(x - h)² + (y - k)² = r²
* Now lets solve the problem
∵ The coordinates of the center of the circle are (-10 , -5)
∵ The standard form of the equation is (x - h)² + (y - k)² = r²
∵ h , k are the coordinates of the center
∴ h = -10 , k = -5
∴ The equation of the circle = (x - -10)² + (y - -5)² = r²
∴ The equation of the circle = (x + 10)² + (y + 5)² = r²
- To find the value of the radius lets use the point (-5 , 5) to
substitute their coordinate instead of x and y in the equation
∵ The circle passes through point (-5 , 5)
∵ (x + 10)² + (y + 5)² = r²
- Use x = -5 and y = 5
∴ (-5 + 10)² + (5 + 5)² = r² ⇒ simplify
∴ (5)² + (10)² = r²
∴ 25 + 100 = r²
∴ r² = 125
* Now lets write the equation in standard form
∴ (x + 10)² + (y + 5)² = 125
* The equation of the circle is (x + 10)² + (y + 5)² = 125 in standard form
In a college parking lot, the number of ordinary cars is larger than the number of sport utility vehicles by 59.3%. The difference between the number of cars and the number of SUVs is 16. Find the number of SUVs in the lot.
Answer:
27 SUVs
Step-by-step explanation:
Let number of ordinary cars be x and SUVs be y
We can write 2 equations and use substitution to solve for the number of SUVs.
"The number of ordinary cars is larger than the number of sport utility vehicles by 59.3%"-
This means that 1.593 times more is ordinary cars (x) than SUVs (y), so we can write:
x = 1.593y
"The difference between the number of cars and the number of SUVs is 16" -
Since we know ordinary cars are "more", we can say x - y = 16
We can now plug in 1.593 y into x of the 2nd equation and solve for y:
x - y = 16
1.593y - y = 16
0.593y = 16
y = 27 (rounded)
Hence, there are 27 SUVs
15 pts. Prove that the function f from R to (0, oo) is bijective if - f(x)=x2 if r- Hint: each piece of the function helps to "cover" information to break your proof(s) into cases. part of (0, oo).. you may want to use this
Answer with explanation:
Given the function f from R to [tex](0,\infty)[/tex]
f: [tex]R\rightarrow(0,\infty)[/tex]
[tex]-f(x)=x^2[/tex]
To prove that the function is objective from R to [tex](0,\infty)[/tex]
Proof:
[tex]f:(0,\infty )\rightarrow(0,\infty)[/tex]
When we prove the function is bijective then we proves that function is one-one and onto.
First we prove that function is one-one
Let [tex]f(x_1)=f(x_2)[/tex]
[tex](x_1)^2=(x_2)^2[/tex]
Cancel power on both side then we get
[tex]x_1=x_2[/tex]
Hence, the function is one-one on domain [tex[(0,\infty)[/tex].
Now , we prove that function is onto function.
Let - f(x)=y
Then we get [tex]y=x^2[/tex]
[tex]x=\sqrt y[/tex]
The value of y is taken from [tex](0,\infty)[/tex]
Therefore, we can find pre image for every value of y.
Hence, the function is onto function on domain [tex](0,\infty)[/tex]
Therefore, the given [tex]f:R\rightarrow(0.\infty)[/tex] is bijective function on [tex](0,\infty)[/tex] not on whole domain R .
Hence, proved.
Imagine a prison consisting of 64 cells arranged like the squares of an 8-by-8 chessboard. There are doors between all adjoining cells. A prisoner in one of the corner cells is told that he will be released, provided he can get into the diagonally opposite corner cell after passing through every other cell exactly once. Can the prisoner obtain his freedom?
Answer:yes
Step-by-step explanation:
Theoretically he could just open the doors leading to the seven cells to the opposing side of the bored, then move forward the seven more, ending in the corner cell diagonal from his orgiu position.
No, I don’t think so. If he passes through each column of cells, he gets to the end cell to the left or right from him instead of the one exactly diagonal from him.
Hope this helps!
In a batch of 8,000 clock radios 7% are defective. A sample of 1313 clock radios is randomly selected without replacement from the 8,000 and tested. The entire batch will be rejected if at least one of those tested is defective. What is the probability that the entire batch will be rejected
Answer: Probability that the entire batch will be rejected is 0.611.
Step-by-step explanation:
Since we have given that
Number of clock radios in a batch = 8000
Probability of defective clock radio = 7%
According to question, we have mentioned that A sample of 13 clock radios is randomly selected without replacement from the 8,000 and tested.
We will use "Binomial distribution":
here, n = 13 and
p (probability of success) = 7% = 0.07
so, we need to find that
P(the entire batch will be rejected) = P(at least one of those test is defected)
So, it becomes,
P(at least one of those tested is defective) = 1 - P(none are defective)
So, P(none are defective ) is given by
[tex](1-0.07)^{13}\\\\=0.93^{13}\\\\=0.389[/tex]
So, P(at least one of those tested is defective) = 1 - P(none are defective)
= 1 - 0.389
= 0.611
Hence, Probability that the entire batch will be rejected is 0.611.
ne angle in a triangle has a measure that is three times as large as the smallest angle. The measure of the third angle is 50 degrees more than that of the smallest angle. Find the measure of the LARGEST angle
Answer:
the smallest angle is x, and then the second angle is 3x and the third angle is 3x+50. I don't know what the smallest angle is though.
Step-by-step explanation:
first I don't see the smallest angle that's why I called it x. Next you said 3 times the smallest angle so 3 times x. Finally you said the third angle is 50 more than the second one so you add 50 to 3x. I can't tell you what the answer is though because you didn't tell me the smallest angles measure.