Dana’s waking rate in miles per hour is 3 mph.
I did 3/4 x 4 = 3 because she walked 1/4 a mile and I needed to figure out the miles per one whole hour.
I hope this made sense and helped you.
!!!!!!!! 50 POINTS !!!!!!!!What are the explicit equation and domain for a geometric sequence with a first term of 2 and a second term of −8?
an = 2(−8)^(n − 1); all integers where n ≥ 1
an = 2(−8)^(n − 1); all integers where n ≥ 0
an = 2(−4)^(n − 1); all integers where n ≥ 0
an = 2(−4)^(n − 1); all integers where n ≥ 1
Answer:
an = 2(−4)^(n − 1); all integers where n ≥ 1
Step-by-step explanation:
The equation has the form ...
an = a1(r)^(n-1) . . . . . where a1 is the first term and r is the common ratio.
The first term is given as 2, and the ratio will be the ratio of the first two terms:
r = (-8)/(2) = -4
Terms are numbered starting with n=1, so the formula is ...
an = 2(-4)^(n-1) for n≥1
6x^2-66x+144=0 solving quadratic equation
Factor out the common term 6
6(x^2 - 11x + 24) = 0
Factor x^2 - 11x + 24
6(x - 8)(x - 3) = 0
Solve for x;
x = 8,3
Using he equation 12(x-3.20)=54 , carolyn determines that she will make a profit of $7.70 on each necklace. Which error did carolyn make?
Answer:
This is correct, Carolyn will make $7.70 for each necklace but she loses $3.20 most likely for the suplices for the making of the necklace so she really earns $4.50.
Step-by-step explanation:
Find the lateral area for the prism.
Will mark Brainiest!!!
The lateral area is the area not including the base.
You have 2 sides that are 6 x 8 = 48 x 2 = 96 square feet.
One side of 4 x 8 = 32 square feet
And the top triangle = 1/2 x 4 x 6 = 12 square feet.
For lateral area you would not include the triangle at the bottom ( base).
Total Lateral area = 96 + 32 + 12 = 140 square feet.
simplify the trigonometric expression. show your work
See the attached picture for the solution:
Answer:
Step-by-step explanation:
1/(1+sinθ) + 1/(1-sinθ)
= (1-sinθ)/[(1+sinθ)(1-sinθ)] + (1+sinθ)/[(1-sinθ)(1+sinθ)]
= [(1-sinθ) + (1+sinθ)] / [(1-sinθ)(1+sinθ)]
= [1 - sinθ + 1 + sinθ] / [1 - sin^2sθ]
= 2 / cos^2θ
What is measure of angle R?
Enter your answer as a decimal in the box. Round only your final answer to the nearest hundredth.
°
P Q R is a right triangle. Q is a right angle. P Q is equal to five centimeters, Q R is equal to twelve centimeters and P R is equal to thirteen centimeters.
Answer:
The measure of angle R is [tex]22.62\°[/tex]
Step-by-step explanation:
we know that
In the right triangle PQR
The cosine of angle R is equal to divide the adjacent side angle R by the hypotenuse
so
[tex]cos(R)=\frac{QR}{PR}[/tex]
substitute the values
[tex]cos(R)=\frac{12}{13}[/tex]
[tex]<R=arccos(\frac{12}{13})=22.62\°[/tex]
see the attached figure to better understand the problem
Mark has a six-sided number cube. Each side is numbered from 1 to 6. What is the probability, expressed as a ratio, that Mark will roll a 3? A) 1 1/2 B) 1/6 C) 2/6 D) 1/2
➷ There is only one '3' on the cube
Therefore, it would be B. 1/6
✽➶ Hope This Helps You!
➶ Good Luck (:
➶ Have A Great Day ^-^
↬ ʜᴀɴɴᴀʜ ♡
Solving Rational Equations. LCD Method. Show work.
[tex]\frac{3}{5x} + \frac{7}{2x} =1[/tex]
Answer: [tex]x=\frac{41}{10}[/tex]
Step-by-step explanation:
Descompose the denominators into their prime factors to calculate the Least Common Denominator (LCD):
[tex]5x=5*x[/tex]
[tex]2=2*x[/tex]
Choose the common and non-common numbers and varibles with the largest exponents and multiply them:
[tex]LCD=5*2*x=10x[/tex]
Divide eac originl denominator by the LCD and multiply the resul by each numerator. Then, make the addition and solve for x:
[tex]\frac{3(2)+7(5)}{10x}=1\\\\\frac{6+35}{10x}=1\\\\\frac{41}{10x}=1\\\\41=10x\\x=\frac{41}{10}[/tex]
Answer:
[tex]x=4.1[/tex]
Step-by-step explanation:
The given equation is;
[tex]\frac{3}{5x}+\frac{7}{2x}=1[/tex]
Multiply through by the Least Common Denominator which is [tex]-10x[/tex]
[tex]10x(\frac{3}{5x})+10x(\frac{7}{2x})=10x[/tex]
Cancel the common factors to obtain;
[tex]2(3)+5(7)=10x[/tex]
[tex]6+35=10x[/tex]
[tex]41=10x[/tex]
Divide by 10
[tex]x=\frac{41}{10}[/tex]
[tex]x=4.1[/tex]
The circumference of a circle is 65?. In terms of pi, what is the area of the circle?
Answer:
1056.25π square units
Step-by-step explanation:
A few formulas an definitions which will help us:
(1) [tex]\pi=\frac{c}{d}[/tex], where c is the circumference of a circle and d is its diameter
(2) [tex]A=\pi r^2[/tex], where A is the area of a circle with radius r. To put it in terms of d, remember that a circle's diameter is simply twice its radius, or mathematically, (3) [tex]d=2r \rightarrow r=\frac{d}{2}[/tex].
We can rearrange equation (1) to put d in terms of π and c, giving us (4) [tex]d = \frac{c}{\pi}[/tex], and we can make a few substitutions in (2) using (3) and (4) to get use the area in terms of the circumference and π:
[tex]A=\pi r^2\\=\pi\left(\frac{d}{2}\right)^2\\=\pi\left(\frac{d^2}{4}\right)\\=\pi\left(\frac{(c/\pi)^2}{4}\right)\\=\pi\left(\frac{c^2/\pi^2}{4}\right)\\=\pi\left(\frac{c^2}{4\pi^2}\right)\\\\=\frac{\pi c^2}{4\pi^2}\\ =\frac{c^2}{4\pi}[/tex]
We can now substitute c for our circumference, 65, to get our answer in terms of π:
[tex]A=\dfrac{65^2}{4\pi}=\dfrac{4225}{4\pi}=1056.25\pi[/tex]
Answer:
Area = 2112.5 / pi
Step-by-step explanation:
They are asking you not to use 3.14 for pi. Just leave it as a symbol.
C = 2*pi*r
C = 65
65 = 2*pi*r
65/(2*pi) = r
The area of a circle is 2*pi * r^2
Area = 2 * pi * (65/2pi)^2
Area = 2 * pi * 65^2/(4*pi^2) Cancel out one of the pi-s in the denominator
Area = 2 * 65^2 / (4 * Pi) Expand the numerator
Area = 8450/(4*pi) Divide by 4
Area = 2112.5 / pi
Consider the infinite geometric series ∑∞ n=1 -4(1/3)^n-1
a. Write the first four terms of the series.
b. Does the series diverge or converge.
c. If the series has a sum, find the sum.
a. The series is
[tex]\displaystyle\sum_{n=1}^\infty-4\left(\frac13\right)^{n-1}=-4-\frac43-\frac4{3^2}-\frac4{3^3}-\cdots[/tex]
(first four terms are listed)
b. The series converges because this is a geometric series with [tex]r=\dfrac13<1[/tex].
c. Let [tex]S_N[/tex] be the [tex]N[/tex]-th partial sum of the series:
[tex]S_N=\displaystyle\sum_{n=1}^N-4\left(\frac13\right)^{n-1}[/tex]
[tex]S_N=-4-\dfrac43-\dfrac4{3^2}-\cdots-\dfrac4{3^{N-1}}[/tex]
Multiplying both sides by [tex]\dfrac13[/tex] gives
[tex]\dfrac13S_N=-\dfrac43-\dfrac4{3^2}-\dfrac4{3^3}-\cdots-\dfrac4{3^N}[/tex]
Subtracting this from [tex]S_N[/tex] gives
[tex]S_N-\dfrac13S_N=\dfrac23S_N=-4+\dfrac4{3^N}[/tex]
[tex]\implies S_N=-6+\dfrac6{3^N}[/tex]
As [tex]N[/tex] gets larger and larger [tex](N\to\infty)[/tex] the rational term converges to 0 and we're left with
[tex]\displaystyle\lim_{N\to\infty}S_N=\sum_{n=1}^\infty-4\left(\frac13\right)^{n-1}=-6[/tex]
Please answer this question only if you know it!
You're answer to the following question: "Which Point Must be the Centre of the Circle?" Is C
What is the solution to this system of linear equations?
7x - 2y = -6
8x + y = 3
A.(-6,3)
B.(0,3)
C.(1,-5)
D.(15,-1)
Answer:
B. (0, 3)
Step-by-step explanation:
Trying the offered solutions in the given equations gets you there pretty quickly.
7·(-6) -2(3) ≠ -6 . . . eliminates choice A
__
7·0 -2·3 = -6
8·0 +3 = 3 . . . . . . . choice B is the solution
Answer:
(0,3)
B is correct
Step-by-step explanation:
Given: The system of equation.
[tex]7x-2y=-6[/tex]
[tex]8x+y=3[/tex]
Now, we solve for x and y using elimination method.
Elimination method: In this method to make the coefficient of one variable same and then cancel out by addition of both equation.
Multiply 2nd equation by 2 and we get
[tex]16x+2y=6[/tex]
[tex]7x-2y=-6[/tex]
Add both equation and eliminate y
[tex]23x=0[/tex]
[tex]x=0[/tex]
Put x=0 into 1st equation, 7x-2y=-6
7(0) - 2y = -6
y = 3
Solution: x = 0 and y = 3
Hence, The solution of the equation would be (0,3)
There are 888 employees on The Game Shop's sales team. Last month, they sold a total of ggg games. One of the sales team members, Chris, sold 171717 fewer games than what the team averaged per employee. How many games did Chris sell?
Answer:
[tex]\frac{g}{8}-17[/tex] games
Step-by-step explanation:
Employees: 8
Sales last month: g
Average: [tex]\frac{totalGames}{Employees}[/tex]
To find out how many games Chris sold, we have to take the average and subtract 17.
[tex]\frac{g}{8}-17[/tex]
Since there are no more values we can use, this is as simple as we can get it. If we knew how many games they sold last month, we could get an exact answer for Chris using this expression.
Answer:
g over 8 -17
Step-by-step explanation:
When you are looking at used cars, you should only look at local lots and newspapers
True
False
Please don't ask me what a true/false question has to do with math, but...
Answer:
The statement is false.
Step-by-step explanation:
When you are looking at used cars, you should only look at local lots and newspapers - This statement is false.
When you are buying a used car, you should look not only in the local lots and newspapers but also online ans various used cars websites.
You can also personally visit the used car market to get wide range of cars and various comparative prices.
A figure is translated using the rule (x, y) → (x – 3, y + 6). Which describes how the figure is moved?
A. left 3 units and down 6 units right
B. 3 units and down 6 units left
C. 6 units and down 3 units left
D. 3 units and up 6 units
Answer:
3 units to the right and 6 units upStep-by-step explanation:
f(x) + n - shift the graph of f(x) n units up → (x, y + n)
f(x) - n - shift the graph of f(x) n units down → (x, y - n)
f(x - n) - shift the graph of f(x) n units to the right → (x - n, y)
f(x + n) - shift the graph of f(x) n units to the left → (x + n, y)
===================================
(x, y) → (x - 3, y + 6)
x - 3 → shift the graph 3 units to the right
y + 6 → shift the graph 6 units up
Answer:
the right answer is left 3 units and up 6 units
Step-by-step explanation:
A rectangle prism has the dimensions 8 feet by 3 feet by 5 feet. What is the surface area of the prism
Answer:
158 square feet
Step-by-step explanation:
The surface area of a prism is found using the following formula: SA = 2(lh+lb+bh). This formula takes the area of each face (6 in total( and adds them together to find a total sum. Substitute l = 8, b = 3 and h = 5 to solve for the surface area.
SA = 2(lh+lb+bh)
SA = 2(8*5+8*3+3*5)
SA = 2(40 + 24 + 15)
SA = 2(79)
SA = 158
Linda saves $1.55 every day. Tyron saves $3.90 every day. Linda started saving 3 days earlier than Tyron. On which day will Tyron's savings be more than Linda's savings?
To determine when Tyron's savings will surpass Linda's, we can set up an inequality using their daily saving rates and the fact that Linda started 3 days earlier. Solving for the number of days Tyron needs to save, we find that Tyron will surpass Linda's savings after saving for 2 full days.
Linda and Tyron are saving money on a daily basis, but Linda started saving 3 days earlier than Tyron. To find out on which day Tyron's savings will be more than Linda's, we need to set up an equation.
Let's define x as the number of days after which Tyron starts saving. Since Linda starts 3 days earlier, she has been saving for x + 3 days by the time Tyron starts saving.
Linda's daily savings: $1.55
Tyron's daily savings: $3.90
Linda's total savings after x + 3 days: $1.55(x + 3)
Tyron's total savings after x days: $3.90x
We want to find the value of x at which Tyron's total savings exceed Linda's, so we set up the inequality:
$3.90x > $1.55(x + 3)
Now, solve for x:
$3.90x > $1.55x + $4.65
$3.90x - $1.55x > $4.65
$2.35x > $4.65
x > $4.65 / $2.35
x > 1.978
Since x represents the number of days and we cannot have a fraction of a day in this context, x must be at least 2 days. Therefore, Tyron's savings will be more than Linda's after he has saved for 2 days.
!!!!!!!!!!!!!!!! HELP 25 points
What is the completely factored form of
x3 + 4x2 – 9x – 36?
(x + 3)(x – 3)
(x2 – 9)(x + 4)
(x + 3)(x – 3)(x + 4)
(x – 3)(x – 3)(x + 4)
x3 + 4x2 - 9x - 36
x2 (x + 4) - 9(x + 4)
(x2 - 9) (x + 4)
(x - 3) (x + 3) (x + 4)
Please help me out with this!
i think its 5x-15 OR -1x-15
Problem:
A non-linear system consists of two functions: f(x)=x²+2x+1 and g(x)=3-x-x². Solve this system in two different ways. Your choices are: Table, Graph, or Algebraically.
A. Make a table of values for the functions. The table may be horizontal or vertical but it must have a minimum of five x-values and the corresponding function values showing each solution, one value lower, one value higher, and one between the two solutions. Indicate the solutions by marking the x-values and the corresponding function values that are equal.
B. Solve the system algebraically. (Hint: set the two functions equal to each other and solve the resulting function.) You should obtain a quadratic equation. Solve it either by factoring or using the quadratic formula. Give the x-values of the solution set, then evaluate the original function to find the corresponding y-values. Give the results as ordered pairs of exact values.
C. Plot a graph of the functions over an interval sufficient to show the solutions. You may carefully sketch or plot your graph manually or use Desmos or other technology. Clearly indicate and label on the graph the x and y values of the solution(s).
Answer:
Part B. see the procedure
Part C. see the procedure
Step-by-step explanation:
we have
[tex]f(x)=x^{2}+2x+1[/tex] -----> equation A
[tex]g(x)=3-x-x^{2}[/tex] -----> equation B
Part B. Solve the system algebraically
equate the equation A and the equation B
[tex]x^{2}+2x+1=3-x-x^{2}[/tex]
[tex]2x^{2}+3x-2=0[/tex]
The formula to solve a quadratic equation of the form [tex]ax^{2} +bx+c=0[/tex] is equal to
[tex]x=\frac{-b(+/-)\sqrt{b^{2}-4ac}} {2a}[/tex]
in this problem we have
[tex]2x^{2}+3x-2=0[/tex]
so
[tex]a=2\\b=3\\c=-2[/tex]
substitute in the formula
[tex]x=\frac{-3(+/-)\sqrt{3^{2}-4a(2)(-2)}} {2(2)}[/tex]
[tex]x=\frac{-3(+/-)\sqrt{25}} {4}[/tex]
[tex]x=\frac{-3(+/-)5} {4}[/tex]
[tex]x1=\frac{-3(+)5} {4}=0.5[/tex]
[tex]x2=\frac{-3(-)5} {4}=-2[/tex]
Find the values of y
For x=0.5
[tex]f(0.5)=0.5^{2}+2(0.5)+1=2.25[/tex]
For x=-2
[tex]f(-2)=(-2)^{2}+2(-2)+1=1[/tex]
the solutions are the points
(0.5,2.25) and (-2,1)
Part C. Solve the system by graph
using a graphing tool
we know that
The solution of the non linear system is the intersection point both graphs
The intersection points are (0.5,2.25) and (-2,1)
therefore
The solutions are the points (0.5,2.25) and (-2,1)
see the attached figure
The calibration of a scale is to be checked by weighing a 10-kg test specimen 25 times. Suppose that the results of different weighings are independent of one another and that the weight on each trial is normally distributed with σ = .200 kg. Let µ denote the true average weight reading on the scale. (a) What hypotheses should be tested? (b) With the sample mean itself as the test statistic, what is the P-value when x = 9.85, and what would you conclude at significance level .01? (c) For a test with α = .01, what is the probability that recalibration is judged unnecessary when in fact µ = 10.2?
Answer:
a: That the mean weight of the trials is 10 kg
b: See attached photo for work
Step-by-step explanation:
We want to see if the scale is weighing properly and are using a 10 kg weight to calibrate it. That means our hypothesis test is that the mean weight of the trails (in this case 25) is 10 kg.
The hypothesis we will use are
H0: µ = 10
Ha: µ ≠ 10
The alternate hypothesis has a not equals to sign because if the scale weighs too much or to little, then it needs to be better calibrated, so it's a two tailed test.
The calibration of the scale follows a normal distribution.
The null and the alternate hypotheses are: [tex]\mathbf{H_o: \mu = 10}[/tex] and [tex]\mathbf{H_a: \mu \ne 10}[/tex]The p-value when [tex]\mathbf{\bar x = 9.85}[/tex] is [tex]\mathbf{p=0.000494}[/tex]The scale needs to be calibratedThe probability that recalibration is judged unnecessary is less than 0.00001The given parameters are:
[tex]\mathbf{\sigma = 0.200}[/tex]
[tex]\mathbf{\mu = 10}[/tex]
[tex]\mathbf{n = 25}[/tex]
[tex]\mathbf{\bar x = 9.85}[/tex]
(a) The null and the alternate hypotheses
The true average weight is to be tested.
So, the null and the alternate hypotheses are:
[tex]\mathbf{H_o: \mu = 10}[/tex]
[tex]\mathbf{H_a: \mu \ne 10}[/tex]
(b) The p-value when [tex]\mathbf{\bar x = 9.85}[/tex]
First, we calculate the test statistic
[tex]\mathbf{t = \frac{\bar x - \mu}{\sigma/\sqrt n}}[/tex]
So, we have:
[tex]\mathbf{t = \frac{9.85 - 10}{0.2/\sqrt{25}}}[/tex]
[tex]\mathbf{t = \frac{9.85 - 10}{0.2/5}}[/tex]
[tex]\mathbf{t = \frac{-0.15}{0.04}}[/tex]
[tex]\mathbf{t = -3.75}[/tex]
Using p-value calculator, we have:
[tex]\mathbf{p=0.000494}[/tex]
The critical regions of [tex]\mathbf{t = -3.75}[/tex] are t >2.797 and t < -2.797
Because -3.75 < -2.797, we reject the null hypothesis.
This means that, the scale needs to be calibrated
(c) Probability that recalibration is judged when [tex]\mathbf{\mu = 10.2}[/tex]
First, we calculate the test statistic
[tex]\mathbf{t = \frac{\bar x - \mu}{\sigma/\sqrt n}}[/tex]
So, we have:
[tex]\mathbf{t = \frac{9.85 - 10.2}{0.2/\sqrt{25}}}[/tex]
[tex]\mathbf{t = \frac{-0.35}{0.04}}[/tex]
[tex]\mathbf{t = -8.75}[/tex]
Using p-value calculator, we have:
[tex]\mathbf{p<0.00001}[/tex]
The probability that recalibration is judged unnecessary is less than 0.00001
Read more about probabilities using test statistics at:
https://brainly.com/question/22783864
2x-3y=-14 3x-2y=-6 if (x,y) is a solution to the system of equations above, what is the value of x-y?
In a system of equations, if the variable x is found to be equal to 2 and y is found to be equal to 4, the value of x - y is -2.
Explanation:To solve for the values of x and y in this pair of linear equations, we can use a method known as substitution or elimination. However, the question asks for the value of x-y, not for the individual values of x and y.
First, let's multiply the first equation by 2 and the second equation by 3:
4x - 6y = -28 (equation 1)
9x - 6y = -18 (equation 2)
If we subtract Equation 2 from Equation 1, we get -5x = -10. Solving for x, we find that x = 2.
Substituting the value of x into the first equation, we get:
2(2) - 3y = -14
Solving for y, we find that y = 4.
Therefore, x - y = 2 - 4 = -2.
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What is the exact value? (Picture provided)
Answer:
b. (√15)/4
Step-by-step explanation:
Since Sin Ф = (opposite side)/Hypotenuse, we have 2 sides of a right triangle.
Use Pythagorean theorem to solve for the missing leg (the adjacent side)
1² + b² = 4²
1 + b² = 16
b² = 15
b = √15
So the adjacent side is √15, so Cos Ф = (√15)/4
Answer:
b. [tex]\frac{\sqrt{15}}{4}[/tex]
Step-by-step explanation:
Given that [tex]\sin(\theta)=\frac{1}{4}[/tex] where [tex]0\:<\: \theta \:<\:\frac{\pi}{2}[/tex].
Recall and use the Pythagorean Identity;
[tex]\sin^2(\theta)+\cos^2(\theta)=1[/tex]
This implies that;
[tex](\frac{1}{4})^2+\cos^2(\theta)=1[/tex]
[tex]\frac{1}{16}+\cos^2(\theta)=1[/tex]
[tex]\cos^2(\theta)=1-\frac{1}{16}[/tex]
[tex]\cos^2(\theta)=\frac{15}{16}[/tex]
Take the square root of both sides;
[tex]\cos(\theta)=\pm \sqrt{\frac{15}{16}}[/tex]
[tex]\cos(\theta)=\pm \frac{\sqrt{15}}{4}[/tex]
Since we are in the first quadrant;
[tex]\cos(\theta)=\frac{\sqrt{15}}{4}[/tex]
Find the length of segment BA.
A) 163.3
B) 128.6
C) 84.7
D) 59.8
Answer:
D) 59.8
Step-by-step explanation:
m<B = 120 deg
Since we know the measure of an angle and the length of the opposite side, we can establish the ratio of the law of sines, so we use the law of sines to find the length of side BA.
[tex] \dfrac{\sin A}{a} = \dfrac{\sin B}{b} = \dfrac{\sin C}{c} [/tex]
[tex] \dfrac{\sin A}{BC} = \dfrac{\sin B}{AC} = \dfrac{\sin C}{AB} [/tex]
[tex] \dfrac{\sin B}{AC} = \dfrac{\sin C}{AB} [/tex]
[tex] \dfrac{\sin 120^\circ}{200} = \dfrac{\sin 15^\circ}{AB} [/tex]
[tex] \dfrac{\sin 120^\circ}{200} = \dfrac{\sin 15^\circ}{AB} [/tex]
[tex] (AB)\sin 120^\circ = 200 \sin 15^\circ [/tex]
[tex] AB = \dfrac{200 \sin 15^\circ}{\sin 120^\circ} [/tex]
[tex] AB = 59.8 [/tex]
Which statement Is a good definition? A. Parallel lines are lines that do not intersect B. Skew lines are lines that do not intersect C. A square is a rectangle with four congruent sides. D. Right angles are angles formed by two intersecting lines
Parallel lines are lines that do not intersect each other at any point.
Explanation:The correct definition of parallel lines is option A: 'Parallel lines are lines that do not intersect.' Parallel lines are two lines in a plane that do not intersect each other at any point, no matter how far they are extended.
Learn more about Definition of Parallel Lines here:https://brainly.com/question/7966281
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Expand the following log:
[tex]log_{3} (x^{4} y)[/tex]
SHOW ALL WORK.
Answer:
[tex]\log_{3}(x^4y)=4\log_{3}(x)+\log_{3}(y)[/tex]
Step-by-step explanation:
The given logarithmic expression is
[tex]log_{3}(x^4y)[/tex]
Recall and use the product property of logarithm: [tex]\log_a(MN)=\log_a(M)+\log_a(N)[/tex];
This implies that;
[tex]\log_{3}(x^4y)=\log_{3}(x^4)+\log_{3}(y)[/tex]
Recall again that; [tex]\log_a(M^n)=n\log_a(M)[/tex];
We apply this property to get;
[tex]\log_{3}(x^4y)=4\log_{3}(x)+\log_{3}(y)[/tex]
The harmonic motion of a particle is given by f(t) = 2 cos(3t) + 3 sin(2t), 0 ≤ t ≤ 8. (a) When is the position function decreasing? (Round your answers to one decimal place. Enter your answer using interval notation.) Correct: Your answer is correct. (b) During how many time intervals is the particle's acceleration positive? 4 Correct: Your answer is correct. time intervals (c) At what time is the particle at the farthest distance away from its starting position in the negative direction? (Round your answer to one decimal place.) t = 5.34 Correct: Your answer is correct. How far away is it from its original position? (Round your answer to the nearest integer.) 7 Correct: Your answer is correct. (d) At what time is the particle moving the fastest? (Round your answer to one decimal place.) t = 4.7 Correct: Your answer is correct. At what speed is the particle moving the fastest? (Round your answer to the nearest integer.) -5 Incorrect: Your answer is incorrect.
For the last part, you have to find where [tex]f'(t)[/tex] attains its maximum over [tex]0\le t\le8[/tex]. We have
[tex]f'(t)=-6\sin3t+6\cos2t[/tex]
so that
[tex]f''(t)=-18\cos3t-12\sin2t[/tex]
with critical points at [tex]t[/tex] such that
[tex]-18\cos3t-12\sin2t=0[/tex]
[tex]3\cos3t+2\sin2t=0[/tex]
[tex]3(\cos^3t-3\cos t\sin^2t)+4\sin t\cos t=0[/tex]
[tex]\cos t(3\cos^2t-9\sin^2t+4\sin t)=0[/tex]
[tex]\cos t(12\sin^2t-4\sin t-3)=0[/tex]
So either
[tex]\cos t=0\implies t=\dfrac{(2n+1)\pi}2[/tex]
or
[tex]12\sin^2t-4\sin t-3=0\implies\sin t=\dfrac{1\pm\sqrt{10}}6\implies t=\sin^{-1}\dfrac{1\pm\sqrt{10}}6+2n\pi[/tex]
where [tex]n[/tex] is any integer. We get 8 solutions over the given interval with [tex]n=0,1,2[/tex] from the first set of solutions, [tex]n=0,1[/tex] from the set of solutions where [tex]\sin t=\dfrac{1+\sqrt{10}}6[/tex], and [tex]n=1[/tex] from the set of solutions where [tex]\sin t=\dfrac{1-\sqrt{10}}6[/tex]. They are approximately
[tex]\dfrac\pi2\approx2[/tex]
[tex]\dfrac{3\pi}2\approx5[/tex]
[tex]\dfrac{5\pi}2\approx8[/tex]
[tex]\sin^{-1}\dfrac{1+\sqrt{10}}6\approx1[/tex]
[tex]2\pi+\sin^{-1}\dfrac{1+\sqrt{10}}6\approx7[/tex]
[tex]2\pi+\sin^{-1}\dfrac{1-\sqrt{10}}6\approx6[/tex]
The correct answer for part (d) is: The particle is moving the fastest at [tex]\( t = 4.7 \)[/tex] with a speed of [tex]5[/tex] units per time period.
To find when the particle is moving the fastest, we need to determine the time at which the velocity of the particle is maximized. The velocity of the particle is given by the derivative of the position function with respect to time. The position function is [tex]\( f(t) = 2 \cos(3t) + 3 \sin(2t) \)[/tex]. Differentiating this with respect to [tex]\( t \)[/tex] gives the velocity function:
[tex]\[ v(t) = \frac{d}{dt}(2 \cos(3t) + 3 \sin(2t)) = -6 \sin(3t) + 6 \cos(2t) \][/tex]
To find the maximum velocity, we need to find the critical points of the velocity function by setting its derivative equal to zero:
[tex]\[ \frac{d}{dt}(-6 \sin(3t) + 6 \cos(2t)) = -18 \cos(3t) - 12 \sin(2t) = 0 \][/tex]
Solving for [tex]\( t \)[/tex] in the interval [tex]\( 0 \leq t \leq 8 \)[/tex] will give us the times at which the velocity is maximized or minimized. Let's solve for [tex]\( t \)[/tex]:
[tex]\[ -18 \cos(3t) = 12 \sin(2t) \][/tex]
This is a transcendental equation and cannot be solved algebraically. We would typically use numerical methods or graphing to find the solutions. However, since we are given that the time when the particle is moving the fastest is [tex]\( t = 4.7 \)[/tex], we can assume that this is the time at which the velocity function reaches its maximum value.
Now, to find the speed at which the particle is moving the fastest, we evaluate the velocity function at [tex]\( t = 4.7 \)[/tex]:
[tex]\[ v(4.7) = -6 \sin(3 \cdot 4.7) + 6 \cos(2 \cdot 4.7) \][/tex]
Calculating the sine and cosine values and then substituting them into the equation will give us the maximum velocity. Since we are looking for the speed, which is the absolute value of the velocity, we take the absolute value of the result.
The speed is given by the magnitude of the velocity vector, so we have:
[tex]\[ |v(4.7)| = |-6 \sin(3 \cdot 4.7) + 6 \cos(2 \cdot 4.7)| \][/tex]
Evaluating this expression will give us the speed at which the particle is moving the fastest. The correct answer, rounded to the nearest integer, is [tex]\( 5 \)[/tex] units per time period, not [tex]\( -5 \).[/tex] The negative sign in the velocity does not affect the speed, as speed is a scalar quantity and is always positive.
Therefore, the particle is moving the fastest at [tex]\( t = 4.7 \)[/tex] with a speed of [tex]\( 5 \)[/tex] units per time period.
Please some help me fast
Answer:
A
Step-by-step explanation:
To find the best equation, we simply substitute the values of a, b, and c into the given equations.
a = 21
b = 5
c = 36
[tex]a=\dfrac{7}{10}b\sqrt{c}[/tex]
[tex]21=\dfrac{7}{10}5\sqrt{36}[/tex]
[tex]21=\dfrac{7}{10}5(6)[/tex]
[tex]21=\dfrac{7}{10}30[/tex]
[tex]21=(0.7)30[/tex]
[tex]21=21[/tex]
There was no more rainfall for the rest of the day. Click on the graph until the graph that best represents the given statement appears.
Answer:
the the third graph
Step-by-step explanation:
this is because the third graph shows a correlation of the time and when the rainfall in a proporational relesho=inshop
Answer:
the third graph :)
Step-by-step explanation:
each hour the rain is increasing by 2 drops.
A large rectangular parking lot is 2/3 km long and 1/2 km wide. What's the area of the raking lot?
Answer:1/3
Step-by-step explanation:
2/3*1/2=1/3
The area of a parking lot measuring 2/3 km long and 1/2 km wide is 1/3 km². This is found by multiplying the length by the width.
Explanation:The question is asking for the area of a rectangular parking lot that measures 2/3 km long and 1/2 km wide. The formula for the area of a rectangle is length times width. Applying this formula, we multiply 2/3 km by 1/2 km.
It's important to remember that when multiplying fractions, you just multiply the numerators (top numbers) and the denominators (bottom numbers) separately. So, (2/3) x (1/2) = 2/6 km², which simplifies to 1/3 km².
So, the area of the parking lot is 1/3 km².
Learn more about Area of Rectangle here:https://brainly.com/question/15218510
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