Which pair of terms can also be used to describe the shape of the sanbox, a square

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]

Answer:

Step-by-step explanation:

r = [tex]\frac{a_n}{a_{n-1}} = \frac{896}{3584} =  \frac{1}{4}[/tex]

Using the geometric series formula for the nth term:

[tex]S_n = a \cdot \frac{1-r^n}{1-r} => S_{6} = 3584 \cdot \frac{1 - (\frac{1}{4})^{6} }{1 - \frac{1}{4} }  = 4777\frac{1}{2}[/tex]

Final answer:

The equation for the nth term of the geometric sequence is a_n = a_1 * r^(n-1), where a_1 is the first term and r is the common ratio. The sixth term of the sequence is 14.

Explanation:

The given sequence is a geometric sequence. In a geometric sequence, each term is found by multiplying the previous term by a common ratio. To find the equation for the nth term of the geometric sequence, we need to find the common ratio. We can do this by dividing any term in the sequence by the previous term.

For example, dividing the second term 896 by the first term 3584 gives us a common ratio of 1/4.

So, the equation for the nth term of the sequence is:

an = a1 × r(n-1)

where a1 is the first term and r is the common ratio.

To find the sixth term, we can substitute n = 6 into the equation:

a6 = 3584 × (1/4)(6-1)

a6 = 3584 × (1/4)5

a6 = 14

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.

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 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:

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Answer:

 √48 could not be M.

Step-by-step explanation:

M must be  between - 2 and 6, so:

a. √48 = 6.93 so this can't be M.

b.   √18 = 4.24 so this could be M.

c and d could also be M.

The coordinate of point M can be any of the given options.

To determine the possible coordinates of point M, we need to find the values between X (-2) and Y (6) on the number line provided. Starting from X (-2) and moving towards Y (6), the numbers that lie between them are -2, 0, 2, 4, and 6. We can see that all the given options for the coordinate of point M can be found on the number line. Therefore, none of the given numbers could not be the coordinate of point M.

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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

Answer:

The area of the triangle = 31.5 feet² ⇒ answer (c)

Step-by-step explanation:

∵ Area of any Δ = 1/2 (side 1)(side 2)(sin the angle between them)

∴ The area of the Δ = 1/2 (b)(c) sin(∠A)

∵ b = 11 feet

∵ c = 10 feet

∵ m∠A = 35°

∴ Area of the Δ = 1/2 (11)(10) (sin(35)) = 31.547 feet²

∴ The area of the triangle = 31.5 feet²

∴ The answer is (c)

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]

Answer:

Step-by-step explanation:

A rectangular prism has 12 edges. In geometry, a prism is a solid figure with parallel ends or bases that are the same size and shape, with each side representing a parallelogram. The parallelograms in a rectangular prism are all rectangles.

The rectangular prism also has six faces, or flat sides. The surface area of a rectangular prism is determined by multiplying the length by the width of each of the six rectangles and by then adding the products together.

Answer:

16 pounds

Step-by-step explanation:

Total cereal she ate = 256 ounces

How many pounds she ate = ?

Given that (16 ounces = 1 pound)

Therefore, we have to divide the total ounces by 16 to calculate the cereal she ate in pounds = 256/16 = 16 pounds

Answer:

A. 16

Step-by-step explanation:

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

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:

Answer:

Step-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:

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:

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]

h = 12.

See the attached picture:

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

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.

Answer:

3rd group of possible answers is the correct set

Step-by-step explanation:

Each of the three points in the 3rd line of possible answers is a solution of the given equation, as can be verified by substitution:

If x = -2, y = -8(-2) = +16

If x = 3, y = -8(3) =  -24

If x = -4, y = -8(-4) = +32

Answer:

c

Step-by-step explanation:

x3 + 4x2 - 9x - 36

x2 (x + 4) - 9(x + 4)

(x2 - 9) (x + 4)

(x - 3) (x + 3) (x + 4)

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]

Answer:

23 Cards were given to your brother.

Step-by-step explanation:

12 + 24 = 36

36 - 13 = 23

Answer:

you gave him 23 cards

Step-by-step explanation:

12+24=36, 36-13=23

Answer:

Step-by-step explanation:

A diameter of a rhombus is an angle bisector. Therefore we have the equation:

[tex]3x-11^o=x+23^o[/tex]              add 11° to both sides

[tex]3x=x+44^o[/tex]                subtract x from both sides

[tex]2x=44^o[/tex]             divide both sides by 2

[tex]x=22^o[/tex]

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)

i think its 5x-15 OR -1x-15

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.

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.

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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.

Answer:

You cant solve this without one more number.

Step-by-step explanation:

You need the length of the blue piece because it could be smaller or larger at this point. I may be wrong but unless I am missing something, and I totally might be if this is a kind of math I've never done, then please feel free to ignore me :) and have a lovely day!

Parallel lines are lines that do not intersect each other at any point.

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.

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