The value of a collector’s item is expected to increase exponentially each year. The item is purchased for $500. After 2 years, the item is worth $551.25. Which equation represents y, the value of the item after x years?y = 500(0.05)xy = 500(1.05)xy = 500(0.1025)xy = 500(1.1025)x

Answer:

  x = 117°, 297°

Step-by-step explanation:

Subtract 1 from both sides of the equation and you have ...

  tan(x) = -2

Then the arctangent function tells you ...

  x = arctan(-2) ≈ 116.5651°, 296.5651°

  x ≈ 117° or 297°

Answer:

[tex]AC=4.3\ in[/tex]

Step-by-step explanation:

see the attached figure to better understand the problem

we know that

The triangle AOC is an isosceles triangle

OA=OC=5/2=2.5 in -----> the radius of the circle

∠AOC=180°-60°=120°

∠CAO=∠ACO=120°/2=60°

Applying the law of cosines find the length of the chord AC

[tex]AC^{2}=OA^{2}+OC^{2}-2(OA)(OC)cos(120\°)[/tex]

substitute

[tex]AC^{2}=2.5^{2}+2.5^{2}-2(2.5)(2.5)cos(120\°)[/tex]

[tex]AC^{2}=18.75[/tex]

[tex]AC=4.3\ in[/tex]

Answer:

  C'(4, 4)

Step-by-step explanation:

We assume dilation is about the origin, so all coordinates are multiplied by the scale factor:

  C' = 2C = 2(2, 2) = (4, 4)

Answer:

Component form of u is (-18,13)

The magnitude of u is 22.2

Step-by-step explanation:

The component form of a vector is an ordered pair that describe the change is x and y values

This is mathematically expressed as (Δx,Δy) where Δx=x₂-x₁ and Δy=y₂-y₁

Given ;

Initial points of the vector as (14,-6)

Terminal point of the vector as (-4,7)

Here x₁=14,x₂=-4, y₁=-6 ,y₂=7

The component form of the vector u is (-4-14,7--6) =(-18,13)

Finding Magnitude of the vector

║u=√(x₂-x₁)²+(y₂-y₁)²

║u=√-18²+13²

║u=√324+169

║u=√493

║u=22.2

[tex]\bf (\stackrel{x_1}{2}~,~\stackrel{y_1}{-5})\qquad (\stackrel{x_2}{4}~,~\stackrel{y_2}{1}) \\\\\\ slope = m\implies \cfrac{\stackrel{rise}{ y_2- y_1}}{\stackrel{run}{ x_2- x_1}}\implies \cfrac{1-(-5)}{4-2}\implies \cfrac{1+5}{2}\implies \cfrac{6}{2}\implies 3[/tex]

Answer:

Time taken by Doreen is 6 hours and speed is 10 miles per hour.

Time taken by Sue is 4 hours and speed is 20 miles per hour.

Step-by-step explanation:

Let the speed of Doreen be x

According to the question  speed of Sue   is = x+10

time  taken By Sue to cover 80 miles = [tex]\frac{80}{x+10}[/tex]

time taken by Doreen to travel 60 miles = [tex]\frac{60}{x}[/tex]

According to question Sue take two hours less than Doreen takes

therefore

[tex]\frac{60}{x}[/tex] - [tex]\frac{80}{x+10}[/tex] =2

[tex]\frac{60(x+10)-80x}{x(x+10)}[/tex] =2

60(x+10) -80x = 2(x(x+10)

60x+600-80x = [tex]2x^2+20x\\[/tex]

simplifying it ,we get

[tex]2x^2+40x-600=0\\[/tex]

Dividing both sides by 2 ,we get

[tex]x^2+20x-300=0\\[/tex]

solving it for x ,we get

(x+30)(x-10) =0

x =-30 which is not possible

x =10 miles per hour

Speed of Doreen = 10 miles per hour

Speed of Sue = 10+10 = 20 miles per hour

Time taken by Doreen = 60 divided by 10 = 6 hours

Time taken  by Sue = 80 divided by 20 = 4 hours

In conclusion, Doreen travels at a speed of 30 mph, taking her 4 hours to travel 60 miles. Sue, on the other hand, travels at a speed of 40 mph, taking her 2 hours to travel 80 miles.

This problem is a classic example of distance, rate, and time relations in mathematics. Let's start by denoting Sue's speed as x mph, the Doreen's speed would then be x-10 mph. We know that time is equal to distance divided by speed. So, the time it takes Sue to travel 80 miles would be 80/x hours and the time it takes Doreen to travel 60 miles would be 60/(x-10) hours. The question states that Sue's travel time is 2 hours less than Doreen's. Therefore, we can form the equation: 60/(x-10) = 80/x + 2. Solving this equation, we find that x equals 40 mph, which is Sue's speed and Doreen's speed is 30 mph. Consequently, the time it takes Sue to travel 80 miles is 2 hours and for Doreen to travel 60 miles is 4 hours.

https://brainly.com/question/4480564

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Answer:

Step-by-step explanation:

As the value of a increases, the radical function sweeps out higher, increasing the range of the function.  The k value moves it up or down.  A "+k" moves up (for example, +3 moves the function up 3 from the origin).  The h value moves it side to side.  A positive h value moves to the right and a negative h value moves to the left.  For example, √x-3 moves 3 to the right and √x+3 moves 3 to the left.

In summary, a and k affect the range of the function, k being the "starting point" and a being the "ending point"; h affects the domain of the function.

Answer:

  (y² +1)/y

Step-by-step explanation:

Invert the denominator fraction and multiply. Factor the difference of squares.

[tex]\displaystyle\frac{\left(\frac{6y^2-6}{8y^2+8y}\right)}{\left(\frac{3y-3}{4y^2+4}\right)}=\frac{6(y^2-1)}{8y(y+1)}\cdot\frac{4(y^2+1)}{3(y-1)}\\\\=\frac{24(y+1)(y-1)(y^2+1)}{24y(y+1)(y-1)}=\frac{y^2+1}{y}[/tex]

Answer:

=20s³+50s²+32s+6

Step-by-step explanation:

We multiply each of the term in the initial expression by the the second expression as follows:

4s(5s²+10s+3)+2(5s²+10s+3)

=20s³+40s²+12s+10s²+20s+6

Collect like terms together.

=20s³+50s²+32s+6

Answer:

Step-by-step explanation:

(1) +(2) ⇒ (x -y -2z) +(-x +3y -z) = (4) +(8)

  2y -3z = 12

__

2(1) +(3) ⇒ 2(x -y -2z) +(-2x -y -4z) = 2(4) +(-1)

  -3y -8z = 7

___

The reduced system of equations is ...

Answer:

2y - 3z = 12.

-3y - 8z = 7.

Step-by-step explanation:

x - y - 2z = 4     (1)

-x + 3y - z = 8    (2)

-2x - y - 4z = -1   (3)

Adding (1) + (2):

2y - 3z = 12.

2 * (1) + (3) gives:

-3y - 8z = 7.

If its acceleration is constant, then it is equal to the jet's average velocity, given by

[tex]a=a_{\rm ave}=\dfrac{\Delta v}{\Delta t}[/tex]

Then it takes

[tex]17.7\dfrac{\rm m}{\mathrm s^2}=\dfrac{233\frac{\rm m}{\rm s}-119\frac{\rm m}{\rm s}}{\Delta t}\implies\Delta t=\boxed{6.44\,\mathrm s}[/tex]

Answer:

The time taken by the jet is 6.44 seconds.

Step-by-step explanation:

It is given that,

Acceleration of the jet, [tex]a=17.7\ m/s^2[/tex]

Initial velocity of the jet, u = 119 m/s

Final velocity of the jet, v = 233 m/s

Acceleration of an object is given by :

[tex]a=\dfrac{v-u}{t}[/tex]

[tex]t=\dfrac{v-u}{a}[/tex]

[tex]t=\dfrac{233-119}{17.7}[/tex]

t = 6.44 seconds

So, the time taken by the jet is 6.44 seconds. Hence, this is the required solution.

Answer:

  $495

Step-by-step explanation:

After the 5% raise, her weekly pay was ...

  $550 × 1.05 = $577.50

If she works 35 hours for that pay, her hourly rate is

  $577.50/35 = $16.50

Then, working 30 hours, her weekly pay will be ...

  30 × $16.50 = $495.00

To find the new amount of weekly pay, multiply the increase in pay by the new number of hours. The new amount is $577.50.

To find the new amount of weekly pay, we need to calculate the increase in pay and then multiply it by the new number of hours.

The employee's pay increased by 5 percent. This means the pay increased by 5% of $550, which is equal to 0.05  imes 550 = $27.50.

Her new hourly rate of pay is the same, so it remains at $550 + $27.50 = $577.50.

Finally, we need to calculate the new amount of weekly pay, taking into account the reduced number of hours. The new pay per hour is $577.50 / 30 = $19.25. Multiply this by the new number of hours to get the new amount of weekly pay: $19.25  imes 30 = $577.50.

Answer:

0.889

Step-by-step explanation:

I think it is multiplied.

10 *9 = 90

which is the total number of ways you can draw 2 cards without replacement.

I think it is easier to figure out how many possibilities there are over 62 and go from there.

10*9

10*8

10*7

===========

9*8

9*7

So there are 5 combinations that are over 62. There are 5 more possibilities because you could draw them in the reverse order

9 * 10

8 * 10

7 * 10  

8* 9

7 * 9

In all there are 10 ways of drawing numbers that are over 62

So what is the probability of drawing 2 cards above 62?

10/90 = 1/9 = 0.111

Therefore, there must be a probability of 1 - 0.111 for under 62 = 0.889

Answer:

3lbs of Cashews

Step-by-step explanation:

lbs of Cashews, and 7 lbs of Peanuts

$4.00P + 6.50C = ($4.75/lbs)(10lbs)

$4.00(7) + $6.50(3) = $47.50

$28.00 + $19.50 = $47.50

$47.50 = $47.50

Therefore it's 3lbs of Cashews

Answer:

3lbs of Cashews

hope it helps! x

Answer:

Given:

Diameter of lake = 2 miles

∴ [tex]Radius = \frac{Diameter}{2}[/tex] = 1 miles  

The circumference of the lake can be computed as :

Circumference = 2πr

Circumference = 2×3.14×1 = 6.28 miles

This circumference is the total distance traveled by Johanna.

We are give the speed at which Johanna jogs, i.e. Speed = 3 miles/hour

∴ Time taken by Johanna to jog around the lake is given as :

[tex]Time = \frac{Distance}{Speed}[/tex]

Time = 2.093 hours

∴ The correct option is (c.)

Answer:

[tex]50xy[/tex] and [tex]40x^2[/tex].

Step-by-step explanation:

The given polynomial is [tex]36x^3-22x^2-144x[/tex].

The prime factorization of each term are;

[tex]36x^2=2^2\times 3^2\times x^3[/tex]

[tex]-22x^2=-2\times 11\times x^2[/tex]

[tex]-144x=-2^4\times 3^2\times x[/tex]

The greatest common factor of these three terms is [tex]2x[/tex].

Now observe that:

The GCF of [tex]2x[/tex] and 11 is 1

The GCF of [tex]2x[/tex] and 50xy is 2x

The GCF of [tex]2x[/tex] and [tex]40x^2[/tex] is 2x

The GCF of [tex]2x[/tex] and 24 is 2

The GCF of [tex]2x[/tex] and 10y is 2

The correct options are [tex]50xy[/tex] and [tex]40x^2[/tex].

Answer:

b. y+7=5x

Step-by-step explanation:

a. 1-3x^2     is a quadratic

b. y+7=5x    is a linear function:  y = 5x - 7

c. x^3 + 4 = y   is a cubic function

d. 9(x^2-y) = 3    is a quadratic function

e.y-x^3=8   is a cubic function

Circle theorem:

The angle at the centre (T) is double the angle at the circumference (Q)

---> That also means that:

The angle at the circumference (Q) is half the angle at the centre (T)

Since T = 130 degrees;

Q = 130 divided by 2

   = 65°

___________________________________

Answer:

∠Q = 65°

Answer:

m<Q = 65°

Step-by-step explanation:

It is given that <T = 130°

To find the <Q

From the figure we can see that <T is the central angle made by the arc RS

And <Q is the angle made by the arc RS on minor arc.

We know that m<Q = (1/2)m<T

We have m<T = 130°

Therefore m<Q = 130/2 = 65°

Answer:

Point-slope form:

[tex]y-0=\frac{1}{3} (x-1)\\f(x)-0=\frac{1}{3}(x-1)[/tex]

Slope-intercept form:

[tex]y=\frac{1}{3}x-\frac{1}{3} \\f(x)=\frac{1}{3} x-\frac{1}{3}[/tex]

Step-by-step explanation:

You have points on that line at (-2, -1) and (1, 0). To find your slope using those points, use the slope formula.

[tex]\frac{y2-y1}{x2-x1} \\\\\frac{0-(-1)}{1-(-2)} \\\\\frac{0+1}{1+2} \\\\\frac{1}{3}[/tex]

Now that we have your slope, you can use your slope and one of your points to write an equation in point-slope form.

[tex]y-y1=m(x-x1)\\y-0=\frac{1}{3} (x-1)\\y=\frac{1}{3} x-\frac{1}{3}[/tex]

To put it in function notation, substitute y for f(x).

[tex]f(x)-0=\frac{1}{3} (x-1)\\f(x)=\frac{1}{3} x-\frac{1}{3}[/tex]

First answer is y+1=(1/3) (x+2)

Second answer is f(x) =(1/3) x-(1/3)

Answer:

E) -3, 5

Step-by-step explanation:

x – 15 / x = 2

x^2 - 15 = 2x

x^2 - 2x - 15 = 0

(x - 5)(x + 3) = 0

x - 5 = 0; x = 5

x + 3 = 0; x = -3

Solutions: -3, 5

For this case we must solve the following equation:

[tex]x- \frac {15} {x} = 2[/tex]

We manipulate the equation algebraically:

[tex]\frac {x ^ 2-15} {x} = 2\\x ^ 2-15 = 2x\\x ^ 2-2x-15 = 0[/tex]

To solve, we factor the equation. We must find two numbers that when multiplied by -15 and when summed by -2. These numbers are:

+3 and -5.

[tex](x + 3) (x-5) = 0[/tex]

So, the roots are:

[tex]x_ {1} = - 3\\x_ {2} = 5[/tex]

Answer:

Option E

Check the picture below.

Answer:

x = 60°

Step-by-step explanation:

From ΔOPQ,

∠OPQ = 120°   [ angle at the center inscribed by arc PQ ]

PQ ≅ OQ

so opposite angles to PQ and OQ will be equal

∠OPQ  ≅ ∠OQP

∠OPQ + ∠OQP + ∠POQ = 180°

∠OPQ + ∠OPQ + 120 = 180°

2∠OPQ = 180 - 120 = 60°

∠OPQ = 30°

Since radius OP is perpendicular to tangent.

so ∠OPQ + Y = 90°

y + 30° = 90°

y = 90 - 30 = 60°

Answer x = 60°

The maximum difference in the lengths of the radii is 1/2.

To solve this problem, let's denote the radius of the smaller circle as ( r ) and the radius of the larger circle as ( R ). We're given that [tex]\( r + R = 10 \)[/tex].

The area of a circle is given by the formula [tex]\( A = \pi r^2 \)[/tex], where ( r ) is the radius.

We want the absolute difference in the areas of the two circles to be less than or equal to [tex]\( 5\pi \)[/tex]. So, we can set up the following inequality:

[tex]\[ |(\pi R^2) - (\pi r^2)| \leq 5\pi \][/tex]

[tex]\[ |(\pi (10-r)^2) - (\pi r^2)| \leq 5\pi \][/tex]

Expanding and simplifying:

[tex]\[ |(100\pi - 20\pi r + \pi r^2) - (\pi r^2)| \leq 5\pi \][/tex]

[tex]\[ |100\pi - 20\pi r| \leq 5\pi \][/tex]

[tex]\[ 100 - 20r \leq 5 \][/tex]

[tex]\[ 100 - 5 \leq 20r \][/tex]

[tex]\[ 95 \leq 20r \][/tex]

[tex]\[ \frac{95}{20} \leq r \][/tex]

[tex]\[ r \geq \frac{19}{4} \][/tex]

So, the maximum difference in the lengths of the radii is when [tex]\( r = \frac{19}{4} \)[/tex] and [tex]\( R = 10 - r = 10 - \frac{19}{4} = \frac{21}{4} \)[/tex].

The maximum difference in the lengths of the radii is [tex]\( \frac{21}{4} - \frac{19}{4} = \frac{2}{4} = \frac{1}{2} \)[/tex].

Answer:

[tex]\frac{\pi }{6}[/tex]

Step-by-step explanation:

The volume of a sphere is [tex]\frac{4}{3} \pi r^{3}[/tex]

Just plug in 1/2 for r

[tex]\frac{4}{3} \pi (\frac{1}{2}) ^{3}[/tex]

The answer is [tex]\frac{\pi }{6}[/tex]

Answer:

The dimensions of the brand name television are [tex]26\frac{2}{3}\ in[/tex] by  [tex]66\frac{2}{3}\ in[/tex]

Step-by-step explanation:

we know that

The generic version was based on the brand name and was 3/5 the size of the brand name

Let

x----> the length of the size of the brand name

y----> the width of the size of the brand name

Find the length of the size of the brand name

we know that

[tex]40=\frac{3}{5}x[/tex] -----> equation A

Solve for x

Multiply by 5 both sides

[tex]5*40=3x[/tex]

Rewrite and divide by 3 both sides

[tex]x=200/3\ in[/tex]

Convert to mixed number

[tex]200/3=(198/3)+(2/3)=66\frac{2}{3}\ in[/tex]

Find the width of the size of the brand name

we know that

[tex]16=\frac{3}{5}y[/tex] -----> equation B

Solve for y

Multiply by 5 both sides

[tex]5*16=3y[/tex]

Rewrite and divide by 3 both sides

[tex]x=80/3\ in[/tex]

Convert to mixed number

[tex]80/3=(78/3)+(2/3)=26\frac{2}{3}\ in[/tex]

Answer:

Option D

Step-by-step explanation:

We have the following variable definitions:

sofas: x

pillows: y

Pillows come in pairs so we have 2y pillows

The total order for all the possible combinations is:

[tex]x+2y[/tex]

The wholesaler requires a minimum of 4 items in each order from its retail customers. This means the retailers can order 4 or more.

Therefore the inequality is:

[tex]x+2y\ge4[/tex]

To graph this inequality, we graph the corresponding linear equation, [tex]x+2y=4[/tex]  with a solid line and shade above.

The correct choice is D

See attachment

Answer:

So we have the two real points (3/2 , 9/2)  and (-1,2).

(Question: are you wanting to use the possible rational zero theorem? Please let me know if I didn't answer your question.)

Step-by-step explanation:

y=2x^2

y^2=x^2+6x+9

is the given system.

So my plain here is to look at y=2x^2 and just plug it into the other equation where y is.

(2x^2)^2=x^2+6x+9

(2x^2)(2x^2)=x^2+6x+9

4x^4=x^2+6x+9

I'm going to put everything on one side.

Subtract (x^2+6x+9) on both sides.

4x^4-x^2-6x-9=0

Let's see if some possible rational zeros will work.

Let' try x=-1.

4-1+6-9=3+(-3)=0.

x=-1 works.

To find the other factor of 4x^4-x^2-6x-9 given x+1 is a factor, I'm going to use synthetic division.

-1   |  4     0     -1     -6    -9

    |         -4     4     -3      9

    |________________ I put that 0 in there because we are missing x^3

        4    -4     3     -9      0

The the other factor is 4x^3-4x^2+3x-9.

1 is obviously not going to make that 0.

Plug in -3 it gives you 4(-3)^3-4(-3)^2+3(-3)-9=-162 (not 0)

Plug in 3 gives you 4(-3)^3-4(-3)^2+3(-3)-9=72 (not 0)

Plug in 3/2 gives you 4(3/2)^2-4(3/2)^2+3(3/2)-9=0 so x=3/2 works as a solution.

Now let's find another factor

3/2  |     4       -4           3        -9

      |                6           3          9

      |________________________

            4          2           6        0

So we have 4x^2+2x+6=0.

The discriminant is b^2-4ac which in this case is (2)^2-4(4)(6). Simplifying this gives us (2)^2-4(4)(6)=4-16(6)=4-96=-92.  This is negative number which means the other 2 solutions are complex (not real).

So the other real solutions that satisfy the system is for x=3/2 or x=-1.

Since y=2x^2 then for x=3/2 we have y=2(3/2)^2=2(9/4)=9/2 and for x=-1 we have y=2(1)^2=2.

So we have the two real points (3/2 , 9/2)  and (-1,2)

Answer:

  see below

Step-by-step explanation:

To find miles per hour, divide miles by hours:

  (5 2/3 mi)/(2 2/3 h) = (17/3 mi)/(8/3 h) = (17/8) mi/h = 2 1/8 mi/h

Hours per mile is the reciprocal of that:

  1/(17/8 mi/h) = 8/17 h/mi

[tex]

\Sigma_{n=1}^{4}-2\cdot(-3)^{n-1} \\

(-2)(-3)^{1-1}+(-2)(-3)^{2-1}+(-2)(-3)^{3-1}+(-2)(-3)^{4-1} \\

-2+6-18+54 \\

\boxed{40}

[/tex]

So the answer is C,

[tex]\Sigma_{n=1}^{4}-2\cdot(-3)^{n-1}=40[/tex]

Hope this helps.

r3t40

The sum of the finite geometric series (-2)(-3)ⁿ⁻¹ for n=1 to n=4 is 40, calculated using the geometric series sum formula.So,option C is correct.

The sum of a finite geometric series with a general term given as (-2)(-3)ⁿ⁻¹ where 'n' ranges from 1 to 4. To find the sum of a geometric series, we need to identify the first term (a) and the common ratio (r), and then use the formula Sₙ = a(1 - rⁿ) / (1 - r), where n is the number of terms.

The first term of the series can be found by substituting n = 1 into the general expression, yielding a = (-2)(-3)¹⁻¹ = -2. The second term, with n = 2, is (-2)(-3)²⁻¹ = -6(-3) = 18, indicating a common ratio of -3.

Thus, the sum of the series for the first four terms can be calculated as:

S₄ = (-2)(1 - (-3)⁴) / (1 - (-3))

S₄= (-2)(1 - 81) / (1 + 3)

S₄= (-2)(-80) / 4

S₄ = 160 / 4

S₄ = 40

Therefore, the sum of the given geometric series is 40.

Answer:

(A)

Step-by-step explanation:

Answer:

Option A is correct.

Step-by-step explanation:

Given  : [tex]f(x) =2^{x}[/tex] and [tex]g(x) =2^{x+4}[/tex].

To find : How will the graph of g(x) differ from the graph of f(x).

Solution : We have given that  

[tex]f(x) =2^{x}[/tex] and g(x)   [tex]g(x) =2^{x+4}[/tex]

By the transformation Rule : If f(x) →→ f(x +h) if mean graph of function shifted to left by h units .

Then  graph of [tex]g(x) =2^{x+4}[/tex] is the graph of [tex]f(x) =2^{x}[/tex] is shifted by 4 unt left.

Therefore, Option A is correct.

Answer:

Length of arc AB is,

= 2πr (angle between AB) /360

=2×3.14×90/360

=1.57 cm

For this case we have that by definition, the arc length of a circle is given by:

[tex]AL = \frac {x * 2 \pi * r} {360}[/tex]

Where:

x: Represents the angle between AB. According to the figure we have that x = 90 degrees.

[tex]r = 7.9 \ cm[/tex]

So:

[tex]AL = \frac {90 * 2 \pi * 7.9} {360}\\AL = \frac {90 * 2 * 3.14 * 7.9} {360}\\AL = \frac {4465,08} {360}\\AL = 12.403[/tex]

Answer:

[tex]12.4\ cm[/tex]