Which input value produces the same output value for the two functions on the graph?

X= -3
X= -1
X= 1
X= 3

Answer: 4 years

Step-by-step explanation:

A(0) has to be amount at start. Assume that's 5mg

Then A(t) = 5×(0.5)^(0.25t) = 5×2^(-t/4),

(also known as 5 exp(-λ t) with λ = ln(2)/4, incidentally).

We need to such that A(t) = 2.5mg, or 2^(-t/4) is 1/2, which happens when -t/4 is -1, or t is 4.

The exponential function given can be used to model how radioactivity decays. Solving this function for when half the material remains, we find it takes approximately 2.8 years for half of the radioactive material to decay.

The amount of radioactive material remaining in a container over time can be modelled by the exponential decay function given in the question: A(t) = 5(0.5)0.25t. Here, A(t) stands for the amount left after time t, and the function gives a general rule for how radioactivity decays over time. Specifically, this rule tells us that after 1 year (t=1), approximately 74% of the radioactive material remains.

However, you're asking for the time it takes for half of the original material to remain. To solve this, we need to set A(t) to half its initial value, which is 2.5 milligrams: 2.5 = 5(0.5)0.25t. This translates to a simple equation 0.5 = (0.5)0.25t. Solving for t, we find that approximately 2.8 years are needed for half of the material to remain. Please note, this relies on the assumptions and limitations of the exponential decay model.

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Answer:

2 and 6

Step-by-step explanation:

Corresponding angles are the angles that occupy the same relative position at each intersection.  

1 and 5

2 and 6

4 and 8

3 and 7

Answer:

12 dogs

Step-by-step explanation:

Let d = dogs

c = cats

d = 3c

c+d = 16

Substitute d =3c into the second equation

c+3c = 16

Combine like terms

4c = 16

Divide by 4

4c/4 = 16/4

c = 4

There are 4 cats

We still need to find the number of dogs

d = 3c

d = 3*4

   = 12

There are 12 dogs

Answer:

cats = 4; dogs = 12

Step-by-step explanation:

There are 3 times more dogs than cats.

There are 16 dogs and cats in all.

Let:

dogs = d ; cats = c

Set the system of equations:

d + c = 16

3c = d

Plug in 3c for d (as d = 3c) in the first equation

(3c) + c = 16

Simplify. Combine like terms:

(3c + c) = 16

4c = 16

Isolate the variable. Note the equal sign, what you do to one side, you do to the other. Divide 4 from both sides

(4c)/4 = (16)/4

c = 16/4

c = 4

Plug in 4 for c in one of the equations.

d + c = 16

d + (4) = 16

Isolate the variable. Note the equal sign, what you do to one side, you do to the other. Subtract 4 from both sides

d + 4 (-4) = 16 (-4)

d = 16 - 4

d = 12

Answers: c = 4; d = 12

~

Answer:

B

Step-by-step explanation:

Let [tex]r[/tex] be the radius of one sphere.

Since radius is r,  the height of the cylinder will be [tex]4r[/tex]

Also, the cylinder has the same radius as the sphere: [tex]r[/tex]

Plugging in the values we get: [tex]V=\pi (r)^{2}(4r)=4\pi r^3[/tex]

Ratio of volume of 1 sphere to volume of cylinder is:

[tex]\frac{\frac{4}{3}\pi r^3}{4\pi r^3}=\frac{\frac{4}{3}}{4}=\frac{4}{3}*\frac{1}{4}=\frac{1}{3}[/tex]

The ratio is 1:3

Answer choice B is right.

Answer:

± 1/2,±1, ±2,± 5/2, ±4 ,±5 , ±10, ± 20

Step-by-step explanation:

We can use the rational root theorem to find all the possible roots

2x^3+5x^2-8x-20=0  

Let the constant term  be called p and the leading term be called q.  Then the possible roots are the positive and negative roots of the factors of p/q

p = 20

q = 2

Factors of p: 1,2,4,5,10,20

Factors of q: 1,2

Possible roots

    1 ,2,4,5,10,20

±   --------------------------------------------------------

      1,2

So we get

±1, ±2, ±4 ,±5 , ±10 ± 20 ± 1/2± 2/2,±4/2,± 5/2,± 10/2,± 20/2

Simplifying

±1, ±2, ±4 ,±5 , ±10, ± 20, ± 1/2,± 1,±2,± 5/2,± 5,± 10

Eliminating repeats

±1, ±2, ±4 ,±5 , ±10, ± 20 ,± 1/2,± 5/2

Putting them in numerical order

± 1/2,±1, ±2,± 5/2, ±4 ,±5 , ±10, ± 20

Answer:

Its Option 1

Step-by-step explanation:

The possible rational roots will have a numerator that divides 20 (the last number) and a denominator that divides 2 (the coefficient of x^3).

For example 20/2, 10/2 and -1/2  =  10, 5 and -1/2.

The correct answer is the first option.

Answer:

B

(x-2) is a factor since remainder is 0.

Step-by-step explanation:

We divide (x-2) into the polynomial [tex]-4x^4+6x^3+8x^2-6x-4[/tex] through long division or synthetic. We choose long division and look for what will multiply with (x-2) to make the polynomial [tex]-4x^4+6x^3+8x^2-6x-4[/tex] .

[tex](x-2)(-4x^3)=-4x^4+8x^3[/tex]

We subtract this from the original [tex]-4x^4-(-4x^4)+6x^3-(8x^3)+8x^2-6x-4[/tex].

This leaves [tex]-2x^3+8x^2-6x-4[/tex]. We repeat the step above.

[tex](x-2)(-2x^2)=-2x^3+4x^2[/tex].

We subtract this from [tex]-2x^3-(-2x^3)+8x^2-(4x^2)-6x-4=4x^2-6x-4[/tex]. We repeat the step above.

[tex](x-2)(4x)=4x^2-8x[/tex].

We subtract this from [tex]4x^2-(4x^2)-6x-(-8x)-4=2x-4[/tex]. We repeat the step above.

[tex](x-2)(2)=2x-4[/tex].

We subtract this from [tex]2x-(2x)-4-(4)=0[/tex]. There is no remainder. This means (x-2) is a factor.

The remainder R when the polynomial p(x) = -4x4 + 6x3 + 8x2 - 6x - 4 is divided by (x - 2) is 0, indicating that (x - 2) is indeed a factor of p(x).

To find the remainder R when the polynomial p(x) = -4x4 + 6x3 + 8x2 - 6x - 4 is divided by (x - 2), we can use the Remainder Theorem. The Remainder Theorem states that the remainder of the division of a polynomial p(x) by a linear term (x - a) is equal to p(a). Therefore, to find R, simply calculate p(2).

p(2) = -4(2)4 + 6(2)3 + 8(2)2 - 6(2) - 4

p(2) = -4(16) + 6(8) + 8(4) - 12 - 4

p(2) = -64 + 48 + 32 - 12 - 4

p(2) = 0

Since the remainder is 0, this means that (x - 2) is a factor of p(x).

The correct answer is therefore R = 0, and yes, (x - 2) is a factor of p(x).

Answer:

It is 15% off

Step-by-step explanation:

30.00-(15%×30.00)= 25.5

Answer:

∠A=40°

Step-by-step explanation:

Equation 1: ∠A+∠B+∠C=180°

Equation 2: ∠B/2+∠C/2+110°=180°

2(Equation 2): ∠B+∠C+220°=360°

                                 ∠B+∠C=140°

Equation 1 - 2(Equation 2): ∠A=40°

Therefore, ∠A=40°

Answer:

Warning:

I think that these are correct, but don't make this your official source :)

7. B

8. D

9. B

I hope this helps :)

l - length

19 ft - width

62 ft - perimeter

l + l + 19 + 19 = 2l + 38 - perimeter

The equation:

2l + 38 = 62    subtract 38 from both sides

2l = 24     divide both sides by 2

l = 12 ft

Answer:

The game costs 32 dollars total, we know this because if you divid 32 by 4 its 8 and . if you multiply 8 times 3 its 24

Step-by-step explanation:

The total cost of the game is $32, calculated using the concept of fractions in mathematics where the given $24 was considered as three-fourths of the total game cost.

The subject of the question involves the concept of fractions in mathematics. Here the information provided states that $24, which Ivanna has saved, represents three-fourths (or 3/4) of the total cost of the game. To find out the total cost of the game, we will consider the $24 as three parts, and we need to find out the value of one part (or one-fourth). So, we divide $24 by 3, which gives us $8. This $8 is one-fourth of the total game cost. Now, to get the total cost, we will multiply this one-fourth value ($8) by 4. So, the total cost of the game is $8 * 4 = $32.

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Answer:

System 2 and system 3, because the second equation in system 3 is obtained by adding the first equation in system 2 to two times the second equation in system 2

Step-by-step explanation:

We are given four systems of equations.  We are asked to find which two sets have the same solution.

We find that in all the 4 systems, first equation 4x-5y =2 remains the same.

I option is wrong because

Equation I + 2times equation 2 in II system

gives 10x-7y =10 but given is 10x-21y =10 hence wrong.

Now compare 2 and 3

In system 2, I equation +2times second equation gives

24x-5y-42y =2+20 or

24x-47y =22

Hence system 2 and 3 are the same set of equations and give the same solution.

Answer:

320.

Step-by-step explanation:

1,000 - the 680 that have been completed gives you 320.

Answer:

$68.32

Step-by-step explanation:

Is 9.76 = x

And the value needed is 8 * this amount, then it can be shown with 8x

8x = 9.76 * 8

8x = 78.08

So the total price is $78.08

To find the money needed, simply subtract the saved money from this amount:

78.08 - 9.76 = 68.32

So she needs $68.32!

Isabelle needs $78.08 to buy the bicycle, which is 8 times her current savings of $9.76.

Isabelle's current savings stands at $9.76. But she needs to have 8 times this amount in order to buy the bicycle. Mathematically speaking, you find how much she needs by multiplying her current savings by 8. So, $9.76 x 8 equals $78.08. Therefore, Isabelle needs a total sum of $78.08 to buy the bicycle.

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Answer:

A. the graph flip over the x - axis...

Option b) the graph flips over the line y=x

When you multiply two function together, you will get a third function as the result,  and that third function as the result, and that third function will be the product of the two original functions . when we multiply the function by -1 ,it becomes y=-f(x)  . Then the coordinates becomes (x, -f(x))

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Answer: Choice D) approaches y = -4

"x increases without bound" is another way of saying "x heads off to positive infinity". Visually, you follow the graph curve going to the right. As the graph shows, the curve steadily gets closer to the horizontal line y = -4, but it never actually gets there. This line is the horizontal asymptote.

Answer:

2/3 - 1/4

=> taking L.C.M of the denominators we get,

=> 8/12-3/12

=> 5/12 is left with her,, ;)

HOPE THIS HELPS YOU...;)

Answer:

C

Step-by-step explanation:

The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles.

The measure of an exterior angle of a triangle is equal to the sum of the measure of the two opposite interior angles.

A triangle is a three-sided closed-plane figure formed by joining three noncolinear points. Based on the side property triangles are of three types they are Equilateral triangle, Scalene triangle, and Isosceles triangle.

We know the sum of all the interior angles in a triangle is 180°.

We also know that the measure of an exterior angle of a triangle is the sum of the measure of two opposite interior angles.

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Answer:

A difference of squares has the following form [tex]a^2-b^2[/tex]. Any two perfect squares connected by subtraction can be factored.

It factors to (a+b)(a-b).

Step-by-step explanation:

A binomial is an expression with only terms where at least one is a term with a variable. When we can factor for difference of squares, we can have two variable terms or just one with a constant.

A difference of squares has the following form [tex]a^2-b^2[/tex]. Any two perfect squares connected by subtraction can be factored.

It factors to (a+b)(a-b).

I think the answer is A, but I could be wrong. I think the answer is A, because it is the only equation that is linear. In mathematics a geometric sequence  is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number, meaning when you graph it the line should be linear. Of all of these A is the only that is Linear the others are not. The slope of line A is 2 and the Y-intercept is 3.

Answer: A

Step-by-step explanation:

f(x) = x³ + 4x² + 7x + 6

possible rational roots are ±{1, 2, 3, 6}

Try x = -2

-2 |   1   4   7   6

   |   ↓  -2  -4  -6

       1    2    3   0 ← remainder is 0 so x = -2 is a root  ⇒  (x + 2) = 0

The factored polynomial x² + 2x + 3 = 0 is not factorable so use the quadratic formula to find the roots.

a=1, b=2, c=3

[tex]x=\dfrac{-b\pm\sqrt{b^2-4ac} }{2a}[/tex]

[tex]x=\dfrac{-(2)\pm\sqrt{(2)^2-4(1)(3)} }{2(1)}[/tex]

[tex]x=\dfrac{-2\pm\sqrt{4-12} }{2}[/tex]

[tex]x=\dfrac{-2\pm\sqrt{-8} }{2}[/tex]

[tex]x=\dfrac{-2\pm2i\sqrt{2}}{2}[/tex]

[tex]x = -1 \pm i\sqrt{2}[/tex]

[tex]x = -1 + i\sqrt{2}[/tex]       [tex]x = -1 - i\sqrt{2}[/tex]

[tex]x - (-1 + i\sqrt{2})=0[/tex]       [tex]x - (-1 - i\sqrt{2})=0[/tex]

The factors are:

[tex](x - 2)[x - (-1 + i\sqrt{2})][x - (-1 - i\sqrt{2})][/tex]

Answer

x = 3, y = -1

Step-by-step explanation:

i used substitution because i find that easier most times so here goes:

x + 6y = -3

    - 6y

x = -6y -3

now we have an "x=" to plug into the "y=" we will get ina second

2x + 3y = 3

-2x

3y = -2x + 3

divide the whole thing by 3 to get y by itself. you get:

y = -2/3x + 1

now plug in what you got for x, which was (-6y-3), into the x. so:

y = -2/3(-6y - 3) + 1

now you multiply the -2/3 by the -6y and the -3. when you distribute that you get

y = 4y + 2 + 1

then combine like terms by adding 2 + 1 to get 3 and subtracting the y on the right from the 4y on the left to get

= 3y + 3

since you cant have an empty side of the equal sign, move the 3y over to the left side by subtracting it. you get

-3y = 3    now divide everything by -3 to get y = -1

now you have your y variable so now just plug that into the original equation to get x + 3

HOPE THIS HELPS!!!

if you have more questions or didnt understand something, please let me know!! ill explain it!!

We can first find the probability of the first person picking Brand A, which would be:

9/6+9

=9/15

=3/5

We can then find the probability of the second person choosing brand A:

(9-1)/(6+9-1)

=8/14

To find the probability they both select Brand A, we could multiply them together:

3/5 x 8/14

=12/35

Therefore the answer would be 12/35.

Hope it helps!

Answer: assuming that this was your equation;

[tex]\frac{1}{16} =64^{4x-3}[/tex]

You answer is x= 7/12

Step-by-step explanation:

First, rewrite 1/16 as 2^-4 and 64^4x-3 as 2^24x-18

This will leave you with;[tex]2^{-4} =2^{24x-18}[/tex]

Since bases are the same, set the exponents equal (eliminate the 2’s);

-4=24x-18

Move 24x to the left to get; -24x-4=-18

Move constant to the right to get -24x=-18+4

Add 4 to -18 to get; -24x=-14

Divide both sides by -24 to get you final answer for x which is 7/12

Hope this helps. Please rate and hit that thanks button. Cheers!

The required value of exponential equation is x =  [tex]\frac{7}{12}[/tex].

Given that,

Exponential equation; [tex]\frac{1}{16} = 64^{4x-3}[/tex]

We have to determine,

The value of x.

According to the question,

To obtain the value of x by solving the exponential equation follow all the steps given below.

Exponential equation;

[tex]\dfrac{1}{16} = 64^{4x-3}[/tex]

Then,

[tex]\dfrac{1}{16} = 64^{4x-3}\\\\\dfrac{1}{2^{4}} = 2^{6(4x-3)}\\\\2^{-4} = 2^{24x-18}\\\\[/tex]

Since, bases are the same, set the exponents equation,

Then,

Equating the power of exponential equation,

[tex]-4 = 24x-18\\\\24x = -4+18\\\\24x = 14\\\\x = \dfrac{14}{24}\\\\x = \dfrac{7}{12}[/tex]

Hence, The required value of exponential equation is [tex]\frac{7}{12}[/tex].

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Answer:

He will make 16.50 per hour after his raise.

Step-by-step explanation:

15 x .10 = 1.50

15.00 + 1.50 = 16.50

Answer:

Step-by-step explanation:

42 < 21+3x < 84

21 < 3x < 63

7 < x < 21

The measure of sides of a triangle lies between 7<x<21.

Given that, Tim is making a fence in the shape of a triangle for his livestock.

The perimeter of a shape is defined as the total distance around the shape. It is the length of the outline or boundary of any two-dimensional geometric shape. The perimeter of different figures can be equal in measure depending upon the dimensions.

Let the measure of one side of a triangle be x meters.

One side of the triangle to be two times as along as another side and the third side to be 21 meters long.

Second side =2x meters and the third side =21 meters

Tim wants the perimeter of the triangle to be more than 42 meters and less than 84meters.

Now, 42<x+2x+21<84

21<3x<63

7<x<21

Therefore, the measure of sides of a triangle lies between 7<x<21.

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Rectangle is the closed shaped polygon with 4 sides. Opposite sides of the rectangle are equal. when the length and the width of the sandbox are doubled the percent of change in the perimeter is 100 percent and the percent of change in the area is 300 percent.

Given-

The length of the sandbox is 10.

The width of the sandbox is 6.

a) The percent of change in the perimeter.

The perimeter of the rectangle is the twice of the sum of its side.The perimeter P of the sandbox is,

[tex]P =2\times (10+6)[/tex]

[tex]P =2\times (16)[/tex]

[tex]P=32[/tex]

When the length and the width of the sandbox are doubled the perimeter [tex]P_d[/tex] of the box is,

[tex]P_d=2\times(20+12)[/tex]

[tex]P_d=2\times(32)[/tex]

[tex]P_d=64[/tex]

Percentage [tex]\DeltaP[/tex][tex]\Delta P[/tex] change in the perimeter,

[tex]\Delta P=\dfrac{64-32}{32}\times 100[/tex]

[tex]\Delta P=100[/tex]

b) The percent of change in the area.

The area of the rectangle is the product of its side.The area A of the sandbox is,

[tex]A = (10\times6)[/tex]

[tex]A=60[/tex]

When the length and the width of the sandbox are doubled the area [tex]A_d[/tex] of the box is,

[tex]A_d=20\times12[/tex]

[tex]P_d=240[/tex]

Percentage [tex]\DeltaP[/tex][tex]\Delta A[/tex] change in the area,

[tex]\Delta A=\dfrac{240-60}{60}\times 100[/tex]

[tex]\Delta P=300[/tex]

Thus when the length and the width of the sandbox are doubled the percent of change in the perimeter is 100 percent and the percent of change in the area is 300 percent.

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Answer: Choice B (1,-2)

note: I'm assuming there is a "less than or equal to" sign as part of the given inequality, rather than a simple "less than" sign (without the "or equal to" part)

============================================

Work Shown:

Plug in x = -2 and y = 4 from the point (x,y) = (-2, 4). Then simplify. If you get a true statement, then this is a solution point

[tex]y - 4x \le -6[/tex]

[tex]4 - 4*(-2) \le -6[/tex]

[tex]4 + 8 \le -6[/tex]

[tex]12 \le -6[/tex]

This statement is false. The value 12 is not to the left of -6 on the number line, nor is 12 equal to -6. So (x,y) = (-2,4) is not a solution

-------------

Plug in x = 1 and y = -2

[tex]y - 4x \le -6[/tex]

[tex]-2 - 4*(1) \le -6[/tex]

[tex]-2 - 4 \le -6[/tex]

[tex]-6 \le -6[/tex]

This statement is true, but only if the inequality sign is a "less than or equal to" sign. Otherwise, the statement is false because -6 cannot be smaller than itself.

-------------

Plug in x = 1 and y = 3

[tex]y - 4x \le -6[/tex]

[tex]3 - 4*(1) \le -6[/tex]

[tex]3 - 4 \le -6[/tex]

[tex]-1 \le -6[/tex]

Like choice A, this is false. So (x,y) = (1,3) is not a solution

-------------

Plug in x = 2 and y = 3

[tex]y - 4x \le -6[/tex]

[tex]3 - 4*(2) \le -6[/tex]

[tex]3 - 8 \le -6[/tex]

[tex]-5 \le -6[/tex]

We end up with another false statement. So (x,y) = (2,3) isn't a solution either.

Since we are given that ABC is congruent to JKL, then AB has to be equal to JK

therefore --- [tex]14x+7=5x+34[/tex]

Subtract 7 from both sides

[tex]14x=5x+27[/tex]

Subtract 5x from each side

[tex]9x=27[/tex]

Divide 9 on each side

[tex]x=3[/tex]

Since, We are asked to find the length of AB and JK

plug in 3 for x

[tex]5(3)+34=15+34=49[/tex]