A sample of 500 g of radioactive​ lead-210 decays to​ polonium-210 according to the function A(t)=500e^-0.032t ​, where t is time in years. Find the amount of radioactive lead remaining after ​(a) ​3yr, ​(b) 8​yr, ​(c) 10 yr. ​(d) Find the​ half-life.

Answer:

A

Step-by-step explanation:

Find the vertex form of the quadratic function below.

y = x^2 - 4x + 3

This quadratic equation is in the form y = a{x^2} + bx + cy=ax  

2

+bx+c. However, I need to rewrite it using some algebraic steps in order to make it look like this…

y = a(x - h)^2 + k

This is the vertex form of the quadratic function where \left( {h,k} \right)(h,k) is the vertex or the “center” of the quadratic function or the parabola.

Before I start, I realize that a = 1a=1. Therefore, I can immediately apply the “completing the square” steps.

STEP 1: Identify the coefficient of the linear term of the quadratic function. That is the number attached to the xx-term.

STEP 2: I will take that number, divide it by 22 and square it (or raise to the power 22).

STEP 3: The output in step #2 will be added and subtracted on the same side of the equation to keep it balanced.

Think About It: If I add 44 on the right side of the equation, then I am technically changing the original meaning of the equation. So to keep it unchanged, I must subtract the same value that I added on the same side of the equation.

STEP 4: Now, express the trinomial inside the parenthesis as a square of a binomial, and simplify the outside constants.

After simplifying, it is now in the vertex form y = a{\left( {x - h} \right)^2} + ky=a(x−h)  

2

+k where the vertex \left( {h,k} \right)(h,k) is \left( {2, - 1} \right)(2,−1).

Visually, the graph of this quadratic function is a parabola with a minimum at the point \left( {2, - 1} \right)(2,−1). Since the value of “aa” is positive, a = 1a=1, then the parabola opens in upward direction.

Example 2: Find the vertex form of the quadratic function below.

The approach to this problem is slightly different because the value of “aa” does not equal to 11, a \ne 1a  



​  

=1. The first step is to factor out the coefficient 22 between the terms with xx-variables only.

STEP 1: Factor out 22 only to the terms with variable xx.

STEP 2: Identify the coefficient of the xx-term or linear term.

STEP 3: Take that number, divide it by 22, and square.

STEP 4: Now, I will take the output {9 \over 4}  

4

9

​  

 and add it inside the parenthesis.

By adding {9 \over 4}  

4

9

​  

 inside the parenthesis, I am actually adding 2\left( {{9 \over 4}} \right) = {9 \over 2}2(  

4

9

​  

)=  

2

9

​  

 to the entire equation.

Why multiply by 22 to get the “true” value added to the entire equation? Remember, I factored out 22 in the beginning. So for us to find the real value added to the entire equation, we need to multiply the number added inside the parenthesis by the number that was factored out.

STEP 5: Since I added {9 \over 2}  

2

9

​  

 to the equation, then I should subtract the entire equation by {9 \over 2}  

2

9

​  

 also to compensate for it.

STEP 6: Finally, express the trinomial inside the parenthesis as the square of binomial and then simplify the outside constants. Be careful combining the fractions.

It is now in the vertex form y = a{\left( {x - h} \right)^2} + ky=a(x−h)  

2

+k where the vertex \left( {h,k} \right)(h,k) is \left( {{{ - \,3} \over 2},{{ - 11} \over 2}} \right)(  

2

−3

​  

,  

2

−11

​  

).

Example 3: Find the vertex form of the quadratic function below.

Solution:

Factor out - \,3−3 among the xx-terms.

The coefficient of the linear term inside the parenthesis is - \,1−1. Divide it by 22 and square it. Add that value inside the parenthesis. Now, figure out how to make the original equation the same. Since we added {1 \over 4}  

4

1

​  

 inside the parenthesis and we factored out - \,3−3 in the beginning, that means - \,3\left( {{1 \over 4}} \right) = {{ - \,3} \over 4}−3(  

4

1

​  

)=  

4

−3

​  

 is the value that we subtracted from the entire equation. To compensate, we must add {3 \over 4}  

4

3

​  

 outside the parenthesis.

Therefore, the vertex \left( {h,k} \right)(h,k) is \left( {{1 \over 2},{{11} \over 4}} \right)(  

2

1

​  

,  

4

11

​  

).

Example 4: Find the vertex form of the quadratic function below.

y = 5x^2 + 15x - 5  

Solution:

Factor out 55 among the xx-terms. Identify the coefficient of the linear term inside the parenthesis which is 33. Divide it by 22 and square to get {9 \over 4}  

4

9

​  

.

Add {9 \over 4}  

4

9

​  

 inside the parenthesis. Since we factored out 55 in the first step, that means 5\left( {{9 \over 4}} \right) = {{45} \over 4}5(  

4

9

​  

)=  

4

45

​  

 is the number that we need to subtract to keep the equation unchanged.

Express the trinomial as a square of binomial, and combine the constants to get the final answer.

Therefore, the vertex \left( {h,k} \right)(h,k) is {{ - \,3} \over 2},{{ - \,65} \over 4}  

2

−3

​  

,  

4

−65

​  

.

Answer:

(x - 1 )^2 - 3

Step-by-step explanation:

( x - 1 )^2 + ( -3)

x^2 - 2x + 1 - 3

x^2 - 2x - 2

Answer: 2

Step-by-step explanation:

Given: Shaquira has made 86  chocolate chip cookies and 42 sugar cookies.

Shaquira wants to create identical packages of cookies to sell, and she must use all of the cookies.

Now, the greatest number of identical packages that Shaquira can make= GCD of 86 and 42

Prime factorization of 86 and 42:

86 = 2 ×43

42 = 2 × 3 × 7

GCD of 86 and 42 = 2   [GCD = greatest common factor]

Hence, the greatest number of identical packages that Shaquira can make =2

Answer:

[tex]\frac{2}{3}[/tex]

Step-by-step explanation:

If we have 3 red spaces, 5 white spaces, and one blank space, there are a total of 9 spaces.

Since there are 3 red spaces, there is a [tex]\frac{3}{9} = \frac{1}{3}[/tex] chance of getting a red. However, the question asks the probability of not getting a red, so the chances of not getting a red are [tex]1 -\frac{1}{3} = \frac{2}{3}[/tex].

Hope this helped!

Answer:

discriminant is b²-4ac

= 2²-4(-8)(0)

= 0

one solution

hope this helps :)

Answer:

Hi, there!!

Hope you mean the answers in the solution.

Hope it helps...

Answer:

Step-by-step explanation:

7)

JKLM is a isosceles trapezium.

KL // JM

∠K + ∠J =  180   {Co interior angles}

50 +∠J = 180

      ∠J = 180 - 50

      ∠J = 130

As it is isosceles, non parallel sides KJ = LM &  

∠L = ∠K

∠L = 50

∠M = ∠J

∠M = 130

8)JKLM is a isosceles trapezium.

KL // JM

∠K + ∠J =  180   {Co interior angles}

100 +∠J = 180

      ∠J = 180 - 100

      ∠J = 80

As it is isosceles, non parallel sides KJ = LM &  

∠L = ∠K

∠L = 100

∠M = ∠J

∠M = 80

Answer:

Difference : 4th option

Step-by-step explanation:

The first thing we want to do here is to factor the expression x² + 3x + 2. This will help us if it is similar to the factored expression " ( x + 2 )( x + 1 ). " The denominators will be the same, and hence we can combine the fractions.

x² + 3x + 2 - Break the expression into groups,

( x² + x ) + ( 2x + 2 ) - Factor x from x² + x and 2 from 2x + 2,

x( x + 1 ) + 2( x + 2 ) - Group,

( x + 2 )( x + 1 )

This is the same as the denominator of the other fraction, and therefore we can combine the fractions.

x - 1 / ( x + 2 )( x + 1 )

As you can see this is not any of the options present, as we have not expanded ( x + 2 )( x + 1 ). Remember previously that ( x + 2 )( x + 1 ) = x² + 3x + 2. Hence our solution is x - 1 / x² + 3x + 2, or option d.

Answer:

a) Suzette ran for 4 hours

b) Suzette biked for 5 hours

Step-by-step explanation:

Speed is rate of distance traveled, it is the ratio of distance traveled to time taken. It is given by:

Speed = distance / time

The total distance ran and biked by Suzette (d) = 80 miles, while the total time ran and biked by Suzette (t) = 9 hours.

For running:

Her speed was 5 miles per hour, let the total hours Suzette ran be x and the total distance she ran be p, hence since Speed = distance / time, therefore:

5 = p / x

p = 5x

For biking:

Her speed was 12 miles per hour, let the total hours Suzette ran be y and the total distance she ran be q, hence since Speed = distance / time, therefore:

12 = q / y

q = 12y

The total distance ran and biked by Suzette (d) = Distance biked + distance ran

d = p + q

80 = p + q

80 = 5x + 12y                 (1)

The total time taken to run and bike by Suzette (t) = time spent to bike + time spent to run

t = x + y

9 = x + y                         (2)

Solving equation 1 and equation 2, multiply equation 2 by 5 and subtract from equation 1:

7y = 35

y = 35/7

y = 5 hours

Put y = 5 in equation 2:

9 = x + 5

x = 9 -5

x = 4 hours

a) Suzette ran for 4 hours

b) Suzette biked for 5 hours

Answer:

The coordinates of A'C'S'T' are;

A'(-7, 2)

C'(-9, -1)

S'(-7, -4)

T'(-5, -1)

The correct option is;

B

Step-by-step explanation:

The coordinates of the given quadrilateral are;

A(-3, 1)

C(-5, -2)

S(-3, -5)

T(-1, -2)

The required transformation is T₍₋₄, ₁₎ which is equivalent to a movement of 4 units in the leftward direction and 1 unit upward

Therefore, we have;

A(-3, 1) + T₍₋₄, ₁₎ = A'(-7, 2)

C(-5, -2) + T₍₋₄, ₁₎ = C'(-9, -1)

S(-3, -5) + T₍₋₄, ₁₎ = S'(-7, -4)

T(-1, -2) + T₍₋₄, ₁₎ = T'(-5, -1)

Therefore, the correct option is B

[tex]\text{Solve for y:}\\\\-4.1=8(y-5)\\\\\text{Use the distributive property}\\\\-4.1=8y-40\\\\\text{Add 40 to both sides}\\\\35.9=8y\\\\\text{Divide by 8}\\\\\boxed{4.4875=y}\\\\[/tex]

Answer:

The answer to your question is given below.

Step-by-step explanation:

1. f(x) = 5x + 1

x = – 5

f(x) = 5x + 1

f(–5) = 5(–5) + 1

f(–5) = –25 + 1

f(–5) = –24

2. f(x) = 5x + 1

x = – 1

f(x) = 5x + 1

f(–1) = 5(–1) + 1

f(–1) = –5 + 1

f(–1) = – 4

3. f(x) = 5x + 1

x = 1

f(x) = 5x + 1

f(1) = 5(1) + 1

f(1) = 5 + 1

f(1) = 6

4. f(x) = 5x + 1

x = 2

f(x) = 5x + 1

f(2) = 5(2) + 1

f(2) = 10 + 1

f(2) = 11

5. f(x) = 5x + 1

x = 2

f(x) = 5x + 1

f(3) = 5(3) + 1

f(3) = 15 + 1

f(3) = 16

Summary

x >>>>>>>> f(x)

–5 >>>>>> – 24

–1 >>>>>> – 4

1 >>>>>>>> 6

2 >>>>>>> 11

3 >>>>>>> 16

Answer:

358.125

Step-by-step explanation:

Answer:

358 3/24

Step-by-step explanation:  

The vertical line test helps us see that we have a function. Note how it is not possible to draw a single straight line through more than one point on the curve. Any x input leads to exactly one y output. This graph passes the vertical line test. Therefore it is a function.

The function is not one-to-one because the graph fails the horizontal line test. Here it is possible to draw a single straight horizontal line through more than one point on the curve. The horizontal line through y = 2 is one example of many where the graph fails the horizontal line test, meaning the function is not one-to-one.

The term "one-to-one" means that each y value only pairs up with one x value. Here we have something like y = 2 pair up with multiple x values at the same time. This concept is useful when it comes to determining inverse functions.

Answer:

  55°

Step-by-step explanation:

Perhaps you want the measure of angle B. (There is no "b" in the figure.)

That measure is half the measure of the intercepted arc:

  m∠B = 110°/2 = 55°

Angle B is 55°.

Answer:

m = -9

Step-by-step explanation:

2m-10=44+8m

Subtract 2m from each side

2m-2m-10=44+8m-2m

-10 = 44+6m

Subtract 44 from each side

-10-44 = 44-44+6m

-54 = 6m

Divide by 6

-54/6 = 6m/6

-9 = m

Answer:

solve by solving the salvation for equation don't be a slave get educated from what's gave

Answer:

vertex(-3,27)

Step-by-step explanation:

f(x)= x^2+ 6x + 36 ( a=1,b=6,c=36)

V(h,k)

h=-b/2a=-6/2=-3

k=f(-3)=3²+6(-3)+36

f(-3)=9-18+36=27

vertex(-3,27)

Answer:

The correct option is;

f(x) = -2·x² + 5·x - 1

Step-by-step explanation:

Given the points

(-1, -8), (0, -1), (1, 2), we have;

The general quadratic function;

f(x) = a·x² + b·x + c

From the given points, when x = -1, y = -8, which gives

-8 = a·(-1)² + b·(-1) + c = a - b + c

-8 =  a - b + c.....................................(1)

When x = 0, y = -1, which gives;

-1 = a·0² + b·0 + c = c

c = -1.....................................................(2)

When x = 1, y = 2, which gives;

2 = a·1² + b·1 + c = a + b + c...............(3)

Adding equation (1) to (3), gives;

-8 + 2 = a - b + c + a + b + c

-6 = 2·a + 2·c

From equation (2), c = -1, therefore;

-6 = 2·a + 2×(-1)

-2·a  = 2×(-1)+6 = -2 + 6 = 4

-2·a = 4

a = 4/-2 = -2

a = -2

From equation (1), we have;

-8 =  a - b + c = -2 - b - 1 = -3 - b

-8 + 3 = -b

-5 = -b

b = 5

The equation is therefore;

f(x) = -2·x² + 5·x - 1

The correct option is f(x) = -2·x² + 5·x - 1.

Answer:

Line S

Step-by-step explanation:

Answer:

line s

Step-by-step explanation:

coz if you extend both the line (line r and line s )

they will not intersect at any point...

plz let me know if it was helpful to you dude!

Answer:

Step-by-step explanation:

Hello, when you try to find the intersection point(s) you need to solve a system like this one

[tex]\begin{cases} y&= m * x + p }\\ y &= a*x^2 +b*x+c }\end{cases}[/tex]

So, you come up with a polynomial equation like.

[tex]ax^2+bx+c=mx+p\\\\ax^2+(b-m)x+c-p=0[/tex]

And then, we can estimate the discriminant.

[tex]\Delta=(b-m)^2-4*a*(c-p)[/tex]

If [tex]\Delta<0[/tex] there is no real solution, no intersection point.

If [tex]\Delta=0[/tex] there is one intersection point.

If [tex]\Delta>0[/tex] there are two real solutions, so two intersection points.

Hope this helps.

[tex]\frac 1c=\frac1{c_1}+\frac1{c_2} [/tex]

$\frac1c=\frac{c_1+c_2}{c_1c_2}$

$\implies c=\frac{c_1c_2}{c_1+c_2}$

Answer:

? = 4.73

Step-by-step explanation:

Since this is a right triangle we can use trig functions

sin theta = opp / hyp

sin 25 = 2 / ?

? sin 25 = 2

? = 2 / sin 25

? =4.732403166

To the nearest hundredth

? = 4.73

Answer:

3.  Intersect at exactly one point.  ( (2/3), (-2/3) )

Step-by-step explanation:

To make the comparison of these lines easier, let's rewrite the 2nd equation into slope-intercept form, as the 1st equation is in slope-intercept form.

[1] y = 2x - 2

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

[2] 6x + 3y = 2 ==> 3y = 2 - 6x ==> y = -2x + (2/3)

[2] y = -2x + (2/3)

So now that we have both equations in slope-intercept form, we can see that the two equations are both linear, have different slopes, and have different y-intercepts.

Since these equations have both different slopes and different y-intercepts, we know that the lines will cross at least one point.  We can confirm that the lines only cross at a single point using the fact that both equations are linear, meaning there will only be one point of crossing.  To find that point, we can simply set the equations equal to each other.

y = 2x - 2

y = -2x + (2/3)

2x - 2 = -2x + (2/3)

4x = (8/3)

x = (8/12) = (2/3)

And plug this x value back into one of the equations:

y = 2x - 2

y = 2(2/3) - 2

y = (4/3) - (6/3)

y = (-2/3)

Thus these lines only cross at the point ( (2/3), (-2/3) ).

Cheers.

Answer:

I don't understand the question

Answer:

d= 6

r= 6/2

r=3

V= π. r². h

V= π . 3². 14

V= π. 9 . 14

V= π 126 cm³

V= 126 π cm³ (π not in number)

hope it helps^°^

Answer:if you use the formula it is 126 pi cm cubed

The answer is c

Step-by-step explanation:

Answer:

last option

Step-by-step explanation:

Let's call the original angle x° and the radius of the circle y. The area of the original sector would be x / 360 * πy². The new angle, which is a 40% increase from x, can be represented as 1.4x so the area of the new sector is 1.4x / 360 * πy². Now, to find the corresponding change, we can calculate 1.4x / 360 * πy² ÷  x / 360 * πy² = (1.4x / 360 * πy²) * (360 * πy² / x). 360 * πy² cancels out so we're left with 1.4x / x which becomes 1.4, signifying that the area of the sector increases by 40%.

Answer:

110° and 70°

Step-by-step explanation:

The angles are supplementary, thus sum to 180°

sum the parts of the ratio, 11 + 7 = 18

divide 180° by 18 to find the value of one part of the ratio

180° ÷ 18 = 10° ← value of 1 part of the ratio

Thus

11 parts = 11× 10° = 110°

7 parts = 7 × 10° = 70°

The angles are 110° and 70°

Answer:

See explanation

Step-by-step explanation:

We have to prove the identity

[tex]tan(\frac{\Theta }{2})=\frac{sin\Theta}{1+cos\Theta }[/tex]

We will take right hand side of the identity

[tex]\frac{sin\Theta}{1+cos\Theta}=\frac{2sin(\frac{\Theta }{2})cos(\frac{\Theta }{2})}{1+[2cos^{2}(\frac{\Theta }{2})-1]}[/tex]

[tex]=\frac{2sin(\frac{\Theta }{2})cos(\frac{\Theta }{2})}{2cos^{2}(\frac{\Theta }{2})}=\frac{sin(\frac{\Theta }{2})}{cos(\frac{\Theta }{2})}[/tex]

[tex]=tan(\frac{\Theta }{2})[/tex] [ Tan θ will be positive since θ lies in 1st quadrant ]

Answer:

1. 8y - 9y = -1y

( 8 - 9 = -1)

3. 8a - 6 +a - 1

( i have showed the like terms here)

8a - 1a= 7a

-6 - 1 = -7

7a - 7

5. -x - 2 + 15x

( i have showed the like terms here)

-x + 15x = 14x

(x = 1)

14x + 2

7.  8d - 4 - d - 2

( i have showed the like terms here)

8d - d = 7d

-4 -2 = -6

7d - 6

8. 9a + 8 - 2a - 3 - 5a

( i have showed the like terms here)

9a - 2a - 5a = 2a

8 - 3= 5

2a + 5

Answer:

[tex]D)\ x + y = 109\\(1.5)x + (3.5)y = 227.50[/tex]

Step-by-step explanation:

Let the items sold with price $1.5 = [tex]x[/tex]

Let the items sold with price $3.5 = [tex]y[/tex]

Initially, total number of items = 150

Items left at the end of the day = 41

So, number of items sold throughout the day = Total number of items - Number of items left

Number of Items sold = 150 - 41 = 109

So, the first equation can be written as:

[tex]\bold{x+y = 109} ....... (1)[/tex]

Now, let us calculate the sales done by each item.

Sales from item with price $1.5 = Number of items sold [tex]\times[/tex] price of each item

= (1.5)[tex]x[/tex]

Sales from item with price $3.5 = Number of items sold [tex]\times[/tex] price of each item

= (3.5)[tex]y[/tex]

Total sales = [tex]\bold{(1.5)x+(3.5)y = 227.50} ....... (2)[/tex]

So, the correct answer is:

[tex]D)\ x + y = 109\\(1.5)x + (3.5)y = 227.50[/tex]

Answer:

360

Step-by-step explanation:

Answer:

perimeter is  4 sqrt(29) + 4pi  cm

area is 40 + 8pi cm^2

Step-by-step explanation:

We have a semicircle and a triangle

First the semicircle with diameter 8

A = 1/2 pi r^2 for a semicircle

r = d/2 = 8/2 =4

A = 1/2 pi ( 4)^2

  =1/2 pi *16

  = 8pi

Now the triangle with base 8 and height 10

A = 1/2 bh

  =1/2 8*10

  = 40

Add the areas together

A = 40 + 8pi cm^2

Now the perimeter

We have 1/2 of the circumference

1/2 C =1/2 pi *d

         = 1/2 pi 8

        = 4pi

Now we need to find the length of the hypotenuse of the right triangles

using the pythagorean theorem

a^2+b^2 = c^2

The base is 4 ( 1/2 of the diameter) and the height is 10

4^2 + 10 ^2 = c^2

16 + 100 = c^2

116 = c^2

sqrt(116) = c

2 sqrt(29) = c

Each hypotenuse is the same so we have

hypotenuse + hypotenuse + 1/2 circumference

2 sqrt(29) + 2 sqrt(29) + 4 pi

4 sqrt(29) + 4pi  cm

Step-by-step explanation:

First we need to deal with the half circle. The radius of this circle is 4, because the diameter is 8. The formula for the circumference of a circle is 2piR.

2pi4 so the perimeter for the half circle would be 8pi/2.

The area of that half circle would be piR^2 so 16pi/2.

Now moving on the triangle part, we need to find the hypotenuse side of AC. We will use the pythagoram theorem. 4^2+10^2=C^2

16+100=C^2

116=C^2

C=sqrt(116)

making the perimeter of this triangle 2×sqrt(116)

The area of this triangle is 8×10=80, than divided by 2 which is equal to 40.

We than just need to add up the perimeters and areas for both the half circle and triangle.

The area would be equal to 8pi+40

The perimeter would be equal to 4pi+4(sqrt(29))