there was a substance of a 450 milligrams of radioactive substance to start a study. Since then the sample has decayed of 5.4% each year. Let t be the number of years since the start of the study. Let Y be the mass of the sample in milligrams. Write an exponential function showing the relationship of the y and t

Answer: A

Step-by-step explanation:

[tex]\overline{AB}[/tex] has a length of 6

[tex]\overline{A'B'}[/tex] has a length of 6 + 12 = 18, which is 3 times the length of [tex]\overline{AB}[/tex]

B = B' so this will be the center point

⇒ center B and a scale factor of 3

Answer:

Option A is the answer.

Step-by-step explanation:

Triangle A'B'C' is a dilation of triangle ABC.

We can apply the formula to get the scale factor as

Scale factor = [tex]\frac{\text{One side of triangle A'B'C'}}{\text{Corresponding side of triangle ABC}}[/tex]

= [tex]\frac{A'B'}{AB}[/tex]

= [tex]\frac{AB'+AA'}{AB}[/tex]

= [tex]\frac{6+12}{6}[/tex]

= [tex]\frac{18}{6}[/tex]

= 3

Therefore, ΔABC has been dilated with a scale factor of 3 about the point B as the center.

Option A is the answer.

Answer:

7x + 9

Yes they are equivalent

Step-by-step explanation:

The first expression given to us is

2x + 3 + 5x + 6

Let us call it equation (i)

Now we have to find equivalent expression to it

the given equation is 2x + 3 + 5x +6

similar terms are terms involving x and not involving x

so solving the equation gives

2x + 5x + 3 +6

= 7x + 9

Now we have two equations one is

2x + 3 +5x + 6              ...............(i)

Other is

7x + 9                            ..............(ii)

To check they are equivalent or not

Put x = 3 in equation (i)

2(3)+3+5(3)+6= 6 + 3 + 15 + 6

                      =30

Now put x=3 in second equation

7(3)+9=21 + 9

          =30

As the answer of both the equations are 30 so they are equivalent

To write an equivalent expression to 2x + 3 + 5x + 6 by combining like terms, the expression can be simplified to 7x + 9. Substituting x with 3 in both expressions shows that they have the same value, which is 30.

To write an equivalent expression to 2x + 3 + 5x + 6 by combining like terms, we add the coefficients of the like terms. In this case, the like terms are the terms with the same variable, which is x. So, 2x + 3 + 5x + 6 simplifies to 7x + 9. To show that this expression is equivalent to the original expression, we can substitute x with 3. So, when we replace x with 3, both expressions 2x + 3 + 5x + 6 and 7x + 9 result in the same value, which is 30.

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Answer:g hvt

Step-by-step explanation:

Answer:

<T = 83

Step-by-step explanation:

In a parallelogram  <M = < P   and < N = <T  (opposite angles are congruent)

We also know that <M + <N = 180  ( consecutive angles are supplementary)

<M + <N = 180

6x+10  + 5x+10.5 = 180

Combine like terms

11x+20.5 = 180

Subtract 20.5 from each side

11x +20.5-20.5 = 180-20.5

11x = 159.5

Divide each side by 11

11x/11 = 159.5/11

x=14.5

We can find <N

<N = 5x+10.5

    = 5(14.5)+105

   =72.5 +10.5

   = 83

<N = <T = 83

Answer:

$1808263    

Step-by-step explanation:

We're given with the below information:-

Initial price of the building = Principal amount (P) = $590000

Rate of interest (r) = 14% = 0.14

Time in years (t) = 2020 - 2012 = 8

The formula to calculate the final amount where interest is continuously  compounded each year is :-

[tex]A = Pe^{rt}[/tex]                              [e is the mathematical const = 2.71828]

Plugging in the values of P, e, r, and t in the above formula, we get

[tex]A = 590000*2.71828^{0.14*8}[/tex]

=> A= 1808262.6

=> A = $1808263     (rounded off to the nearest whole dollar)

So, she expect to sell the building in 2020 for an amount of $1808263

Answer:\

180000

Step-by-step explanatii took

Answer:

The period is [tex]\pi[/tex].

Step-by-step explanation:

The answer is determined by finding the length of 1 cycle of the function. Trig functions sine and cosine have a wavy form. One starts and ends int he middle - sine. The other starts above and stops above- cosine. Regardless of the function, the period will be the same because the period asks "how fast does this function complete 1 wave?"

If cosine we start at one of the top points, trace the graph till we reach the same y-value again. The period is the time or difference in x-values. For instance if I started at the x-value [tex]\frac{\pi }{2}[/tex] and ended at [tex]\frac{3\pi }{2}[/tex] then I'd subtract them.

[tex]\frac{3\pi }{2}[/tex]-[tex]\frac{\pi }{2}[/tex]=[tex]\frac{2\pi }{2}[/tex]=[tex]\pi[/tex]

The period is [tex]\pi[/tex].

Answer:

Option d is correct.

Domain = all real number

range = positive real numbers

Step-by-step explanation:

Given the function: [tex]y=f(x) = a^x[/tex]

Domain of the function is all real numbers except where the expression is undefined.

In this case, there is no real number that makes the expression undefined.

Domain of f(x) = [tex](-\infty , \infty)[/tex] = {a | a∈R}

Range is the set of all valid  f(x) values.

[tex](0,1) \cup (1, \infty)[/tex]

[tex]\{y | y\neq 1 , y>0\}[/tex]

Therefore, Domain is all real number and the range of function is positive real number

Answer:

1.  16

2.  -4

Step-by-step explanation:

First factor the expression

lim x→1  (x^3 + 5x^2 + 3x-9)/(x-1)

lim x→1  (x-1) (x+3)^2/(x-1)

Canceling the x-1  in the top and bottom

lim x→1   (x+3)^2

Let x=1

lim x→1   (1+3)^2 = 4^2 = 16

2.  lim x→0 (x^2 -6x+8) /(x-2)          

First factor the expression

    lim x→0 (x-4) (x-2) /(x-2)        

Canceling the x-2  in the top and bottom

lim x→0 (x-4)

Let x=0

lim x→0   (0-4) = -4

Filling in the table

x     -.1      -.01       -.001     .001     .01     .1

f(x) -4.1  -4.01    -4.001   -3.999   -3.99  -3.9

I know this isn't helpful, but I REALLY need help on this question...

Given: f(x) = x^2 + 7x + 10. From the Rational Root Theorem, we know p=10 and q=1. List all potential roots using the ratio p/q.

Answer:

7.15x + 3.55y + 20 = weekly income

Step-by-step explanation:

7.15x + 3.55y + 20 = weekly income

Answer:

The experimental probability is 7/12 or 58.3%

Step-by-step explanation:

In order to find the experimental probability, we just look at the amount of times it happened in our test. The fact that the actual probability is 50% does not matter when looking at experimental numbers.

Answer:

If the shape is a square,then it is a rectangle.

Step-by-step explanation:

The hypothesis of the statement is the one which follows after if,

If the shape is a square.

The conclusion of the statement is the sentence that follows after then,

Then it is a rectangle.

The statement 'All squares are rectangles' translates into an if-then form as 'If a shape is a square, then it is a rectangle'. The hypothesis is 'a shape is a square' and the conclusion is 'it is a rectangle'.

The geometrical statement 'All squares are rectangles' can be translated into an if-then form as follows: 'If a shape is a square, then it is a rectangle'. In this statement, the hypothesis is 'a shape is a square' and the conclusion is 'it is a rectangle'. This means that if we start with a shape and it happens to be a square (which is our condition or hypothesis), then it must also necessarily be a rectangle (which is our end result or conclusion). This is because all the properties of a rectangle - having all angles being 90 degrees and opposite sides being equal - are also properties of a square.

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Answer:

First I'm not exactly sure what you're asking but the equation for slope is:

Y2-Y1/X2-X1=M

Step-by-step explanation:

M is your slope

X1 and Y1 represent your first point

X2 and Y2 represent your second point

To translate you are finding the rate of change (slope) between two points

Answer: Slope = ∆y/∆x = y2-y1/x2-x1

Step-by-step explanation:

The slope of a line which is also known as the gradient can be gotten by taking the ratio of the change in coordinate of y-axis to the x-axis of the line.

Change in y axis = y2-y1

Where y2 and y1 are final length and initial length respectively on the y axis

Change in x axis = x2-x1

Where x2 and x1 are final length and initial length respectively on the x axis.

The slope of the line therefore gives;

Slope = ∆y/∆x = y2-y1/x2-x1

Answer:

(f + g)(x) = 5x^2 + 4x + 6

Step-by-step explanation:

The given functions are f(x) = 4x + 6 and g(x) = 5x^2

(f + g)(x) = f(x) + g(x)

= 4x + 6 + 5x^2

= 5x^2 + 4x + 6

Thank you.

C it the answer because since heads hit 14 times it haves a higher percent of hitting heads again.

Answer:

x=6

Step-by-step explanation:

If ABCD is a parallelogram, then AB = CD

AB=CD

10x-15 = 6x+9

Subtract 6x from each side

10x-6x -15 = 6x-6x+9

4x-15 = 9

Add 15 to each side

4x-15+15 = 9+15

4x = 24

Divide by 4 on each side

4x/4 =24/4

x=6

Look at the picture.

The equation:

10x - 15 = 6x + 9       add 15 to both sides

10x = 6x + 24      subtract 6x from both sides

4x = 24    divide both sides by 4

2. CD = CD , Reflexive Prop.

3. BC = DE, Subtraction Prop.

Answer:

In quadrilateral LMNO, LO ∥ MN.

ML ∥ NO

LO ≅ MN

Answer:

1 & 3 is correct on edg.

For [tex]f[/tex] to be differentiable, it must be continuous, so we need to have

[tex]\displaystyle\lim_{x\to3^-}f(x)=\lim_{x\to3^+}f(x)=f(3)[/tex]

By its definition, [tex]f(3)=3^2=9[/tex]. The one-sided limits are

[tex]\displaystyle\lim_{x\to3^-}f(x)=\lim_{x\to3}x^2=9[/tex]

[tex]\displaystyle\lim_{x\to3^+}f(x)=\lim_{x\to3}mx+b=3m+b[/tex]

so we require [tex]3m+b=9[/tex].

In order for [tex]f[/tex] to be differentiable at [tex]x=3[/tex], we also need to have [tex]f'(3)[/tex] exist, which requires that [tex]f'[/tex] also be continuous at [tex]x=3[/tex]. First, compute the derivatives of all pieces of [tex]f[/tex]:

[tex]f'(x)=\begin{cases}2x&\text{for }x<3\\?&\text{for }x=3\\m&\text{for }x>3\end{cases}[/tex]

[tex]f'[/tex] is continuous at [tex]x=3[/tex] if

[tex]\displaystyle\lim_{x\to3^-}f'(x)=\lim_{x\to3^+}f'(x)=f'(3)[/tex]

The one-side limits are

[tex]\displaystyle\lim_{x\to3^-}f'(x)=\lim_{x\to3}2x=6[/tex]

[tex]\displaystyle\lim_{x\to3^+}f'(x)=\lim_{x\to3}m=m[/tex]

so we need to have [tex]m=6[/tex], and moreover [tex]f[/tex] will be differentiable if we set [tex]f'(3)=6[/tex].

So with [tex]m=6[/tex], we must have [tex]3m+b=9\implies b=-9[/tex].

For the piecewise function f(x) = x^2 when x <= 3 and mx + b when x > 3, values m = 6 and b = -9 ensure differentiability at x = 3.

To make the piecewise function f(x) differentiable at x = 3, we need the two pieces, x^2 and mx + b, to smoothly connect at x = 3. This requires the values of m and b to ensure continuity of both function values and derivatives.

First, evaluate both pieces at x = 3:

x^2 when x <= 3 and mx + b when x > 3

For continuity, set these expressions equal to each other:

3^2 = m * 3 + b

This yields 9 = 3m + b. To ensure differentiability, the derivatives of both pieces must also match at x = 3:

f'(x) = 2x when x <= 3 and f'(x) = m when x > 3

For continuity of derivatives, set the derivatives equal to each other at x = 3:

2 * 3 = m

This gives m = 6. Substituting m = 6 into the continuity equation 9 = 3m + b gives b = -9.

Therefore, the values m = 6 and b = -9 make the piecewise function f(x) differentiable at x = 3.

For more such information on: piecewise function

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Answer:

The first one, because the x values are not repeated in a data set.

Answer:

A

Step-by-step explanation:

given that y varies directly with x then the equation relating them is

y = kx ← k is the constant of variation

to find k use the given condition y = 2 when x = 10

k = [tex]\frac{y}{x}[/tex] = [tex]\frac{2}{10}[/tex] = [tex]\frac{1}{5}[/tex]

y = [tex]\frac{1}{5}[/tex] x ← equation of variation

when x = 40, then

y = [tex]\frac{1}{5}[/tex] × 40 = 8 → A

The value of y when x = 40 with the same proportion will be 8 thus option (A) is correct.

The ratio is the division of the two numbers.

For example, a/b, where a will be the numerator and b will be the denominator.

Proportion is the relation of a variable with another. It could be direct or inverse.

As per the given,

If y varies directly with x.

y ∝ x

y = kx

k = y/x

Thus, k will be the same for both conditions.

2/10 = y/40

1/5 = y/40

40/5 = y

8 = y

Hence "The value of y when x = 40 with the same proportion will be 8 ".

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Answer: Correct Option is "C"

( - ∞ , 3) U (3, ∞ )

Step-by-step explanation:

The function f/g is defined as

( x + 8) / (x -3)

Domain of the function is the set of values which the independent variable can assume.

Clearly in the above function x cannot assume the value 3, otherwise the function would become undefined.

So domain of the function is

( - ∞ , 3) U (3, ∞ )

Hope it helps.

Thank you.

Answer: 2x + y

Step-by-step explanation:

logₐ(3) = x

logₐ(5) = y

logₐ(45) = logₐ(3²· 5)

             = logₐ(3)² + logₐ(5)

             = 2 logₐ(3) + logₐ(5)

             = 2     x      +   y        substituted given values (stated above)

Step-by-step explanation:

Here we make use of the laws of logarithms:

log_a(PQ) = log_a(P)+log_a(Q)

which implies the following corollary

log_a(P^2) = log_a(P)+log_a(P) = 2log_a(P)

Notice how the log of a product is reduced to the sum of the log of the factors.  (Advantage is taken of this fact in the use of logarithm tables before the wide-spread use of electronic calculators (pre-70's) )

So substituting

x=log_a(3)

y=log_a(5)

we have

log_a(45) = log_a(3^2 * 5) = log_a(3^2) + log_a(5)=2log_a(3)+log_a(5)

=2x+y

Answer:

A

Step-by-step explanation:

Your first step is to figure out what f(x) is. You should start with x = 0. f(x) has a value of - 4.  That only tells you that whatever x is or what it is coupled with, the constant term is - 4.

What you know so far is that y = x something - 4

Now go to work on some of the other numbers. It really doesn't look linear so don't try it.

What happens when x = 16? Somehow f(x) winds up being 0. What can cause that?

First of all it can't be y = ax - 4. We've already establish the constant must be  - 4

What about y = ax^2 - 4

That cannot be either. y = 16^2*a - 4  ?? y = 256ax  - 4. No other value will fit the bill.

By looking at g(x) you get the idea that y = a*sqrt(x) - 4 might have something to do with f(x). The most obvious value for a = 1, so try it first.

Now try some of the other values.

And that's what the table says.

The red graph is y = sqrt(x) - 4

The blue graph is y = 4*sqrt(x) - 8

Answer

It looks like A is the answer.

g(x) has a y intercept of - 8

f(x) has a y intercept of - 4

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

The two intercepts are not equal. If they were they would start from the same point.   B is incorrect

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

C is wrong. the x intercepts are distinct points. One is 4 and the other is 16 I think.

0 = sqrt(x) - 4

4 =  sqrt(x)                Square both sides.

16 = x                         This is the x intercept of the red line

The other intercept = 4 (Blue line). I'll leave you to work it out. Leave a not if you can't.

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

A and D can't  both be true. Since A is true, D can't be.

Adam has 6,000 points.

Let's use algebra to solve this problem. Let's assume that Adam's points are represented by 'x'. According to the problem, Nick has 20% more points than Adam, so Nick's points can be represented as 'x + 20% of x' or '1.2x'. Given that Nick has 7,200 points, we can set up the equation 1.2x = 7,200 and solve for 'x'. Divide both sides of the equation by 1.2 to isolate 'x' and find that Adam has 6,000 points.

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Answer:

P =44 + 4xy

Step-by-step explanation:

To find the perimeter of a triangle, add up the three sides

P = 29+15+4xy

P =44 + 4xy