The table of values below represent an exponential function. Write an exponential equation that models the data.


x y

-2 21

-1 14.7

0 10.29

1 7.203

2 5.0421


a. y = 21(0.7) b. y = 14.7(1.7) c. y = 10.29(0.7) d. y = 10.29(1.3)

Answer: c

Answer:

0.48

Step-by-step explanation:

The problem at hand

[tex]500 = (2)(10^{5x})[/tex]

First divide out 2 on both side of the = sign

[tex]500 = (2)(10^{5x})[/tex]

[tex]\frac{500}{2} = \frac{(2)(10^{5x})}{2}[/tex]

[tex]250 = 10^{5x}[/tex]

Now take the base 10 log on each side of the = sign

[tex]\log_{10} (250) = \log_{10} (10^{5x})[/tex]

Now move the exponent to the left like so

[tex]\log_{10} (250) = 5x \log_{10} (10)[/tex]

Rule you need to know. [tex]\log_a (a) = 1[/tex]

[tex]\log_{10} (250) = 5x (\log_{10} (10))[/tex]

[tex]\log_{10} (250) = 5x(1)[/tex]

[tex]\log_{10} (250) = 5x[/tex]

Now divide 5 from both sides of the equation

[tex]\frac{\log_{10} (250)}{5} = \frac{5x}{5}[/tex]

[tex]\frac{\log_{10} (250)}{5} = x[/tex]

[tex]x = \frac{\log_{10} (250)}{5}[/tex]

[tex]x = 0.4795880017[/tex]

Round to nearest hundredths

[tex]x = 0.48[/tex]

Side Note:

You could approach this problem in a different manner.

For instance, we could have done the following

[tex]x = \frac{\log_{10} (250)}{5(\log_{10} (10))}[/tex]

Also, you did not have to use log with a base 10. You could have used the natural log, which is [tex]\ln_e (x)[/tex]

[tex]x = \frac{\ln_{e} (250)}{5(\ln_{e} (10))}[/tex]

Solving the equation gives 0.48 when rounded to the nearest hundredth.

To solve the equation [tex]500 = (2)(105x)[/tex] , we will first isolate the exponential term. The steps are as follows:

Divide both sides of the equation by 2 to get 250 = 105x.

Take the common logarithm (log) of both sides of the equation to get [tex]log(250) = log(105x).[/tex]

Using logarithmic properties, we have log(250) = 5x log(10)

Now we simplify further since log(10) is 1, so log(250) = 5x.

Divide both sides by 5 to solve for x, which gives [tex]x = log(250) / 5.[/tex]

Using a calculator with a LOG button, we find log(250) approximately equals 2.39794 and, therefore, x is approximately 0.47959 when rounded to five decimal places.

However, we are asked to round x to the nearest hundredth, so the final answer is x = 0.48.

Answer:

Option D

Step-by-step explanation:

Its is the last option because all the others fail the vertical line test. The vertical line test says that if you can draw a vertical line through the graph which intersects it at more than 1 point then its NOT a function. For example Option A, a circle, the  vertical axis passes through the graph  at 2 points.

Answer:

D. Option B: Coordinate grid with graph of a curve that passes through point (zero, zero) and is in the first and third quadrants

Step-by-step explanation:

The vertical line test says if a line going straight up and down passes through more than one point of a relation, than the graph is not  a function.

A. Option C: Coordinate grid with graph of a circle centered at point (zero, zero)

A circle fails the vertical line test so it cannot be a function

B. Option D: Coordinate grid with graph of a curve that passes through point (zero, zero) and is in the first and fourth quadrants

This will fail the vertical line test.  The first and fourth quadrants are above each other.

C. Option A: Coordinate grid with graph of vertical line at x equals three

A vertical line will fail the vertical line test

D. Option B: Coordinate grid with graph of a curve that passes through point (zero, zero) and is in the first and third quadrants

This will pass the vertical line test.  The first and third quadrants are diagonal from each other.

Answer:

D

Step-by-step explanation: 153 divided by 3 = 51

Answer:

D. 153

Step-by-step explanation:

Which number is divisible by 3?

A number is divisible by 3 if you add all the digits and it is divisible by e.

A. 35

3+5 = 8

8/3  has a remainder so it is not divisible by 3

B. 133

1+3+3 = 7

7/3 has a remainder so it is not divisible by 3

C. 103

1+0+3 =4

4/3 has a remainder so it is not divisible by 3

D. 153

1+5+3 =9

9/3 =3  153 is divisible by 3

1. Area of the original patio: 35 ft^2

The original patio has is a rectangle with length = 7 feet and width = 5 feet. The area of a rectangle is given by the product between length and width:

[tex]A=L \cdot W[/tex]

Therefore, since in this case L=7 and W=5, the area of the original patio is

[tex]A=(7 ft)(5 ft)=35 ft^2[/tex]

2. Area of section A: 7x ft^2

Section A is also a rectangle, with length = 7 feet and width = x. Therefore, the area of this section is equal to:

[tex]A=L\cdot W=(7 feet)(x)=7x[/tex]

3. Area of section B: 5x ft^2

Section B is also a rectangle, with length = x and width = 5 feet. Therefore, the area of this section is equal to:

[tex]A=L\cdot W=(x)(5 feet)=5x[/tex]

4. Area of section C:  [tex]x^2 ft^2[/tex]

Section C is a square, with side equal to x. The area of a square is equal to the square of the length of the side:

[tex]A=L^2[/tex]

therefore, in this case, since L = x, the area of this section is

[tex]A=(x)^2 = x^2[/tex]

5. Total area of the new patio using addition: [tex]x^2 +12x+35[/tex] ft^2

The total area of the new patio is equal to the sum of the four areas calculated in the previous sections:

[tex]A=35 +7x +5x+x^2 = x^2 +12x+35[/tex] ft^2

6. Total area of the new patio using multiplication: [tex]x^2+12x+35[/tex]

The total area of the new patio is equal to the product between the length (7+x) and the width (5+x):

[tex]A=(7+x)(5+x)=35+7x+5x+x^2=x^2+12x+35[/tex] ft^2

7. Yes

As we can see by comparing the area calculated in 5. and the area calculated in 6., the two areas are equal.

8. 80 ft^2

We already have the formula for the area of the new patio:

[tex]A=x^2+12x+35[/tex]

If we substitute x=3, we find the value of the area:

[tex]A=(3)^2+12\cdot 3+35=9+36+35=80[/tex]

The axis of symmetry is x=-3 and the minimum or maximum y value is 5.

The vertex is the extreme point of the quadratic function. The graph is left/right symmetrical about the vertex, so the x-value defines the axis of symmetry. The y-value is the extreme, the maximum or minimum.

_____

Comment on the attachment

The graph shows two quadratic functions (red, blue), each with its vertex at (-3, 5). You can see that the line x=-3 is the axis of symmetry of each of them. You can also see that y=5 is the extreme value of the function (maximum or minimum).

Answer:

39

Step-by-step explanation:

Use the Pythagorean theorem! So 15^2+36^2=x^2. So x^2=1521 or x=39

Answer:

√5

Step-by-step explanation:

We suppose the vertices are named clockwise around the top of the cube, then clockwise around the bottom (looking down from above the cube), with vertex E below vertex D. Then line AD is in plane ADEF, and line BM is in plane BCHG.

The distance between the named parallel planes is the distance between the lines. That distance is AB, which is given as √5.

_____

A diagram helps.

Answer: t < 4  hours

Step-by-step explanation:

Renting = $37 + t x $5.70

If $37 + t$5.70 < $59.80

t$5.70 < $59.80 - $37  

t$5.70 < $22.80

t < $22.80/$5.70  

t < 4 hours

[tex]\textit{\textbf{Spymore}}[/tex

x = 77.56 cm

y = 69.71 cm

The mnemonic SOH CAH TOA reminds you that ...

... Cos = Adjacent/Hypotenuse

so ...

... cos(64°) = (34 cm)/x

... x = (34 cm)/cos(64°) ≈ 77.56 cm

___

It also reminds you ...

... Tan = Opposite/Adjacent

so ...

... tan(64°) = y/(34 cm)

... (34 cm)·tan(64°) = y ≈ 69.71 cm

Answer:

C) 115 feet

Step-by-step explanation:

Assuming your setup is correct and the question is asking the height, given that the length of the slant rope is 150 ft, you find x by multiplying the equation by 150:

... 150*sin(50°) = x ≈ 115 ft

Almost any on-line calculator will compute this value. If you don't specifiy the angle is degrees, you will need to verify that it is properly interpreted. (It may be interpreted as radians. Different calculators make different assumptions.)

The Google or Bing search boxes are pretty reliable calculators. Usually * is suitable as the multiplication indicator, and / works for division. They follow the order of operations rigorously, so parentheses are needed for numerators, denominators, and exponents when more than a single number is involved.

_____

There are on-line equation solvers, too. I've found them to be more fussy and less useful. One likes to give irrational results as the ratio of two (large) integers, for example.

Answer:

115 feet

Step-by-step explanation:

Given : A hot-air balloon is tied to the ground with two taut ropes. One rope is directly under the balloon and makes a right angle with the ground. The other rope forms an angle of 50º with the ground.

To Find: What is the height x, to the nearest foot, of the balloon?

Solution:

We are given a set up :

[tex]sin 50 ^{\circ}=\frac{x}{150}[/tex]

Using scientific calculator find the value of sin 50

0.76604

[tex]0.76604=\frac{x}{150}[/tex]

[tex]0.76604 \times 150=x[/tex]

[tex]114.906=x[/tex]

Hence the height of the balloon is 115 feet.

Option C is correct.

Answer:

2x

Step-by-step explanation:

You want the sum ...

... Jamie's age + Ana's age

... = x + x

... = 2x

_____

Comment on answer form

For this sort of problem, it isn't clear whether the preferred answer is the sum (x+x) or the simplified sum (2x).

Answer:

2 + 3i, midpoint is (2,3)

Step-by-step explanation:

we need to find the midpoint between (-1+9i) and B=(5-3i)

To find the midpoint of two points (a+bi)  and (c+di) in a complex plane,  

we apply formula

[tex]\frac{a+c}{2} + \frac{b+d}{2} i[/tex]

A = (-1+9i) and B=(5-3i)

Midpoint for AB is

[tex]\frac{-1+5}{2} + \frac{9+(-3)}{2}i[/tex]

[tex]\frac{4}{2} + \frac{6}{2}i[/tex]

2 + 3i , so midpoint is (2,3)

Answer:

Step-by-step explanation:

The specified set is fairly limited in size, so we can simply check all the choices and see which works:

For n ∈ {1, 2, 3, 4}

... 4n ∈ {4, 8, 12, 16}

Of these values, only the first three {4, 8 12} are less than 16. (16 is equal to 16, not less than 16.)

The corresponding values of n are {1, 2, 3}.

Answer:

C. 26 + 11 - S

Step-by-step explanation:

She had 26 shells

She collected 11 more shells ( that is addition)

26 + 11

Then she gave S shells away ( that is subtraction)

26+11-S

She has 26+11-S

Answer:

C. 26 + 11 - S

Step-by-step explanation:

Answer:

50

Step-by-step explanation:

The length of the midsegment is half the length of the base segment, so you have ...

... 2(6x +2) = 2x +84

... 10x = 80 . . . . . . . . . simplify, subtract 2x+4

... x = 8

The length of the midsegment is 6·8+2 = 50.

1 27/28 ≈ 1.964 gallons/hour

You want gallons in the numerator of your unit rate, but that unit is in the denominator of the mileage rate. So, the computation must involve division by 28 mpg. Hours is already in the denominator of 55 mph, so the computation will involve multiplication by that rate.

... (55 mi/h)/(28 mi/gal) = (55 mi/h)·(1 gal/(28 mi)) = 55/28 gal/h

... = 1 27/28 gal/h

Final answer:

To calculate the gas usage per hour, divide the speed of 55 mph by the car's fuel efficiency of 28 mpg, resulting in approximately 1.9643 gallons of gas used per hour.

Explanation:

To determine the amount of gas used every hour by a car that gets 28 miles per gallon (mpg) and is travelling at 55 miles per hour (mph), we can use unit analysis as follows:

Therefore, the car uses about 1.9643 gallons of gasoline per hour when driving at a constant speed of 55 mph.

The confidence interval for confidence level of [tex]1-\alpha[/tex] is

[tex]\left(\overline x-Z_{\alpha/2}\dfrac\sigma{\sqrt n},\overline x+Z_{\alpha/2}\dfrac\sigma{\sqrt n}\right)[/tex]

where [tex]\overline x[/tex] is the sample mean, [tex]Z_{\alpha/2}[/tex] is the critical value for the given confidence level, [tex]\sigma[/tex] is the standard deviation of the population, and [tex]n[/tex] is the sample size. The margin of error is the [tex]Z_{\alpha/2}\dfrac\sigma{\sqrt n}[/tex] term.

a) For a confidence level of [tex]1-\alpha=0.90[/tex], we have [tex]Z_{\alpha/2}=Z_{0.05}\approx1.64[/tex]. So in order to have a margin of error of at most 25 points, we have

[tex]1.64\dfrac{250}{\sqrt n}=25\implies n\approx268.96[/tex]

so Raina should collect a sample of at least 269 students.

b) A confidence interval with a higher confidence level would more closely approximate and reflect the population, so it stands to reason that Luke should collect a larger sample than Raina to meet his 99% confidence spec.

c) For a confidence level of [tex]1-\alpha=0.99[/tex], we have [tex]Z_{\alpha/2}=Z_{0.005}\approx2.58[/tex]. Then the margin of error would at most satisfy

[tex]2.58\dfrac{250}{\sqrt n}=25\implies n\approx665.64[/tex]

so that Luke should collect a sample of at least 666 students.

To calculate the sample size required for a given margin of error in a confidence interval, you can use the formula: n = (z * σ) / E. For Raina's 90% confidence interval with a margin of error of 25, she should collect a sample size of 17. Luke's sample size for a 99% confidence interval should be larger than Raina's, and he should collect a minimum sample size of 26.

To calculate the sample size required for a given margin of error in a confidence interval, you can use the formula:

n = (z * σ) / E

Where:
n = sample size
z = z-score corresponding to the desired confidence level
σ = standard deviation of the population
E = margin of error

(a) To find the sample size for Raina's 90% confidence interval with a margin of error of 25, we plug the values into the formula:

n = (1.645 * 250) / 25 = 16.45

Since sample sizes must be whole numbers, Raina should collect a sample size of 17.

(b) Luke's 99% confidence interval will require a larger sample size because the z-score for 99% confidence is larger than for 90% confidence. Therefore, without calculating the actual sample size, we can determine that Luke's sample size should be larger than Raina's.

(c) To calculate the minimum required sample size for Luke's 99% confidence interval, we use the same formula and plug in the values:

n = (2.576 * 250) / 25 = 25.76

Rounding up, Luke should collect a minimum sample size of 26.

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Answer:

225 people.

Step-by-step explanation:

It is 225 people because if you imagine that those 250 people are all made up of groups of ten, there are 25 groups. One person in each group does not prefer David Bright's and there are 25 groups so there are 25 people in the group of 250 people that do not prefer David Bright's. 250-25=225 people.

Answer:

The correct option is D.

Step-by-step explanation:

It is given that events X and Y are two independent events.

Two events are called independent events if the occurrence of one does not affect the probability other. It means

[tex]P(X\text{ and }Y)=P(X)\cdot P(Y)[/tex]

[tex]P(X\cap Y)=P(X)\cdot P(Y)[/tex]          ..... (1)

Therefore option C is correct.

We know that

[tex]P(X|Y)=\frac{P(X\cap Y)}{P(Y)}[/tex]

Using equation (1),

[tex]P(X|Y)=\frac{P(X)\cdot P(Y)}{P(Y)}[/tex]

[tex]P(X|Y)=P(X)[/tex]

Therefore option A is correct.

We know that

[tex]P(Y|X)=\frac{P(X\cap Y)}{P(X)}[/tex]

Using equation (1),

[tex]P(Y|X)=\frac{P(X)\cdot P(Y)}{P(X)}[/tex]

[tex]P(Y|X)=P(Y)[/tex]

Therefore option C is correct.

Since options A, B and C are correct, therefore we can say that correct option is D.

Final answer:

If events X and Y are INDEPENDENT, then option C) P(X and Y) = P(X) x P(Y) is correct.

Explanation:

If events X and Y are INDEPENDENT, then option C) P(X and Y) = P(X) x P(Y) is correct.

For two events to be independent, the probability of their intersection (P(X and Y)) must be equal to the product of their individual probabilities (P(X) x P(Y)).

In other words, if X and Y are independent events, the chance of both X and Y occurring is equal to the chance of X occurring multiplied by the chance of Y occurring.

Look for any interval where the curve is going uphill as you read it from left to right. The interval your teacher will want is the x interval that corresponds to this upward motion.

Nicole has already spent 17 dollars, and will spend c more dollars, for a total of [tex] 17+c [/tex] dollars.

We want this total to be more than 30, so we have

[tex] 17+c>30 [/tex]

Solving this inequality for c yields

[tex] c > 13 [/tex]

The question asks about the additional amount Nicole will need to spend to exceed a total of $30. Given she has already spent $17, the inequality c > 13 represents the additional amount, in dollars, Nicole must spend to meet this requirement. Therefore, Nicole must spend more than $13.

The question is about calculating the possible additional spending of Nicole in order to ensure her total spent will exceed $30. Nicole has already spent $17. As per the condition, set up the inequality which indicates that Nicole will spend more than $30. It's mentioned that we can use c for the additional amount Nicole will spend. Therefore:

17 + c > 30

This inequality can be solved for c by subtracting 17 from both sides:

c > 30 - 17

So,

c > 13

This means Nicole will have to spend more than $13 additionally to ensure her total spending is more than $30.

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Answer:

Acute

Step-by-step explanation:

The last angle is 71 which is less than 90

Answer:

acute triangle

Step-by-step explanation:

The sum of the two angles given is more than 90°, so the remaining angle is less than 90°. All angles are acute angles, so the triangle is an acute triangle.

A. -2

D. 2

"Expansion" means the magnitude of the scale factor is more than 1. Your choices for that are -2 and 2.

(0.75 and |-0.75| are less than 1.)

Answer: A. −2​  and D. 2

Step-by-step explanation:

A scale factor (k) is number which is used to scale a figure.

It is basically used to either reduce or enlarge the size of the figure

A dilation is transformation which uses scale factor to produce the images of same shape as original but of different size.

From the given options , |-2|>1 and |2|>1 where as |-0.75|=0.75<1 and |0.75|<1

Therefore the scale factors produce a expansion under a dilation of the original image are : -2 and 2.

Hence, the correct options are A. −2​  and D. 2  .

[tex]3(x-3)=(3)(x)+(3)(-3)=3x-9\\\\A.\ 3x-6\qquad NOT\\\\B.\ 3x-8-1=3x-9\qquad YES\\\\C.\ x+2x-3=3x-3\qquad NOT\\\\D.\ x-3+x-3+x-3=3x-9\qquad YES[/tex]

about 0.380435 hours . . . approximately 22 minutes 49.6 seconds

Start with this equation ...

... 10.5 hours = 27.6 miles

And divide both sides by 27.6

... 10.5/27.6 hours = 1 mile

... 0.380435 hours = 1 mile

Answer: Z.

Step-by-step explanation:

They want you to figure formula then find right graph, but do it fastest way. If it doubles each year, graph can't be straight line, eliminate Y.

A(0) has to be $5, eliminate W. A(1) has to be double A(0), eliminate X. Check: Z shows A(1) is 10, A(2) is 20.

Answer:

The equation is A(x) = 5(2)ˣ and the graph is Graph Z.

Step-by-step explanation:

Equations on compounding interest (which this essentially is) are of the form

A(x) = p(1+r)ˣ, where p is the amount of principal, r is the interest rate and x is the amount of time.

Since Stella begins with $5, the principal is 5.

Since the amount of money doubles each year, this means we add an extra 100%; this makes r equal to 1.

This gives us the equation

A(x) = 5(1+1)ˣ = 5(2)ˣ

This graph will not be linear, as there is not a constant rate of change (there are different amounts added every year).

She begins with $5.  This means the y-intercept of the graph will be 5.

After 1 year, the amount of money doubles; this makes it 10.  The only graph that is not linear and goes through (0, 5) and (1, 10) is graph Z.

Answer:

the answer is 0.760

Answer:

4 1/2

Step-by-step explanation:

The point is marked at -4 1/2 on the number line. The distance from there to the origin is 4 1/2 units.

(Distances and lengths are always positive—as are values derived from them, such as perimeter or area.)

The distance from 0 to any given point P on a number line is simply the absolute value of the point's coordinate.

The distance from a point to zero on a number line is determined by the absolute value of the point's coordinate. For example, let's say Point P is situated at 5 on a number line. The distance from 0 to Point P would then simply be the absolute value of 5, which is 5 units.

Another example could be if Point P is located at -3 on the number line. In this case, the distance from 0 to Point P is the absolute value of -3, which again, equals 3 units.

So basically, the distance from 0 to any point P on a number line is just the absolute value of the coordinate of that point.

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