For what values of b will f(x)= log b^x be an increasing function?

Answers

Answer 1
[tex]log(b^x) = xlog(b)[/tex]

This is basically a line with slope [tex]log(b)[/tex]

For this to be an increasing function, the slope has to be greater than 0.

[tex]log(b) \ \textgreater \ 0\\ b \ \textgreater \ 10^0\\ b \ \textgreater \ 1[/tex]

So f(x) will be an increasing function for all b > 1
Answer 2

Using logarithmic function concepts, it is found that for values of b > 1, [tex]f(x) = \log_b{x}[/tex] will be an increasing function.

What is an logarithmic function?

A logarithmic function is modeled by:

[tex]f(x) = \log_b{x}[/tex]

The coefficient b determines the behavior of the function, as follows:

b > 1: The function is increasing.b < 1: The function is decreasing.

Hence, for values of b > 1, [tex]f(x) = \log_b{x}[/tex] will be an increasing function.

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Related Questions

Write a research problem that would be best studied using a probability sample.

Answers

A research problem could be of any topic. For example, you could make a research study based on the social status of people in the capital region. This would make a correlation with the country's economic performance. So, you gather around 1,000 respondents and you ask them some social class-determining questions. From your finding, you find that 823 of them belong to the lower class. Thus, the probability that a person in the capital region belongs to the lower class is equal to 823/1000 or 0.823.

On a busy day you clock into work at 6:45 a.m .You clock out for lunch at 12:30 p.m how long did you work before lunch

Answers

Final answer:

The student worked for 5 hours and 45 minutes before taking a lunch break, calculated by finding the difference between the clock-in time of 6:45 a.m. and the lunchtime of 12:30 p.m.

Explanation:

The student worked for a certain number of hours before taking a lunch break. To calculate the duration of work before lunch, we subtract the start time from the end time. The student clocks in at 6:45 a.m. and clocks out at 12:30 p.m. for lunch.

First, we convert the time worked to a 24-hour format: 6:45 a.m. remains the same but 12:30 p.m. is 12:30 in 24-hour time. Now, we calculate the time difference:

From 6:45 a.m. to 7:45 a.m. is 1 hour.7:45 a.m. to 12:30 p.m. is 4 hours and 45 minutes.

Adding up the hours and minutes, we get a total of 5 hours and 45 minutes worked before lunch.


What are the intercepts of the graphed function?

x-intercept = (–1, 0)
y-intercept = (–3, 0)
x-intercept = (0, –1)
y-intercept = (0, –3)
x-intercept = (0, –1)
y-intercept = (–3, 0)
x-intercept = (–1, 0)
y-intercept = (0, –3

Answers

x-intercept: (-1,0)
y-intercept: (0,-3) 

we know that

The x-intercept is the value of x when the the value of y is equal to zero

and

The y-intercept is the value of y when the the value of x is equal to zero

In the graphed function we have that

the value of x when the the value of y is equal to zero is [tex]-1[/tex]

therefore

the x-intercept is equal to the point [tex](-1,0)[/tex]

the value of y when the the value of x is equal to zero is [tex]-3[/tex]

therefore

the y-intercept is equal to the point [tex](0,-3)[/tex]

the answer is

x-intercept = (–1, 0)

y-intercept = (0, –3)

csc(-x)/1+tan^2x) = ?

Answers

[tex]\bf 1+tan^2(\theta)=sec^2(\theta)\qquad \qquad sin(-\theta )=-sin(\theta ) \\\\\\ cot(\theta)=\cfrac{cos(\theta)}{sin(\theta)}\qquad \qquad csc(\theta)=\cfrac{1}{sin(\theta)} \qquad \qquad % secant sec(\theta)=\cfrac{1}{cos(\theta)}\\\\ -------------------------------\\\\[/tex]

[tex]\bf \cfrac{csc(-x)}{1+tan^2(x)} \implies \cfrac{csc(-x)}{sec^2(x)}\implies \cfrac{\frac{1}{sin(-x)}}{\frac{1^2}{cos^2(x)}} \\\\\\ \cfrac{1}{-sin(x)}\cdot \cfrac{cos^2(x)}{1}\implies -\cfrac{cos^2(x)}{sin(x)}\implies cos(x)\cfrac{cos(x)}{sin(x)} \\\\\\ \boxed{cos(x)cot(x)}[/tex]

Jalil and Victoria are each asked to solve the equation ax – c = bx + d for x. Jalil says it is not possible to isolate x because each x has a different unknown coefficient. Victoria believes there is a solution, and shows Jalil her work:  ax – c = bx + d  ax – bx = d + c  x (a – b) = d + c  x =   How can Victoria justify Step 3 of her work?

Answers

Rewrite the expression on the left using the distributive property. In other words choice 1.

IT is A: Rewrite the expression on the left using the distributive property.

A hardware store customer requests a square slab of tile that measures 12.8 feet wide. The width of each side of the slab of tile is __________ inches.

Answers

1 foot = 12 inches

12.8 x 12 = 153.6 inches each side

What happens when you apply the power rule for integration to the function f(x)=1/x?

Answers

The power rule that applies to [tex]f(x)= \frac{1}{x} [/tex] is [tex]f(x)= x^{-1} [/tex]

Integrating [tex] \int\ {x^{-1} } \, dx [/tex] will give the effect of
[tex] \frac{x^{-1+1} }{-1+1} = \frac{ x^{0} }{0} [/tex], which is undefined since we cannot divide by '0'

The conclusion is that to integrate [tex]f(x)= \frac{1}{x} [/tex] we don't use the power rule. We use instead
[tex] \int\ { \frac{1}{x} } \, dx =ln(x)[/tex]

Shannon Perfumeries sells two fragrances. The table contains the price corresponding to the number of bottles of fragrance A. Bottles Price($) 3 78 6 156 9 234 The graph represents the relationship of the price with respect to the number of bottles of fragrance B. The unit rate of fragrance A is $ , and the unit rate of fragrance B is $ . Fragrance has the greater unit rate.

Answers

Answer:

since he missed b ill answer it for your b is 24 because when you look at the grragh  it goes from 0 then to 24 so its unit rate would be 24

Step-by-step explanation:

The unit rate of fragrance A is; $ 26, and the unit rate of fragrance B is; $ 24. Hence Fragrance A has the greater unit rate.

What is the unitary method?

The unitary method is a method for solving a problem by the first value of a single unit and then finding the value by multiplying the single value.

According to the condition the rate of the fragrance A will be;

78/3 = 156/6

= 234/9

= 26 $ per bottle

According to the graph the price of the fragrance B will be;

24/4 = 48/2

=24 $ per bottle

Therefore, the unit rate of fragrance A is; $ 26, and the unit rate of fragrance B is; $ 24.

Hence, Fragrance A has the greater unit rate.

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Which should equal 105 to prove that f // g ?

A
B
C
D
Please hurry !!

Answers

since you have the 75, we know that a would equal 105 for line g , since a line = 180 degrees

 so to make line f parallel with g it needs the same angles with line n as line g has

so if a = 105, then angle d would also need to be 105

 The answer is D

If your monthly net (after-tax) income is $1,500, what should be your maximum amount spent on credit payments? A. $200 B. $300 C. $400 D. $500

Answers

Final answer:

According to budgeting guidelines often referred to as the 20/30/50 rule, no more than 20% of your net income should be spent on debt repayments. Therefore, the maximum amount spent on credit payments for a $1,500 monthly income should be $300.

Explanation:

The best practice for budgeting recommends that no more than 20% of your net monthly income should be spent on debt repayments, including credit payments. This is commonly referred to as the 20/30/50 rule. Therefore, given a monthly net income of $1,500, the maximum amount spent on credit payments should be 20% of $1,500, which equals to $300.

Here's how you would calculate it:

Convert 20% to decimal form by dividing it by 100. 20/100 = 0.2.Multiply the decimal by your net income to get your maximum spend on credit payments. 0.2 x $1,500 = $300.

So, the correct answer is B. $300.

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4a + 6b=10
2a - 4b =12
What is 12a?

Answers

Hi!

4a + 6b = 10
2a - 4b = 12

First make both a terms equal
2a · 2 - 4b · 2 = 12 · 2
4a - 8b = 24

Subtract both expressions to cancel out the a term.
4a + 6b = 10
-4a - 8b = 24
14b = -14
b = -1

Now put the value in one of the equations and solve
2a - 4 · -1 = 12
2a + 4 = 12
2a = 8
a = 4

Since a = 4, 12a = 48.

The answer is 48

Hope this helps! :)

Let x denote the distance (m) that an animal moves from its birth site to the first territorial vacancy it encounters. suppose that for banner-tailed kangaroo rats, x has an exponential distribution with parameter λ = 0.01357. what is the value of the median distance?

Answers

To solve this problem, all we have to do is to use the formula below, plug in the value of the parameter λ, then calculate for the median distance. The formula is:

Median = ln 2 / λ

Substituting:

Median = ln 2 / 0.01357

Median = 51.08 m

The t value for a 99% confidence interval estimation based upon a sample of size 10 is

Answers

Answer:
For a sample size of 10, the t-value is about 3.25 (from tables) at a 99% confidence interval.

Explanation:
When the standard deviation for the population is not known, the confidence interval estimate for the population mean is performed with the Student's t-distribution.
The confidence interval for the mean is calculated as
[tex](\Bar{x}- t\frac{s}{\sqrt{n}} , \, \Bar{x}+ t\frac{s}{\sqrt{n}} [/tex]
where
 [tex]\Bar{x}[/tex] = sample mean,
s = sample standard deviation,
t = t-value (provided in tables),
n =  sample size.

When reading the t-value, (n-1) is called the df or degrees of freedom.

The [tex]\( t \)[/tex]-value for a 99% confidence interval based on a sample size of 10 is 3.2498.

To find the [tex]\( t \)[/tex]-value for a 99% confidence interval estimation based on a sample size of 10, we need to use the [tex]\( t \)[/tex]-distribution table or a calculator. The [tex]\( t \)[/tex]-distribution is used when the sample size is small (typically [tex]\( n < 30 \)[/tex]) and the population standard deviation is unknown.

Given:

- Confidence level: 99%

- Sample size [tex](\( n \)): 10[/tex]

The degrees of freedom [tex](\( df \))[/tex] are calculated as:

[tex]\[ df = n - 1 = 10 - 1 = 9 \][/tex]

To find the critical [tex]\( t \)[/tex]-value for a 99% confidence interval with 9 degrees of freedom, we look for the [tex]\( t \)[/tex]-value that corresponds to the area in the tails of the distribution. For a 99% confidence interval, the area in each tail is:

[tex]\[ \frac{1 - 0.99}{2} = 0.005 \][/tex]

So we need the [tex]\( t \)[/tex]-value such that 0.5% of the distribution is in each tail.

Using a [tex]\( t \)[/tex]-distribution table or a calculator, we find the [tex]\( t \)[/tex]-value for 9 degrees of freedom and a 99% confidence interval (or 0.5% in each tail).

The [tex]\( t \)[/tex]-value for 9 degrees of freedom at the 99% confidence level is approximately:

[tex]\[ t_{0.005, 9} \approx 3.2498 \][/tex]

Thus, the [tex]\( t \)[/tex]-value for a 99% confidence interval based on a sample size of 10 is approximately 3.2498.

What is the axis of symmetry and vertex for the function f(x) = 3(x – 2)2 + 4?
x =

Answers

hello : 
 the axis of symmetry is the line for equation : x = 2
the vertex  is the point :  A (2 , 4)

Answer:

x= 2 vertex: (2,4)

Step-by-step explanation:

just did the assignment

Upper a 18a 18?-footfoot ladder is leaning against a building. if the bottom of the ladder is sliding along the pavement directly away from the building at 22 ?feet/second, how fast is the top of the ladder moving down when the foot of the ladder is 44 feet from the? wall?

Answers

The foot of the ladder cannot be 44 feet from the wall, that would be larger than the length of the ladder, which means the ladder has crawled a few feet :)

So we are assuming a distance of 4 feet, similarly a rate of change in x equal to 2ft/s.

check the picture.

let [tex]h(x)= \sqrt{ 18^{2}- x^{2} } = (18^{2}- x^{2})^{ \frac{1}{2}} [/tex]

be the function of the height of the ladder with respect to x, the distance of the bottom of the ladder to the wall.

We want [tex] \frac{dh}{dt} [/tex], the rate of change of h with respect to t.

h is a function of x and x is a function of t, so we keep this in mind as we derivate h with respect to t:

[tex] \frac{dh}{dt}= \frac{dh}{dx} \frac{dx}{dt}= \frac{1}{2} (18^{2}- x^{2})^{ -\frac{1}{2}}(-2x) \frac{dx}{dt} [/tex]

we substitute [tex] \frac{dx}{dt}=2[/tex] and x=4:

[tex]\frac{dh}{dt}=\frac{1}{2} (18^{2}- 4^{2})^{ -\frac{1}{2}}(-2)*(4)*2= \frac{-8}{ \sqrt{18^{2}- 4^{2}} } = \frac{-8}{17.5}= -0.46[/tex] ft/s




In the triangle XYZ, IF WZ=24, then WY is:

12.
24.
48.
None of the choices are correct.

Answers

WZ is congruent to WY based on the picture, so WY is also 24.

A researcher computes a 2 x 3 factorial anova. in this example, how many interactions can be observed?

Answers

 

The one-way ANOVA or one – way analysis of variance is used to know whether there are statistically substantial dissimilarities among the averages of three or more independent sets. It compares the means between the sets that is being examined whether any of those means are statistically pointedly dissimilar from each other. If it does have a significant result, then the alternative hypothesis can be accepted and that would mean that two sets are pointedly different from each other. The symbol, ∑ is a summation sign that drills us to sum the elements of a sequence. The variable of summation is represented by an index that is placed under the summation sign and is often embodied by i. The index is always equal to 1 and adopt values beginning with the value on the right hand side of the equation and finishing it with the value over head the summation sign.

Suppose Sn is defined as 2 + 22 + 23 + . . . + 2n . What is the next step in your proof of Sn = 2(2n - 1), after you verify that Sn is valid for n = 1?
 A. Show that Sn is valid for n = k + 2.
B. Assume that Sn is valid for n = k .
C. Verify that Sn is valid for n = 1.
D. Show that Sn is valid for n = k.

Answers

Remark:

[tex]S_n=2*1+2*2+2*3+...+2*n=2(1+2+3+...+n)[/tex]

[tex]1+2+3+...+n= \frac{n(n+1)}{2} [/tex], by the famous Gauss formula.

So the formula for [tex]S_n[/tex] is:

[tex]S_n=2*\frac{n(n+1)}{2}=n(n+1)[/tex]



these types of formulas are proven by Induction.

The first step is proving for n=1,

then the next step is assuming Sn is valid for n=k.



Answer: B. Assume that Sn is valid for n = k .

A cylinder has a diameter of 14 cm and a height of 20 cm.

a. Find the total surface area of the cylinder.
b. If gift wrap cost $3 per square centimeter, how much will it cost to cover the cylinder with gift wrap? Use 3.14 for π.
c. Find the volume of the cylinder.

Answers

The total surface area of the cylinder is approximately 1187.72 cm², the volume is approximately 3077.2 cm³, and the cost to cover it with gift wrap at $3 per square centimeter is $3,558.76.

Finding the Surface Area and Volume of a Cylinder

The surface area of a cylinder is calculated using the formula: Surface Area = 2πr(height) + 2πr². With a diameter of 14 cm, the radius (r) is half of that, which is 7 cm. Plugging in the values, the surface area is 2π(7 cm)(20 cm) + 2π(7 cm)².

For part b, once we have calculated the surface area, we can determine the cost to cover the cylinder using the given price per square centimeter. If S represents the total surface area, the cost will be $3 times S.

The volume of the cylinder can be found with the formula V = πr²h, and using the radius of 7 cm and a height of 20 cm, we get the volume V = π(7 cm)²(20 cm).

Performing these calculations:

Surface Area = 2π(7 cm)(20 cm) + 2π(7 cm)² = 2π(7 cm)(20 cm) + 2π(49 cm²) = 2π(140 cm²) + 2π(49 cm²) = 280π cm² + 98π cm² = 378π cm².Volume = π(7 cm)²(20 cm) = π49 cm²20 cm = 980π cm³.Cost = $3 × 378π cm² = $1134π.

Using 3.14 for π, we get:

Surface Area = 378π cm² = 1187.72 cm² (approximately).Volume = 980π cm³ = 3077.2 cm³ (approximately).Cost = $1134π = $3,558.76 (approximately).

A congested computer network has a 0.010 probability of losing a data packet and packet losses are independent events. a lost packet must be resent. round your answers to four decimal places (e.g. 98.7654). (a) what is the probability that an e-mail message with 100 packets will need any resent?

Answers

The probability of losing a data packet is 0.010.
Therefore the probability of successfully sending a data packet (not losing a data packet) is
p = 1 - 0.010 = 0.99

In 100 packet transmissions (independent events), the probability of success is
0.99¹⁰⁰ = 0.3660

The probability of losing a data packet is
1 - 0.366 = 0.6340

Answer:
The probability of resending a data packet is 0.6340

Find the particular solution of the differential equation dydx+ycos(x)=5cos(x) satisfying the initial condition y(0)=7.

Answers

[tex]\dfrac{\mathrm dy}{\mathrm dx}+y\cos x=5\cos x[/tex]
[tex]e^{\sin x}\dfrac{\mathrm dy}{\mathrm dx}+ye^{\sin x}\cos x}=5e^{\sin x}\cos x[/tex]
[tex]\dfrac{\mathrm d}{\mathrm dx}\left[e^{\sin x}y\right]=5e^{\sin x}\cos x[/tex]
[tex]e^{\sin x}y=5\displaystyle\int e^{\sin x}\cos x\,\mathrm dx[/tex]
[tex]e^{\sin x}y=5e^{\sin x}+C[/tex]
[tex]y=5+Ce^{-\sin x}[/tex]

With [tex]y(0)=7[/tex], we have

[tex]7=5+Ce^{-\sin 0}\implies 7=5+C\implies C=2[/tex]

so that the particular solution is

[tex]y=5+2e^{-\sin x}[/tex]
Final answer:

The provided differential equation is a first-order linear differential equation, which can be solved using an integrating factor. After solving, the particular solution satisfying the initial condition y(0)=7 is y=e^(-sin(x))(5sin(x)+7).

Explanation:

The differential equation provided is a first-order linear differential equation, which can be solved using an integrating factor. In this case, dy/dx + ycos(x) = 5cos(x), the integrating factor is e^(∫ cos(x) dx) = e^sin(x). Multiplying everything by the integrating factor, we get (ye^sinx)' = 5cos(x)e^sin(x).

Then we can integrate on both sides to get ye^sin(x) = 5sin(x) + C, where C is the constant of integration. To find the particular solution, we can use the initial condition y(0)=7. By substituting these values, we can solve for C. Substituting x=0 and y=7 yields C=7. Thus, the particular solution is y=e^(-sin(x))(5sin(x)+7).

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A lawn sprinkler sprays water 8 feet at full pressure as it rotates 360 degrees. If the water pressure is reduced by 50%, what is the difference in the area covered?

Answers

check the picture below

so, if the pressure is halfed, then the radius covered would be halfed

now, if 64π is the 100%, what is 16π in percentage?

[tex]\bf \begin{array}{ccllll} amount&\%\\ \text{\textemdash\textemdash\textemdash}&\text{\textemdash\textemdash\textemdash}\\ 64\pi &100\\ 16\pi &x \end{array}\implies \cfrac{64\pi }{16\pi }=\cfrac{100}{x}\implies \cfrac{4}{1}=\cfrac{100}{x}\implies x=\cfrac{1\cdot 100}{4}[/tex]

Answer:

[tex]150.72 feet^2[/tex] is the difference in the area covered.

Step-by-step explanation:

A lawn sprinkler sprays water 8 feet at full pressure, P.

A lawn sprinkler rotates 360 degree which means area covered by sprinkler is of circular shape. Since the sprinkler is in center and sprays the the water 8 feet away in all the direction while rotating.

Radius of the circle = 8 feet

Maximum pressure = P

As we know that higher the pressure higher will the force by which water will move out of the sprinkler. And with more force, sprinkler will able to spray water farther.

So we this we can say that pressure of the sprinkler is directly proportional to the radius of the circle in which water sprayed

[tex]pressure\propto Radius[/tex]

[tex]P\propto r[/tex]

[tex]\frac{P_1}{r_1}=\frac{P_2}{r_2}=constant[/tex]

[tex]P_1=P.P_2=P-50\%\times P=0.5 P[/tex]

[tex]r_1=8 feet.r_2=?[/tex]

[tex]r_2=\frac{0.5 P\times 8 feet}{P}=4 feet[/tex]

Area when , [tex]r_1= 8 feet[/tex] (Area of circle=[tex]\pi (radius)^2[/tex])

[tex]A=\pi r_1^{2}=\pi (8 feet)^2[/tex]

Area when ,[tex]r_2= 4 feet[/tex]

[tex]A'=\pi r_1^{2}=\pi (4 feet)^2[/tex]

Difference in Area = A- A'

[tex]\pi (8 feet)^2-\pi(4 feet)^2=\pi(48 feet^2)=150.72 feet^2[/tex]

[tex]150.72 feet^2[/tex] is the difference in the area covered.

Two similar regular hexagons have a common center. If each side of the big hexagon is twice the side of the small one and the area of the small hexagon is 3 sq. in, what is the area of the big hexagon?

Answers

Final answer:

The area of the larger square is 4 times larger than the area of the smaller square. The area of the big hexagon is 12 sq. in.

Explanation:

The area of the larger square is 4 times larger than the area of the smaller square. This is because the area of a square is proportional to the square of its side length.

In this case, the side length of the larger square is twice the side length of the smaller square, so the area of the larger square is 2² times greater than the area of the smaller square.

Given that the area of the small hexagon is 3 sq. in, the area of the big hexagon can be found by multiplying the area of the small hexagon by the square of the scale factor:

Area of big hexagon = (scale factor)² * Area of small hexagon = 2² * 3 sq. in = 12 sq. in

Use an Addition or Subtraction Formula to simplify the equation. sin(3θ) cos(θ) − cos(3θ) sin(θ) = Square root 2/2 Find all solutions in the interval [0, 2π). (Enter your answers as a comma-separated list.)

Answers

[tex]\sin3\theta\cos\theta-\cos3\theta\sin\theta=\sin(3\theta-\theta)=\sin2\theta=\dfrac{\sqrt2}2[/tex]
[tex]\sin2\theta=\dfrac1{\sqrt2}[/tex]
[tex]\implies2\theta=\dfrac\pi4+2n\pi,\,2\theta=\dfrac{3\pi}4+2n\pi[/tex]
[tex]\implies\theta=\dfrac\pi8+n\pi,\,\theta=\dfrac{3\pi}8+n\pi[/tex]

where [tex]n[/tex] is any integer. To take only the solutions within the interval [tex]0\le\theta<2\pi[/tex], we solve

[tex]0\le\dfrac\pi8+n\pi<2\pi\implies\dfrac18+n<2\implies n<\dfrac{15}8\implies n=0,\,n=1[/tex]
[tex]\implies\theta=\dfrac\pi8,\,\theta=\dfrac\pi8+\pi=\dfrac{9\pi}8[/tex]

[tex]0\le\dfrac{3\pi}8+n\pi<2\pi\implies \dfrac38+n<2\implies n<\dfrac{13}8\implies n=0,\,n=1[/tex]
[tex]\implies\theta=\dfrac{3\pi}8,\,\theta=\dfrac{11\pi}8[/tex]

Answer: For 0 ≤Ф≥ 2π (where π= 180°)

∴ Ф = 22.5°, 67.5°, 112.5°, 157.5°, 202.5°, 247.5°, 292.5°, 337.5°

Step-by-step explanation:

sin(3Ф)cos(Ф) - cos(3Ф)sin(Ф) = √2/2

sin(3Ф - Ф) =√2/2

3Ф -Ф = sin∧-1{√2/2}

 2Ф = 45°

∴ Ф = 22.5°

identify the real and imaginary parts of the complex number. -5 + 6i

Answers

-5 is the real part and 6i is the imaginary part. This can be determined by looking which number has the "i" attached to it. 

A total of
564
tickets were sold for the school play. They were either adult tickets or student tickets. The number of student tickets sold was three times the number of adult tickets sold. How many adult tickets were sold?

Answers

The answer will be 141 adults

What is the domain of the function f(x) = x2 + 3x + 5?

Answers

Domain: -∞<x<∞ since it's infinitely going both ways of the graph on the x-axis
It's all real numbers

the slope of a line is -2 and the line contains the points (7,4) and (x,12). what is the value of x?

Answers

Slope= y2-y1/x2-x1.

-2= (12-4)/(x-7)
-2= 8/(x-7)
-2(x-7)=8
-2x+14=8
-2x=-6
x=3

Final answer: x=3

The sales tax for an item was $22.50 and it cost $450 before tax. Find the sales tax rate. Write your answer as a percentage.

Answers

22.5 / 450 = 0.05 = 5% <== the rate

Final answer:

The sales tax rate is found by dividing the amount of sales tax by the cost of the item before tax and then multiplying by 100. In this case, the sales tax rate is 5%.

Explanation:

To find the sales tax rate of an item, you need the amount of sales tax paid and the cost of the item before tax. The formula to calculate the sales tax rate is:

sales tax rate = (amount of sales tax \/ cost of the item before tax) \ 100

Applying the formula, we have:

sales tax rate = ($22.50 \/ $450) \ 100

sales tax rate = 0.05 \ 100

sales tax rate = 5%

Therefore, the sales tax rate for the item is 5%.

The volume of a box(V) varies directly with its length(l). If a box in the group has a length of 30 inches, and the girth of 20 inches (perimeter of the side formed by the width and height), what is its height? Use k = 24. (Hint: Volume = length • width • height. Solve for length, and substitute into the equation for constant of proportionality.)?

Answers

[tex]\bf \textit{V varies directly with l}\implies V=kl\qquad k=24\implies V=24l \\\\\\ \textit{volume of the box, or a rectangular prism}\\\\ V=lwh\qquad \begin{cases} l=length\\ w=width\\ h=height\\ ----------\\ k=24 \end{cases}\implies \boxed{lwh=24l} \\\\\\ \textit{girth of the box }\\\\ w+w+h+h=20\implies 2w+2h=20\implies 2(w+h)=20 \\\\\\ thus\qquad w+h=\cfrac{20}{2}\implies \boxed{w=10-h}[/tex]

[tex]\bf \\\\ -------------------------------\\\\ lwh=24l\implies wh=24\implies (10-h)h=24\implies 10h-h^2=24 \\\\\\ 0=h^2-10+24\implies 0=(h-6)(h-4)\implies h= \begin{cases} 6\\ 4 \end{cases}[/tex]

now, notice, we didn't use the length of 30inches.... since the "l"'s cancel each other anyway, so it doesn't weight much on what the value for "h" is, by simply doing the substution of "w" from the Girth.
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