could you please help me with this equation?

The distance between both houses is: 3839.75 ft

The Law of Cosines is simply a numerical formula that relates the side lengths and points of any triangle. It expresses that the square of any side of a triangle is equivalent to the number of squares of the other different sides short two times the result of those sides and the cosine of the point between them.

Numerically, the Law of Cosines can be communicated as:

c² = a² + b² - 2abcos(C),

where:

c is the length of the side inverse to the point C, and an and b are the lengths of the other different sides.

Using law of cosines, the length AB is:

AB = 50² + 100² - 2(50 * 100) * cos 30

AB = 3839.75 ft

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The salesman traveled at an average speed of 55 miles per hour on his trip.

To find the average miles per hour, we need to find the total distance traveled and divide it by the total time taken.

Total distance traveled = 52 miles + 64 miles + 49 miles = 165 miles

Total time taken = 1 hour + 1 hour + 1 hour = 3 hours

Average speed = Total distance traveled / Total time taken

= 165 miles / 3 hours

= 55 miles per hour

Therefore, the salesman averaged 55 miles per hour on his trip.

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The rate of change of the average height in feet of the trees on the farm with respect to the number of years since the trees were planted would be = 7ft/year. That is option C.

To calculate the rate of change of the average height of the trees in the farm the following is carried out.

The total heights of the tress that was recorded;

= 10+24+45+80+108

= 267 ft

The total number of years that was recorded;

= 1+3+6+12+15

= 37 years

Therefore the rate of change of the average height of the tress would be = 267/37 = 7 ft/yr.

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Answer:

3.14

Step-by-step explanation:

The value of pi to two decimal places is 3.14. It is a mathematical constant that represents the ratio of the circumference of a circle to its diameter. Pi is an irrational number, which means it cannot be expressed as a finite decimal or fraction. It is an important constant in many fields of mathematics and science, including geometry, trigonometry, and physics. The decimal representation of pi goes on infinitely without repeating, making it an interesting and challenging topic for mathematicians to explore.

The height of the water increasing at a rate of 0.05 m/min.

The Volume of a cylinder is

V = [tex]\pi r^{2} h[/tex]

where r is the radius and h is the height of the cylinder.

We are given that the radius r of cylinder is 5m and the water is filled at the rate of 4 ;

V = 25πh  ...(1)

Now, differentiating (1) with respect to time t,

dV/dt = 25πdh/dt

Substituting the value of dV/dt ;

4 = 25πdh/dt

dh/dt = 0.05 m/min

Hence, the water is increasing at the rate of 0.05 m/min.

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The prove of expression is shown below.

Now, It is given that;

P = aˣ , q = [tex]a^{y}[/tex] and a⁴ = [tex](p^4^y q^4^x)^z[/tex]

And, we have to prove that xyz = 1/2.

Hence, We can simplify as;

Since, here a⁴ = [tex](p^4^y q^4^x)^z[/tex]

⇒ a⁴ = [tex][(a^x)^{4y} . (a^y)^4^x ]^z[/tex]

⇒a⁴ = [tex][a^{4xy} . a^{4xy} ]^z[/tex]

⇒a⁴ = [tex][a^{4xy + 4xy} ]^z[/tex]

⇒a⁴ = [tex](a^{8xy} )^{z}[/tex]

⇒a⁴ = [tex]a^{8xy} ^{z}[/tex]

⇒ 4 = 8xyz

⇒ xyz = 1/2

Thus, It can prove as shown.

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Question is,

If p = a^x, q = a^y and a^4 = (p^4y.q^4x)^z prove that xyz =1/2​

The expression xyz = 1/2 is proved.

The given exponential expressions are,

[tex]p = a^{x}[/tex]

[tex]q = a^{y}[/tex]

And [tex]a^{4} = (p^{4y} q^{4x}) ^{z}[/tex]

Now proceed the third expression,

[tex]a^{4} = (p^{4y} q^{4x}) ^{z}[/tex]

⇒[tex]a^{4} = ((a^{x} )^{4y} (a^{y} )^{4x}) ^{z}[/tex]

⇒[tex]a^{4} = (a^{4xy} a^{4xy}) ^{z}[/tex]

⇒[tex]a^{4} = (a^{8xy}) ^{z}[/tex]

⇒[tex]a^{4} = a^{8xyz}[/tex]

Now equate the exponents

⇒ 8xyz = 4

⇒   xyz = 1/2

Which was required.

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The complete question is :

If [tex]p = a^{x}[/tex], [tex]q = a^{y}[/tex] and  [tex]a^{4} = (p^{4y} q^{4x}) ^{z}[/tex] then prove that xyz =1/2​.

Answer:

Same, same, same, different.

Step-by-step explanation:

Same x-coordinate of vertex because no horizontal movement.

Same y-coordinate of vertex because no vertical movement.

Same direction of opening because the exponents are equal and both coefficients are positive.

Different y value at x = 1. y value for x^2 is 1 while y value for 2x^2 is 2.

There does not appear to be enough evidence to back the claim that individuals taking Lipitor have a higher risk of infection compared to those on the placebo.

To test the claim that the rate of infection for those treated with Lipitor was more than the rate of infection for those given a placebo at the .025 significance level, we will perform a hypothesis test using a two-proportion z-test.

Claim: p1 > p2

Opposite: p1 ≤ p2

P1 is the Lipitor group

P2 is the placebo group

The test is a right-tailed test.

Let's calculate the proportions and pooled proportion:

p1 = 86/220 = 0.3909

p2 = 88/260 = 0.3385

p = (86 + 88) / (220 + 260) = 0.3615

Now, we'll find the test statistic (z):

Using a calculator, we see that

z ≈ 1.255

The critical value (Zα) for a right-tailed test at a .025 significance level is approximately 1.96.

Since the test statistic (1.255) is below the critical value (1.96), we fail to reject the null hypothesis.

In conclusion: There does not appear to be enough evidence to back the claim that individuals taking Lipitor have a higher risk of infection compared to those on the placebo.

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Answer:

$17.8

Step-by-step explanation:

C= 15.00 + 0.07(40)

The range of possible values for the expression -2a + b is -3 ≤ -2a + b ≤ -1.

.Given the inequalities 3 < a < 5 and 5 < b < 7, we need to find all possible values of the expression -2a + b. We can start by substituting the given values of a and b in the expression.

Since a is between 3 and 5, the maximum value of -2a will be -2(3) = -6.

Similarly, since b is between 5 and 7, the minimum value of b will be 5, giving us a possible minimum value of -2a + b as -6 + 5 = -1. On the other hand, the minimum value of -2a will be -2(5) = -10, and the maximum value of b will be 7, giving us a possible maximum value of -2a + b as -10 + 7 = -3.

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Answer: [tex]\frac{10\sqrt{3}}{3}[/tex]

Step-by-step explanation:

To find this use TOA which is Tan = Opposite/Adjacent

so we have tan(30°) = y/10 and to isolate the "y" we multiply both sides by 10 to get 10tan(30°) = y

where y = [tex]\frac{10\sqrt{3}}{3}[/tex]

We can see here that option C is the correct option that gives us the percentage bar graph that best represents the data.

A bar graph, also called a bar chart, is a form of graph that uses rectangular bars of equal width to depict data. Each bar's height or length reflects the value or frequency of the data it conveys. Discrete or categorical data are frequently compared and analyzed using bar graphs.

We see that from the data given, the total number of students =

44 + 55 + 55 + 66 = 220

Reading = 44/220 × 100% = 20%

Sports =  55/220 × 100% = 25%

Video Games =  55/220 × 100% = 25%

Music = 66/220 × 100% = 30%

Thus, option C is the correct answer.

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The correct answer is option D. 250 milliliters, is the capacity of a small cup. This is a proportion exercise.

It is given that, the  total capacity of two large cups and two small cups is 1,500 milliliters. If the total capacity of one large cup and two small cups is 1,000 milliliters.

To find the capacity of a small cup

Let small cup capacity  be 's' and large cup capacity be 'l'

From the given information we can write,

2l + 2s = 1500

Then l + s = 750

one large cup and two small cups is 1,000 milliliters.

l + 2s = 1000

l + s + s = 1000

(l + s) + s = 1000

750 + s = 1000

s = 1000 - 750 = 250

The correct answer is option D. 250 milliliters

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Full Question:

The total capacity of two large cups and two small cups is 1,500 milliliters. If the total capacity of one large cup and two small cups is 1,000 milliliters, what is the capacity of a small cup?

A. 375 milliliters

B. 200 milliliters

C. 500 milliliters

D. 250 milliliters

The surface area of the rectangular prism is 26/3 times larger than the surface area of the cube.

The surface area of a rectangular prism of height h, width w and length l is given by:

S = 2(hw + lw + lh).

This means that the area of each rectangular face of the prism is calculated, and then the surface area is given by the sum of all these areas.

For the cube, the dimensions are given as follows:

x, x and x.

Hence the surface area is:

S = 2 x 3x² = 6x².

For the rectangular prism, the dimensions are given as follows:

2x, 3x and 4x.

Hence the surface area is:

S = 2 x (2x(3x) + 3x(4x) + 2x(4x)) = 52x².

Hence the ratio is given as follows:

52x²/6x² = 26/3.

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To solve this problem, we need to convert all the measurements to a consistent unit. Let's convert the height to inches since the height of the bounce is given in inches.

64 ft = 64 * 12 inches = 768 inches

Now, we can set up an equation to represent the height of each bounce. Let's use "b" to represent the number of bounces, and "h" to represent the height of each bounce in inches.

The height of each bounce is three-fourths (3/4) the height of the previous bounce. So, we can write the equation as:

h = (3/4) * h_previous

where h_previous is the height of the previous bounce.

We know that the initial height of the ball is 768 inches, and we want to find the number of bounces when the height of the bounce is less than 9 inches. We can set up an inequality to represent this situation:

h < 9

Substituting the expression for h from the equation above, we get:

(3/4) * h_previous < 9

Now, we can start with the initial height of 768 inches and keep applying the equation for each bounce until the height of the bounce is less than 9 inches.

1st bounce:

h = (3/4) * 768 = 576 inches

2nd bounce:

h = (3/4) * 576 = 432 inches

3rd bounce:

h = (3/4) * 432 = 324 inches

4th bounce:

h = (3/4) * 324 = 243 inches

5th bounce:

h = (3/4) * 243 = 182.25 inches

6th bounce:

h = (3/4) * 182.25 = 136.6875 inches

7th bounce:

h = (3/4) * 136.6875 = 102.515625 inches

8th bounce:

h = (3/4) * 102.515625 = 76.88671875 inches

9th bounce:

h = (3/4) * 76.88671875 = 57.6650390625 inches

10th bounce:

h = (3/4) * 57.6650390625 = 43.248779296875 inches

So, the ball will bounce up to a height less than 9 inches after 10 bounces.

A hypothesis test was conducted to determine if the proportion of online applications in 2004 was more than in 2009, using a significance level of .005. The test results show that there is not enough evidence to support the claim.

The null hypothesis is that the proportion of online applications in 2004 is equal to or less than the proportion in 2009. The alternative hypothesis is that the proportion of online applications in 2004 is greater than the proportion in 2009. Symbolically, we can write

H o: p2004 ≤ p2009

H a: p2004 > p2009

The significance level is given as 0.005. Since this is a one-tailed test (we are testing for a greater proportion in 2004), we will use a one-tailed critical value. Using a normal distribution table or calculator, we can find the critical value to be 2.576.

The test statistic for this hypothesis test is a z-score, which we can calculate using the formula

z = (P₁ - P₂) / √(P(1-P)(1/n₁ + 1/n₂))

where P₁ and P₂ are the sample proportions, n₁ and n₂ are the sample sizes, and P is the pooled proportion. In this case, we have

P₁ = 123/373 = 0.33

P₂ = 70/776 = 0.09

n₁ = 373, n₂ = 776

P = (123 + 70) / (373 + 776) = 0.157

Plugging these values into the formula, we get

z = (0.33 - 0.09) / √(0.157(1-0.157)(1/373 + 1/776))

z ≈ 8.48

Since the test statistic (8.48) is greater than the critical value (2.576), we can reject the null hypothesis. This means there is enough evidence to support the alternative hypothesis that the proportion of online applications in 2004 is greater than the proportion in 2009.

Based on the results of the hypothesis test, we can conclude that there is enough evidence to support the claim that the proportion of online applications in 2004 was more than the proportion of online applications in 2009. Therefore, the opposite answer in the given question is incorrect.

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You will spend 27 minutes in the doctor's office.

An exponential function has the definition presented as follows:

[tex]y = ab^x[/tex]

In which the parameters are given as follows:

The initial amount is of 14 mg, hence:

[tex]y = 14b^x[/tex]

After 16 minutes, the amount is of 4.5 mg, hence the parameter b is obtained as follows:

[tex]4.5 = 14b^{16}[/tex]

[tex]b^{16} = \frac{4.5}{14}[/tex]

[tex]b = \left(\frac{4.5}{14}\right)^{\frac{1}{16}}[/tex]

b = 0.931521.

Hence:

[tex]y = 14(0.931521)^x[/tex]

You will have 2 mg when:

[tex]2 = 14(0.931521)^x[/tex]

[tex](0.931521)^x = \frac{1}{7}[/tex]

x = log(1/7)/log(0.931521)

x = 27 minutes.

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a) The function that models the number of bacteria is n(t) = 8900[tex]e^{(0.126t)[/tex].

b) Rounding to the nearest hundred, we get the answer of 12100 bacteria after 2 hours.

c) It would take approximately 5.5 hours for the number of bacteria to double.

To model the growth of the bacteria population, we can use an exponential growth function of the form n(t) = n₀[tex]e^{(rt)[/tex], where n₀ is the initial population, r is the growth rate, and t is the time in hours.

(a) To find the growth rate, we can use the formula r = ln(N/N₀)/t, where N is the final population, N₀ is the initial population, and t is the time interval. Plugging in the given values, we get:

r = ln(10000/8900)/1 = 0.126

(b) To find the number of bacteria after 2 hours, we can simply plug in t=2 into the exponential function:

n(2) = 8900[tex]e^{(0.126*2)[/tex] = 12099.92

Rounding to the nearest hundred, we get the answer of 12100 bacteria after 2 hours.

(c) To find the time it takes for the number of bacteria to double, we can use the fact that the population doubles when n(t) = 2n₀. Substituting these values and solving for t, we get:

2n₀ = n₀[tex]e^{(rt)[/tex]

2 = [tex]e^{(rt)[/tex]

ln(2) = rt

t = ln(2)/r = 5.5 hours (rounded to one decimal place)

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To know the area of a figure, you would like to know its shape and estimations. The steps to take on how to draw a figure with the use of a ruler and a pencil is given below.

Begin by selecting the kind of figure you need to draw. Make use of a ruler to draw the sides of the figure. Make sure that the lines are straight and have the right length.

E.g. If the figure may be a rectangle or a parallelogram, you wish to create beyond any doubt that inverse sides are parallel to each other. If the figure is triangle, you would like to draw three lines that shows at three distinctive points. At last, name the sides and vertices of the figure.

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Answer:

$600

Step-by-step explanation:

We Know

1kg of rice costs $1.20

How much is 500kg?

We Take

1.20 x 500 = $600

So, 500kg of rice cost $600

Joey could need to pay $3,825.00 per year in total Social security and Medicare tax if he operates his personal construction company as a self-employed individual. Therefore Option D is correct.

If Joey is self-employed and operates his personal construction organization, he would need to pay both the employee and organization portions of the Social safety and Medicare taxes. those taxes are together referred to as self-employment taxes.

The current self-employment tax charge for Social security is 12.4% on profits up to $142,800 (as of 2021) and the price for Medicare is two.9% and not using a earnings limit.

But, for the reason that Joey is each the worker and the employer, he might need to pay both portions, resulting in a total self-employment tax rate of 15.3%.

To calculate the full amount of Social security and Medicare tax that Joey would pay, we can multiply his profits of $25,000 via the self-employment tax fee of 15.3%:

$25,000 x 0.153 = $3,825.00

Therefore, Joey could need to pay $3,825.00 per year in total Social security and Medicare tax.

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The account will have $598.91 after 50 years.

Given that Kathy deposited $100 in an account for 40 years, the account has a 4.5% effective annual interest rate,

Therefore,

[tex]Amount = Deposit(1+r/4)^{4t[/tex]

[tex]= 100(1+0.1125)^{160[/tex]

= 598.91.

Hence, the account will have $598.91 after 50 years.

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In order to achieve the shortest distance from starting vertex (A) to F on a network, utilize Dijkstra's algorithm following these steps:

Initiate every vertex with a tentative distance value: assign zero to the starting point and infinity to other vertices.

Establish the initiating vertex as the current vertex.

Analyze all neighboring vertices of the current vertex and determine their respective tentative path length via current vertex. Proceed to compare it with its already assigned value; upgrade such distance if shorter than before.

Subsequent to analyzing adjacent points of the present node, designate it as 'visited'. Futuristic reference will disregard this chosen and visited vertex altogether.

Examine unexplored vertices marked as having the most reduced tentative distances, declare them 'current' while formulating new calculations for other contenders until pertinent conclusion is reached.

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Answer:

Step-by-step explanation:

y = -2/3(x) + 1, which is the line through the point (3, -1) and perpendicular to the line y = 3/2(x) + 3.

Since the diameter of this circle is 18 kilometers, the circle's area is 254.34 square kilometers.

In Mathematics and Geometry, the area of a circle can be calculated by using this formula:

Area = πr²

Where:

r represents the radius of a circle.

Note: Radius = diameter/2 = 18/2 = 9 kilometers.

By substituting the radius into the formula for the area of a circle, we have the following;

Area of circle = 3.14 × 9²

Area of circle = 254.34 square kilometers.

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Answer:

8

Step-by-step explanation:

multiply 80 by 0.10 to get 8