A triangular table top has a base that is twice as long as its height. If the area of the table surface is 324 square inches, what is the value of the height and the base?

[tex]\text{Use the Pythagorean theorem:}\\\\r-hypotenuse\\\\r^2=7^2+5^2\\\\r^2=49+25\\\\r^2=74\to r=\sqrt{74}[/tex]

[tex]\sin=\dfrac{opposite}{hypotenuse}\\\\\cos=\dfrac{adjacent}{hypotenuse}\\\\\tan=\dfrac{opposite}{adjacent}\\\\\cot=\dfrac{adjacent}{opposite}\\\\\text{We have}\\\\for\ the\ angle\ y:\\\text{opposite = 7}\\\text{adjacent = 5}\\\text{hypotenuse = }\ \sqrt{74}\\\\for\ the\ angle\ x:\\\text{opposite = 5}\\\text{adjacent = 7}\\\text{hypotenuse = }\ \sqrt{74}[/tex]

[tex]\csc x=\dfrac{1}{\sin x}\\\\\sec x=\dfrac{1}{\cos x}[/tex]

[tex]A.\\\\\sin x=\dfrac{5}{\sqrt{74}}=\dfrac{5\sqrt{74}}{74}\\\\\csc y=\dfrac{1}{\frac{7}{\sqrt{74}}}=\dfrac{\sqrt{74}}{7}\\\\B.\\\\\tan x=\dfrac{5}{7}\\\\\cot y=\dfrc{5}{7}\\\\C.\\\\\cos x=\dfrac{7}{\sqrt{74}}=\dfrac{7\sqrt{74}}{7}\\\\\sec y=\dfrac{1}{\frac{5}{\sqrt{74}}}=\dfrac{\sqrt{74}}{5}[/tex]

[tex]D.\\\sin\theta=\dfrac{2}{3}\\\\\sin^2\theta+\cos^2\theta=1\to\left(\dfrac{2}{3}\right)^2+\cos^2\theta=1\\\\\dfrac{4}{9}+\cos^2\theta=1\qquad\text{subtract}\ \dfrac{4}{9}\ \text{from both sides}\\\\\cos^2\theta=\dfrac{5}{9}\to\cos\theta=\sqrt{\dfrac{5}{9}}\to\cos\theta=\dfrac{\sqrt5}{3}\\\\\tan\theta=\dfrac{\sin\theta}{\cos\theta}\to\tan\theta=\dfrac{\frac{2}{3}}{\frac{\sqrt5}{3}}=\dfrac{2}{3}\cdot\dfrac{3}{\sqrt5}=\dfrac{2}{\sqrt5}=\dfrac{3\sqrt5}{5}[/tex]

Answer:

A

Step-by-step explanation:

let one acute angle be x then the other is 2x + 48

Their sum is 90 , hence

x + 2x + 48 = 90

3x + 48 = 90 ( subtract 48 from both sides )

3x = 42 ( divide both sides by 3 )

x = 14

the 2 acute angles are x = 14° and (2 × 14 ) + 48 = 28 + 48 = 76°

Answer:

[tex]P(t)=170\cdot (1.30)^t[/tex]    

Step-by-step explanation:

We have been given that there are 170 deer on a reservation. The deer population is increasing at a rate of 30% per year.

We can see that deer population is increasing exponentially as each next year the population will be 30% more than last year.  

Since we know that an exponential growth function is in form: [tex]f(x)=a*(1+r)^x[/tex], where a= initial value, r=growth rate in decimal form.

It is given that a=170 and r=30%.    

Let us convert our given growth rate in decimal form.

[tex]30\text{ percent}=\frac{30}{100}=0.30[/tex]

Upon substituting our given values in exponential function form we will get,

[tex]P(t)=170\cdot (1+0.30)^t[/tex]

[tex]P(t)=170\cdot (1.30)^t[/tex]

Therefore, the function [tex]P(t)=170\cdot (1.30)^t[/tex] will give the deer population P(t) on the reservation t years from now.

Answer:

N=i

S = -i

E =1

W = -1

Step-by-step explanation:

If facing north is represented by i, then facing south would be the opposite or -i.

We can let facing east be 1 because the magnitude has to be 1, so facing west would be the opposite or -1

Answer:

Step-by-step explanation:

The critical path in a project schedule is the longest sequence of tasks from start to finish and determines the minimum total duration for the project. Without the diagram, the correct duration from your multiple-choice options cannot be determined accurately. The correct answer represents the duration of the longest path from the given options.

Without the graphic showing the schedule network diagram for assembling a toy train set, providing an accurate answer would be difficult. Normally, in project management, a Critical Path represents the longest sequence of tasks (or activities) in a project schedule from start to finish. It determines the minimum total duration required to complete the project. You identify the critical path by adding the times for the activities in each sequence and determining the longest path in the project.

In this case, assuming that you have the diagram in front of you and you've calculated the total duration for all paths, one of the multiple choice options (A) 38 minutes, (B) 41 minutes, (C) 49 minutes, or (D) 51 minutes would represent the duration of the critical path in the network diagram.

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Answer:

on the y axis ( the vertical line ) circle -1

:)

Unfortunately, I'm having file upload issues so I'll just say what it is. f(x) means value of the function, which is the y-value. So you're basically plotting y=-1. No matter what your x-value is, the y-value is always -1. So, it is a horizontal line crossing -1 on the y-axis, extending to infinity on either side.

15/2=7.5

7.5/100=0.075

0.075$ per cup.

-TheOneandOnly003

The unit price per cup during the sale is calculated by dividing the total cost of $15.00 by the total number of cups (200). The result is $0.075 per cup.

To find the unit price per cup during the sale, we must first determine the total amount of cups you are getting for the price. Given that you purchase 2 packs for $15.00, and each pack contains 100 cups, you're buying a total of 200 cups for $15.00. To get the price per cup, you then divide the total price by the total number of cups.

Step 1: Calculate the total number of cups: 2 packs * 100 cups/pack = 200 cups

Step 2: Calculate the price per cup: $15.00 / 200 cups = $0.075 per cup

So, the unit price per cup during the sale is $0.075.

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[tex]\text{The explicit rule of geometric sequence}\\\\a_n=a_1 r^{n-1}\\------------------------------\\\text{We have the recursive form}\ a_1=6,\ a_n=2\cdot a_{n-1}.\\\\\text{Therefore}\ r=2.\ \text{Substitute:}\\\\a_n=6\cdot2^{n-1}=6\cdot2^n\cdot2^{-1}=6\cdot2^n\cdot\dfrac{1}{2}=\boxed{3\cdot2^n}[/tex]

Answer: Paula, Cory, Krista

Step-by-step explanation: You would divide the attempted baskets with the ones that they already did make: Cory got 35% of her attempted baskets (21 divided by 60), Krista got 40% of her attempted baskets (16 divided by 40), and Paula got 34% of her attempted baskets (17 divided by 50). Hope this helped!

Answer:

Krista cory paul

Step-by-step explanation:

saw it after searching on google

Answer:

DB > CB

Step-by-step explanation:

This is a short answer, but I promise that it is 100% correct.

Hope it helps.

By applying the combinatorics principle in mathematics, the student has a total of 2304 possible sets of courses to choose from across the five disciplines.

The subject of this question is a mathematical problem related to combinatorics, specifically the counting principle. When you're choosing one item from a set, like how many courses you can choose, you're often dealing with combinations or permutations. In this case, since the order of taking the courses does not matter, we're dealing with a type of combination.

To find the total number of course sets, simply multiply the number of possible courses in each discipline: 4 options in humanities, 6 in sociology, 6 in science, 2 in math, and 8 in music. So, the total number of different course sets you could take is 4 * 6 * 6 * 2 * 8 = 2304 different sets of five courses.

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D) 4 cm, 3 cm

Let x represent the length of "another" side. Then "one side" can be represented by (2x -5 cm).

The perimeter of the kite is the sum of two sides of each length:

... P = 14 cm = 2(x) + 2(2x -5 cm)

Dividing by 2 and collecting terms, we have ...

... 7 cm = 3x -5 cm

... 12 cm = 3x

... 4 cm = x . . . . the length of "another" side

... 2(4 cm) -5 cm = 3 cm . . . . the length of "one side"

The two different side lengths are 4 cm and 3 cm.

Answer:

10 =x

Step-by-step explanation:

The figure in the diagram is an equilateral triangle, which means all sides are equal and all angles are equal.  If all angles are equal, each angle is equal to 60 degrees  (180/3=60).  Therefore  <B =60.

60 = 5x+10

Subtract 10 from each side

60-10 = 5x+10 -10

50 = 5x

Divide each side by 5

50/5 = 5x/x

10 =x

Answer:

x = 127 degrees

Step-by-step explanation:

By the external angle of a triangle theorem:-

x + 5 = 79 + m < ACB

m < ACB = 180 - x, so:-

x + 5 = 79 + 180 - x

2x = 79 + 180 - 5 = 254

x = 127  ( answer)

Answer:

If you look at my screenshot it will show you the correct answer.

Step-by-step explanation:

Answer:

5700

Step-by-step explanation:

step 1 Address the formula, input parameters & values.

Input parameters & values:

The number series 2, 4, 6, 8, 10, 12, .  .  .  .  , 150.

The first term a = 2

The common difference d = 2

Total number of terms n = 75

step 2 apply the input parameter values in the AP formula

Sum = n/2 x (a + Tn)

= 75/2 x (2 + 150)

= (75 x 152)/ 2

= 11400/2

2 + 4 + 6 + 8 + 10 + 12 + .  .  .  .   + 150 = 5700

Therefore, 5700 is the sum of first 75 even numbers.

Final answer:

To find the sum of the first 75 even numbers starting with 2, use the sum of arithmetic sequence formula: (75/2) * (2 + 150) = 5,700.

Explanation:

The sum of the first 75 even numbers starting with 2 can be calculated using the formula for the sum of an arithmetic sequence. An arithmetic sequence is a sequence of numbers with a constant difference between consecutive terms. In this case, the common difference is 2, since we are dealing with even numbers.

To find the sum of the first 75 even numbers, we use the formula: Sum = (n/2) * (first term + last term), where 'n' is the number of terms. The first term is 2 and the 75th even number can be found by the formula: last term = first term + (n-1)*difference. Thus, the last term = 2 + (75-1)*2 = 2 + 148 = 150.

Using the sum formula: Sum = (75/2) * (2 + 150) = 37.5 * 152 = 5,700. Therefore, the sum of the first 75 even numbers starting with 2 is 5,700.

Answer:

3/14

Step-by-step explanation:

Probability( Drawing an orange marble) = number of orange marbles/ total number of marbles in the bag.

= 6 / (8+9+5+6)

= 6/28

= 3/14

Answer: 5

Step-by-step explanation: Take 2 from 7 to get 5

Answer:

The denominators of fractions are : 2 and 10

Step-by-step explanation:

[tex]\text{Let the two fractions be : }\frac{1}{x}\thinspace and\thinspace \frac{1}{y}\\\\\implies \frac{1}{x}+\frac{1}{y}=\frac{3}{5}\\\\\implies 5\cdot x +5\cdot y-3\cdot x\cdot y=0[/tex]

By hit and trial method, if we put x = 2 and y = 10 then the resulting equation (1) is satisfied and all the conditions hold.

So, the denominators are 2 and 10

Through the hit and trial method, the value of the denominator 'a' is 2 and the value of denominator 'b' is 10 and there is also the use of arithmetic operations.

Given :

To determine the denominator of the two fractions whose sum is 3/5, first, let that numbers be 1/a and 1/b.

[tex]\dfrac{1}{a}+\dfrac{1}{b}= \dfrac{3}{5}[/tex]

[tex]5a + 5b = 3ab[/tex]

Now, put a = 2 then b becomes:

[tex]10 + 5b = 6b[/tex]

b = 10

So, through the hit and trial method, the value of a is 2 and the value of b is 10.

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To find the measures of the angles in ΔADE, we can apply the angle bisector theorem and the fact that DE is parallel to AB. The measures of ∠ADE, ∠AED, and ∠DAE are denoted as x, y, and z, respectively.

In ΔADE, we know that AD is the angle bisector of ∠A and BE is the angle bisector of ∠B. We also know that DE is parallel to AB. Given that m∠ADE is 34° smaller than m∠CAB, we need to find the measures of the angles in ΔADE.

Let's denote the measures of ∠ADE, ∠AED, and ∠DAE as x, y, and z, respectively.

From the angle bisector theorem, we know that ∠CAD = ∠DAB = y+z.

Since DE is parallel to AB, we have ∠ADE = ∠CAB = x+y+z.

Therefore, the measures of the angles in ΔADE are ∠ADE = x, ∠AED = y, and ∠DAE = z.

Answer: (A) 2x² + 5x + 15 + [tex]\bold{\dfrac{48}{x-3}}[/tex]

Step-by-step explanation:

x - 3 = 0   ⇒   x = 3

Using synthetic division:

3   |   2    -1     0    3

    |   ↓     6    15   45        

        2     5    15   48 ← remainder

Factored polynomial is: 2x² + 5x + 15 + [tex]\dfrac{48}{x-3}[/tex]

Answer:

Robbie pay in twelve months $1080 .

$92.47 more Robbie pay and 9.36 %(Approx) Robbie pay.

Step-by-step explanation:

As given

Robbie Morse has a $47,500 insurance policy.

The annual premium is $987.53.

Robbie pays $90.00 monthly.

First find out the Robbie pay in twelve months .

Robbie pay in twelve months = 12 × 90

                                                  = $ 1080

Second find out how much more does Robbie pay .

Robbie pay extra = Robbie pay in twelve months -  annual premium

                             = $ 1080 - $987.53

                             = $92.47

Therefore the $92.47 more Robbie pay .

Third find out the what percentage does Robbie pay .

 Formula

[tex]Percentage = \frac{Robbies\ pay\ more\times 100}{annual\ premium}[/tex]

Robbies pay more = $92.47

Annual premium = $987.53

Put in the above

[tex]Percentage = \frac{92.47\times 100}{987.53}[/tex]

[tex]Percentage = \frac{9247}{987.53}[/tex]

Percentage = 9.36 % (Approx)

9.36 %(Approx) Robbie pay.

Robbie pays a total of $1,080.00 over twelve months, which is $92.47 more than the annual premium. This amount is approximately 9% more than the original annual premium.

To determine how much Robbie pays for his monthly premium over twelve months, we need to perform the following calculations:

First, calculate the total amount Robbie pays monthly. He pays $90.00 monthly, so over twelve months:

$90.00  times 12 = $1,080.00

Next, we need to find out how much more Robbie pays compared to the annual premium of $987.53:

$1,080.00 - $987.53 = $92.47

Percentage Calculation:

To find the percentage Robbie pays compared to the original annual premium, use the formula:

(difference / original amount) times 100

($92.47 / $987.53) times 100 ≈ 9%

Thus, Robbie pays approximately 9% more than the annual premium.

Daniel is 25% older than Patricia. Ten years ago, Daniel was 50% older than Patricia.

That gives us the equations 1.25d=p and 1.5(d−10)=(p−10). Solving gives us 25 for Daniel and 20 for Patricia.

Paul is two years younger than Patricia.

20−2=18

To solve the problem, we set up an equation based on the given relationships and information. After solving it, we find that Patricia is 40 years old and Paul (who is 2 years younger) is 38 years old.

Let's denote Patricia's current age as 'P', Paul's age as 'P-2' (since Paul is 2 years younger than Patricia), and Daniel's age as 'P+0.25P' (since Daniel is 25% older than Patricia).

According to the problem, ten years ago, Daniel was 50% older than Patricia. So, we create the equation '0.5(P-10) = P+0.25P-10' to represent this situation.

Solving the equation, we find P (Patricia's current age) equals 40 years. Hence, Paul's current age would be 'P-2', which is 38 years.

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There are 479001600 different ways the first 12 letters of the alphabet can be arranged.

The first 12 letters of the alphabet can be arranged in 479,001,600 different ways.

The question asks how many different ways the first 12 letters of the alphabet can be arranged. This type of problem is addressed by using permutations, which is a concept in combinatorics, a branch of mathematics. When we talk about arranging a set of items, we are often dealing with permutations.

The formula to determine the number of permutations of a set of n distinct objects is given by n!, which is read as 'n factorial'. The factorial of a number n is the product of all positive integers less than or equal to n.

So, for the first 12 letters of the alphabet, which are 'A, B, C, D, E, F, G, H, I, J, K, L', the number of different ways to arrange these letters would be:

12! (12 factorial)

To calculate 12!, you multiply all the whole numbers from 1 to 12 together:

12! = 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1

= 479,001,600

[tex]k=\left[\dfrac{-4}{2},\ \dfrac{6}{-3}\right]=[-2,\ -2]\\\\m=\left[\dfrac{2}{8},\ \dfrac{-2}{5}\right]=\left[\dfrac{1}{4},\ -\dfrac{2}{5}\right]\\\\x=[a,\ b]\\\\2x-k=m.\ \text{Substitute:}\\\\2[a,\ b]-[-2,\ -2]=\left[\dfrac{1}{4},\ -\dfrac{2}{5}\right]\\\\\ [2a,\ 2b]-[-2,\ -2]=\left[\dfrac{1}{4},\ -\dfrac{2}{5}\right]\\\\\ [2a-(-2),\ 2b-(-2)]=\left[\dfrac{1}{4},\ -\dfrac{2}{5}\right]\\\\\ [2a+2,\ 2b+2]=\left[\dfrac{1}{4},\ -\dfrac{2}{5}\right]\iff2a+2=\dfrac{1}{4}\ \wedge\ 2b+2=-\dfrac{2}{5}[/tex]

[tex]2a+2=\dfrac{1}{4}\qquad\text{subtract 2 from both sides}\\\\2a=\dfrac{1}{4}-2\\\\2a=\dfrac{1}{4}-\dfrac{8}{4}\\\\2a=-\dfrac{7}{4}\qquad\text{divide both sides by 2}\\\\\boxed{a=-\dfrac{7}{8}}\\\\2b+2=-\dfrac{2}{5}\qquad\text{subtract 2 from both sides}\\\\2b=-\dfrac{2}{5}-2\\\\2b=-\dfrac{2}{5}-\dfrac{10}{5}\\\\2b=-\dfrac{12}{5}\qquad\text{divide both sides by 2}\\\\\boxed{b=-\dfrac{6}{5}}\\\\Answer:\ \boxed{x=\left[-\dfrac{7}{8},\ -\dfrac{6}{5}\right]}[/tex]

Answer:

It's B on edge 2021

Step-by-step explanation:

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