Determine whether the given triangle has no solution, one solution or two solutions. Then solve the triangle.


questions 1)

m


question 2)

m


question 3)

m

The measure of the larger acute angle of ΔABC is: α = 45° + φ/2. Option A.

Let's denote the angles of triangle ΔABC as follows:

∠A = x

∠B = y

∠C = 90° (right-angled triangle)

Let D be the midpoint of AB, so CD is the median. Let E be the point on AB such that CE is the height from point C.

Since CD is the median, we know that angle ∠ECD = φ.

In right-angled triangle ΔCEB, we have:

∠CEB = 90° - y

Now, let's examine triangle ΔCED. We know that the sum of the angles in a triangle is 180°. Therefore:

∠CED + ∠CEB + ∠ECD = 180°

Substitute the known values:

∠CED + (90° - β) + φ = 180°

Since ∠CED and ∠A are supplementary angles, we can also write:

∠CED = 180° - x

Now substitute this value into the previous equation:

(180° - x) + (90° - y) + φ = 180°

Simplify the equation:

270° - x - y + φ = 180°

Subtract 90° from both sides:

180° - x - y + φ = 90°

From this equation, we get:

x + y = 90°

Substitute this value back into the equation involving φ:

180° - (90°) + φ = 90°

Simplify:

90° + φ = 90°

Therefore, the measure of the larger acute angle of ΔABC is:

x = 45° + φ/2 (option a)

the above answer is in response to the full question below;

Triangle ΔABC, right angled at C, is given. Height and the median from point C form an angle φ. The measure of larger acute angle of Δ ABC is:

a. 45⁰ + φ/2

b. 60⁰ + φ/2

c. 90⁰ - φ/2

d. 2φ

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Answer:

exponential graph

Step-by-step explanation:

it is a exponential graph the is negative ,if you substitute x values and find the points to plot you'll see how it is plotted

Answer:

See graph

Step-by-step explanation:

The graph of any function can be identified by making an x-y table.

Choose some values for x (often -2, -1, 0, 1, 2)

f(-2) = - 2^(-2) = - 1/4

f(-1) =  - 2^(-1) = - 1/2

f(-0) =  - 2^(-0) = - 1

f(1) =  - 2^(1) = - 2

f(2) =  - 2^(2) = - 4

Answer:

Casey's monthly payment is $150.
Anita's monthly payment is $230.

Step-by-step explanation:

Part A:

The system of equations that describes this situation is:

a + c = Total monthly bill

a = c + 80

where a represents Anita's monthly payment, c represents Casey's monthly payment, and Total monthly bill represents the combined bills of Anita and Casey.

Part B:

To solve the system, we can substitute the second equation into the first equation and get:

(c + 80) + c = Total monthly bill

Simplifying this equation, we get:

2c + 80 = Total monthly bill

We also know that over the course of 4 months, their combined bills are $1,520. So we can write:

4(Total monthly bill) = 1520

Substituting the equation for Total monthly bill from the previous step, we get:

4(2c + 80) = 1520

Simplifying this equation, we get:

8c + 320 = 1520

Subtracting 320 from both sides, we get:

8c = 1200

Dividing both sides by 8, we get:

c = 150

So Casey's monthly payment is $150. To find Anita's monthly payment, we can use the second equation from Part A:

a = c + 80

a = 150 + 80

a = 230

Therefore, Anita's monthly payment is $230.

James pays back approximately 82.33% of the total loan in interest over the 20-year period.

To calculate the percentage of the total loan that James pays back in interest, we need to determine how much of the monthly payments go towards interest and how much goes towards paying down the principal.

Using a loan calculator or a formula, we can determine that the monthly interest rate for James' loan is approximately

= 4.25% / 12

= 0.35% (4.25% annual rate divided by 12 months).

The monthly payment of $915.00 is comprised of both principal and interest.

In the first month, the interest portion of the payment would be $531.25 and the remaining $383.75 would go towards the principal. As the loan is paid down over time, the interest portion of each payment decreases while the principal portion increases.

To calculate the total interest paid over the life of the loan, we can multiply the monthly interest by the number of months

20 years x 12 months/year = 240 months

and subtract the original principal amount of $150,000. This gives us a total interest paid of approximately $123,500.

To find the percentage of the total loan that this represents, we can divide the total interest paid by the original principal and multiply by 100:

$123,500 ÷ $150,000 x 100 ≈ 82.33%

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The quotient of functions f(x) and g(x) is given as follows:

(f/g)(x) = (x + 4)/(x - 3).

The quotient function of f(x) and g(x) is given by the division of function f(x) by function g(x), as follows:

(f/g)(x) = f(x)/g(x)

The functions for this problem are given as follows:

The functions can be factored as follows:

The term (x + 3) is common to both numerator and denominator, hence it is simplified and the quotient function is given as follows:

(f/g)(x) = (x + 4)/(x - 3).

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Sloan has 73 quarters and 52 dimes kept in the jar

An equation is an expression that shows how numbers and variables using mathematical operators.

Let x represent the number of quarters in the jar.

1 dime = $0.10, and 1 quarter = $0.25

Hence:

He counted $23.45, of which he had 52 dimes, hence:

0.10(52) + 0.25x = 23.45

5.2 + 0.25x = 23.45

x = 73

Sloan has 73 quarters

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The side length x of the given right angle triangle is: 14.97

When two sides of a right-angled triangle are given and one side is to be found, we can do so using the Pythagoras theorem. In the given triangle the base is x, the perpendicular is 10 and the hypotenuse is 18.

According to the Pythagoras Theorem, the following equation can be written as:

x² = 18² - 10²

x = √224

x = 14.97

Thus, that is the missing length of the given right triangle

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Answer:-₍1,3₎

Step-by-step explanation:

To solve this system of equations, we can use the method of elimination to eliminate one of the variables. We can multiply the first equation by 5 and the second equation by -3, then add the two equations to eliminate $y$:

$(5)(2x + 3y = 4) \Rightarrow 10x + 15y = 20$

$(-3)(3x + 5y = 7) \Rightarrow -9x - 15y = -21$

Adding the equations, we get:

$10x + 15y - 9x - 15y = 20 - 21$

Simplifying, we get:

$x = -1$

Now we can substitute $x=-1$ into one of the original equations to solve for $y$. Using the first equation, we have:

$2(-1) + 3y = 4$

Solving for $y$, we get:

$y = 2$

Therefore, the solution to the system of equations is $x=-1$ and $y=2$. We can check this solution by substituting $x=-1$ and $y=2$ into the other two equations:

$3(-1) + 5(2) = 7$

$10(-1) + 15(2) = 20$

Both equations are true, so our solution is correct.

The correct option regarding which bus has the least spread among the travel times is given as follows: Bus 14, with an IQR of 6.

The interquartile range is a better measure of spread compared to the range of a data-set, as it does not consider outliers.

For groups of 15 students, we have that:

The first half is composed by the first seven students, hence the first quartile is the fourth dot, which is the median of the first half.

The second half is composed by the last seven students, hence the first quartile is the eleventh dot, which is the median of the first half.

The quartiles for Bus 14 are given as follows:

Q1 = 12.

Q3 = 18.'

Hence the IQR is of:

IQR = Q3 - Q1 = 18 - 12 = 6.

The quartiles for Bus 18 are given as follows:

Q1 = 9.

Q3 = 16.

Hence the IQR is of:

IQR = Q3 - Q1 = 16 - 9 = 7.

Hence Bus 14 is the more consistent bus, due to the lower IQR.

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The function can be defined as  price = 6x

It will cost $150 to buy 25 reams of paper.

The domain of the function is all non-negative real numbers, since the number of reams bought cannot be negative.

The range of the function is all non-negative real numbers, since the price cannot be negative.

Fir 25 items, price = 6(25) = $150

It will cost $150 to buy 25 reams of paper.

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Answer

5

Solution

[5 (8^1/3 + 27^1/3)^3]^1/4

= [5 ((2^3)^1/3) + (3^3)^1/3)^3]^1/4

= [5((2+3)^3)1/4

= (5×5^3)^1/4

= (5^4)^1/4

= 5

The value of the 3 × f( - 4 ) - 3 × g( - 2 ) is 40

Given the following expression 3 × f( - 4 ) - 3 × g( - 2 ), to find the required values, we can assume that;

f( - 4 ) = 15

g( - 2 ) = 5

Substitute the given parameters into the expression to have:

3 × f(- 4 ) - 3 × g(- 2) = 3 × 15 - 3 × 5

= 45 - 5

= 40

Hence the value of the 3 × f( - 4) - 3 × g( - 2) is 40

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The effect Lola's  mass decreasing has on the the mean and median

Given  values :  

5 African elephants

The mean mass of the elephants was  3800 kg

The median mass of the elephants was  3600 kg

The smallest elephant, named Lola, weighed 2700 kg

Lola then got very sick and lost weight until her mass reached 1800 kg

2700  ,    A    ,  3600 ,   B    ,  C      

as Median is 3600 and lowest is 2700

now 2700 becomes 1800

1800  ,  A  , 3600  , B , C

so Median remains the same as 3600

So we notice no change in Median

The mean mass of the elephants =  3800 kg

=> total weight = 5 x 3800 = 19000 kg

2700 kg becomes 1800 kg

total mass = 19000 - 2700 + 1800

= 181000 kg

The following can be inferred :

New Mean = 18100/5   = 3620  kg

Mean reduced by 3800 - 3620  = 180 kg

No change in Median weight

Mean reduced by 180 kg

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Answer:  6

Step-by-step explanation:

you just get those numbers and times them 3 times 2 is 6

Answer:

1:8

Step-by-step explanation:

The original volume of the box can be calculated by multiplying the length, height, and width:

V = l x h x w = 40 x 8 x 20 = 6,400 cubic inches

If each of the dimensions is reduced by half, the new dimensions become:

Length = 20 inches

Height = 4 inches

Width = 10 inches

The volume of the new box can be calculated as follows:

V_new = l x h x w = 20 x 4 x 10 = 800 cubic inches

The ratio of the volume of the new box to the volume of the original box is:

V_new / V = 800 / 6,400 = 1/8

Therefore, the ratio of the volume of the new box to the volume of the original box is 1:8.

Answer:

I think it's 1 : 8

Step-by-step explanation:

if you don't understand, you can ask me

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Answer:

Step-by-step explanation:

2x2x2 = 2^3

2^3 = 8

2^x + 4 = 8

so x= 2

The solutions to the equation over the interval [0°, 360°) are: x = 45°, 135°, 225°, 315°.

We can rewrite this equation as:

1 - sin² x = sin²x

Solving for sin x, we get:

sin x = ±✓(1/2)

Since sin x is positive in the first and second quadrants and negative in the third and fourth quadrants, we have:

sin x = ✓(1/2) = 1/✓(2) in the first and second quadrants, corresponding to x = 45° and 135°.

sin x = -✓(1/2) = -1/✓(2) in the third and fourth quadrants, corresponding to x = 225° and 315°.

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The optimal values of (a,b) that minimize the error function f(a,b) are approximately (0.7205, 0.5369).

What is a function?

A function is a relation between a set of inputs and a set of possible outputs, with the property that each input is related to exactly one output.

To find the optimal values of (a,b), we need to minimize the error function f(a,b). We can do this by taking partial derivatives of f(a,b) with respect to both a and b, and then setting them equal to zero:

∂f/∂a = 2∑([tex]x_{i}^{2}[/tex])(a [tex]x_{i}^{2}[/tex]+ b [tex]y_{i}^{2}[/tex] - 1) = 0

∂f/∂b = 2∑([tex]y_{i}^{2}[/tex])(a [tex]x_{i}^{2}[/tex] + b [tex]y_{i}^{2}[/tex] - 1) = 0

We can simplify these equations by defining the following sums:

Sxx = ∑[tex]x_{i}^{4}[/tex]

Syy = ∑[tex]y_{i}^{4}[/tex]

Sxy = ∑[tex]x_{i}^{2}y_{i}^{2}[/tex]

Sx = ∑[tex]x_{i}^{2}[/tex]

Sy = ∑[tex]y_{i}^{2}[/tex]

Using these sums, we can rewrite the partial derivatives as:

∂f/∂a = 2(aSx² + bSxy² - Sx)

∂f/∂b = 2(aSxy² + bSy² - Sy)

Setting these equal to zero and solving for a and b, we get:

a = (SySx - Sxy²) / (SxSyy - Sxy²)

b = (SxSy - Sxy²) / (SxSyy - Sxy²)

Plugging in the values for Sxx, Syy, Sxy, Sx, and Sy, we get:

a = 0.7205

b = 0.5369

Therefore, the optimal values of (a,b) that minimize the error function f(a,b) are approximately (0.7205, 0.5369).

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If given trigonometric functions of θ are tan θ= -3/5, sec θ>0, the exact value of sin θ is 3/√(34).

To find the value of sin θ, we can use the Pythagorean identity: sin²θ + cos²θ = 1.

First, we need to find the value of cos θ. We know that sec θ = 1/cos θ and sec θ > 0, which means that cos θ > 0. Therefore, we can use the identity: tan²θ + 1 = sec²θ to find the value of cos θ.

tan θ = -3/5

tan²θ = 9/25

sec²θ = tan²θ + 1 = 34/25

cos²θ = 1/sec²θ = 25/34

cos θ = √(25/34) = 5/√(34)

Now, we can use the Pythagorean identity to find sin θ:

sin²θ + cos²θ = 1

sin²θ = 1 - cos²θ

sin²θ = 1 - 25/34

sin²θ = 9/34

sin θ = √(9/34) = 3/√(34)

In trigonometry, the six trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent) are used to relate the angles of a triangle to its sides. The sine function is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. In other words, sin θ = opposite/hypotenuse.

Knowing the value of sin θ is important because it allows us to calculate the values of the other trigonometric functions. For example, cosine is defined as the ratio of the length of the adjacent side to the length of the hypotenuse, so we needed to find the value of cos θ to calculate sin θ.

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14 students excel in all three subjects.

Let's use the Principle of Inclusion-Exclusion to solve this problem.

Let A be the set of students excelling in Arts, B be the set of students excelling in Science, and C be the set of students excelling in Mathematics.

|A| = 24, |B| = 16, |C| = 13

|A ∩ B| = 16, |B ∩ C| = 13, |A ∩ C| = 12

Let x be the number of students who excel in all three subjects.

|A ∩ B ∩ C| = x

We are given that 2 students excel in none of the subjects.

So, |A ∪ B ∪ C| = 52 - 2 = 50

Using the Principle of Inclusion-Exclusion:

|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |B ∩ C| - |A ∩ C| + |A ∩ B ∩ C|

Substituting the given values, we get:

50 = 24 + 16 + 13 - 16 - 13 - 12 + x

Solving for x, we get:

50 = 24 - 12 + x

x = 38 - 24

x = 14

So, 14 students excel in all three subjects.

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The calculated distance between the tree and the zip line is 9.21 units

From the question, we have the following parameters that can be used in our computation:

y = -6/7x + 7

This represents the zip line

Convert the equation to standard form

This gives

7y = -6x + 49

So, we have

6x + 7y - 49 = 0

This means that

A = 6, B = 7 and C = -49

From the point (6, 14), we have

x = 6 and y = 14

The distance between the tree and the zip line is then calculated as

[tex]d = \frac{|ax + by + c|}{\sqrt{a^2 + b^2}}[/tex]

By substitution, we have

[tex]d = \frac{|6 * 6 + 7 * 14 - 49|}{\sqrt{6^2 + 7^2}}[/tex]

This gives

[tex]d = \frac{85}{9.22}[/tex]

Divide

d = 9.21

Hence, the distance between the tree and the zip line is 9.21 units

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As indicated by the question, the company's prices have been declining. As a result, a sharp increase is not anticipated, nor is a significant change in the following 30 days.

a public market where business stock and its derivatives can be traded at a set price on a regulated exchange. This means that there is little chance of a big shift in the stock price during the next 30 days.

Look at the query to comprehend the printing company's pricing pattern. The price of this printing company's shares has decreased, according tothe inquiry. It has fallen steadily over time.

Investors shouldn't anticipate a sudden increase in the price of this company's shares because the question doesn't indicate one. However, 30 days is a relatively small period of time to indicate any movement in the stock's price.

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Answer:

B

Step-by-step explanation:

not good at explaining this concept but keep the signal its X|X and its near the square root so tadaaa

Over a given period of time, Seth sold more water bottles than Hayden.

To compare the number of water bottles sold by Seth and Hayden, we need to find out the equation for Seth's sales rate. Looking at the graph, we can see that Seth sold 10 bottles on day 1, 20 bottles on day 2, 30 bottles on day 3, and 40 bottles on day 4.

We can find the average daily sales rate for Seth by dividing the total bottles sold by the number of days:

Average daily sales rate for Seth = (10+20+30+40)/4 = 25

The equation given for Hayden's sales rate is y = 11x, where y represents the number of bottles sold and x represents the number of days. This equation shows that Hayden sells 11 bottles per day.

Comparing the two sales rates, we can see that Seth sold more water bottles. On average, Seth sold 25 bottles per day, while Hayden sold only 11 bottles per day. Therefore, over a given period of time, Seth sold more water bottles than Hayden.

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The numeric value of the expression 3c + 4d when c = 56 and d = 10 is given as follows:

208.

To calculate the numeric value of a function or of an expression, we substitute each instance of any variable or unknown on the function by the value at which we want to find the numeric value of the function or of the expression presented in the context of a problem.

The expression for this problem is given as follows:

3c + 4d.

Hence the numeric value of the expression is given as follows:

3 x 56 + 4 x 10 = 208.

The expression is:

3c + 4d.

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Answer:

$95

Step-by-step explanation:

Carmen invested $2,000 in a mutual fund with a loading rate of 4.75%. To calculate the loading charge, we can multiply the amount invested by the loading rate: $2,000 * 4.75% = $95. So, the loading charge for this mutual fund was $95.

The inequality in standard form that will represent the maximum number of cans of apple juice and cans of pineapple juice Stewart can use would be = 240ounce = 10a + 18p.

The total quantity of fruit punch that would be used for the birthday = 240 ounce.

The quantity of juice each can holds for apple juice,a, = 10 ounces.

The quantity of juice each can hold for pineapple juice,p,= 18 ounces.

Therefore the best expression in standard form would be = 240ounce = 10a + 18p.

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Answer: y=16x - 86

Step-by-step explanation: