A piggy bank contains some dimes and nickels. There are 8 more dimes than nickels in the bank. There is a total of $1.40. How many of each type of coin are in the bank?

Answer:

Step-by-step explanation:

To find the reference angle for an angle given in degrees, you can follow these steps:

Determine the absolute value of the given angle.

If the angle is more significant than 360 degrees, subtract the largest possible multiple of 360 degrees to bring it within the range of 0 to 360 degrees.

If the angle is negative, convert it to a positive angle by adding 360 degrees.

The reference angle is the acute angle formed between the terminal side of the angle and the x-axis.

Let's apply these steps to the given angle t = -216 degrees:

Absolute value of -216: | -216 | = 216 degrees

216 degrees is already within the 0 to 360-degree ranges, so there is no need to subtract any multiple of 360 degrees.

Since the angle is negative, convert it to a positive angle: 216 degrees

The reference angle is the acute angle formed with the terminal side of the angle, which is 216 degrees.

Therefore, the reference angle for t = -216 degrees is 216 degrees.

Answer:

Option C is correct.

Step-by-step explanation:

-x+3y=2

4x-2y=22

In matrix form is represented as:

[tex]\left[\begin{array}{cc}-1&3\\4&-2\end{array}\right] \left[\begin{array}{c}x&y\end{array}\right] =\left[\begin{array}{c}2&22\end{array}\right][/tex]

AX=B

[tex]X = A^{-1}B[/tex]

[tex]A^{-1} = |A|/Adj A[/tex]

|A| = (-1)(-2)-(3)(4)

|A| = 2-12

|A| = -10

Adj A = [tex]\left[\begin{array}{cc}-2&-3\\-4&-1\end{array}\right][/tex]

A^-1 = -1/10[tex]\left[\begin{array}{cc}-2&-3\\-4&-1\end{array}\right][/tex]

A^-1 = 1/10[tex]\left[\begin{array}{cc}2&3\\4&1\end{array}\right][/tex]

X= A^-1 B

X = 1/10[tex]\left[\begin{array}{cc}2&3\\4&1\end{array}\right][/tex][tex]\left[\begin{array}{c}2&22\end{array}\right][/tex]

X=1/10[tex]X=1/10\left[\begin{array}{c}2*2+3*22\\4*2+1*22\end{array}\right]\\X=1/10\left[\begin{array}{c}4+66\\8+22\end{array}\right]\\X=1/10\left[\begin{array}{c}70\\30\end{array}\right]\\X=\left[\begin{array}{c}70/10\\30/10\end{array}\right]\\X=\left[\begin{array}{c}7\\3\end{array}\right][/tex]

So, x = 7 and y =3

Hence Option C is correct.

Answer:

7

Step-by-step explanation:

right on edge

Let r = amount customer gets back

r = 1000 - (1000)(0.03)

r = 1000 - 30

r = $970

Answer:

see explanation

Step-by-step explanation:

a

The tangent and the radius at the point of contact form a right angle

Using Pythagoras' identity on the right triangle formed.

Let x be the distance from the centre to P, then

x² = 4² + 10² = 16 + 100 = 116 ( take the square root of both sides )

x = [tex]\sqrt{116}[/tex] ≈ 10.77 cm (to 2 dec. places )

b

let the required angle be Θ, then

Using the sine or cosine ratio in the right triangle.

cosΘ = [tex]\frac{adjacent}{hypotenuse}[/tex] = [tex]\frac{10}{\sqrt{116} }[/tex]

Θ = [tex]cos^{-1}[/tex] ( [tex]\frac{10}{\sqrt{116} }[/tex] ) ≈ 21.8°

The distance from point P to the center of the circle is approximately 10.77 cm, and the angle between the tangent at P and the line joining P to the center of the circle is 90 degrees.

Let's address each part of the question about a tangent to a circle and its properties:

We can visualize a right triangle where one leg is the radius (4 cm), the other leg is the tangent (10 cm), and the hypotenuse is the line from point P to the center of the circle. Using the Pythagorean theorem (a² + b² = c²), we compute the hypotenuse: c² = 4² + 10², so c² = 16 + 100, which means c = √116, and c ≈ 10.77 cm. So, the distance from P to the center of the circle is approximately 10.77 cm.

An important property of a tangent to a circle is that it is perpendicular to the radius at the point of contact. Therefore, the angle between the tangent and the radius is 90 degrees. Because we are looking for the angle between the tangent and the line joining P to the center, which is the hypotenuse and also includes the radius, the angle remains 90 degrees.

Answer:

Step-by-step explanation:

_____

Good evening ,

_______________

Look at the photo below for the answer.

___

:)

Answer: Observe the image attached.

Step-by-step explanation:

You have  the  points (4, 14), (22, 6), and (16, 18).

It is important to remember that he first number of each point is the x-coordinate of that point and the second number of each one of them is the y-coordinate of that point.

Therefore, knowing the above, you can mark each point, as you can observe in the image attached, and then you can connect the points to form the triangle shown in the image.

Answer:

[tex]\large\boxed{y>\dfrac{5}{2}x-3}[/tex]

Step-by-step explanation:

<, > - dotted line

≤, ≥ - solid line

<, ≤ - shaded region below the line

>, ≥ - shaded region above the line

We have dotted line (<, >) and shaded region above the line (>, ≥).

Therefore your answer is:

[tex]y>\dfrac{2}{5}x-3[/tex] or [tex]y>\dfrac{5}{2}x-3[/tex]

Calculate the slope.

The formula of a slope:

[tex]m=\dfrac{y_2-y_1}{x_2-x_1}[/tex]

Put the coordinates of the given points from the graph:

(0, -3) and (2, 2):

[tex]m=\dfrac{2-(-3)}{2-0}=\dfrac{5}{2}[/tex]

Step-by-step explanation:

5^(3b−1) = 5^(b−3)

Since the bases are equal, the exponents must also be equal.

3b − 1 = b − 3

2b = -2

b = -1

Answer:

see below

Step-by-step explanation:

41

-4 ≤2+4x<0

Subtract 2 from all sides

-4-2 ≤2-2+4x<0-2

-4 ≤2+4x<0

Divide all sides by 4

-6/4 ≤4x/4<-2/4

-3/2 ≤x <-1/2

graph is attached

45

2x-3 ≤-4  or 3x+1 ≥4

Lets solve the left side first

2x-3≤-4

Add 3 to each side

2x-3+3 ≤-4+3

2x ≤-1

Divide by 2

2x/2  ≤-1/2

x  ≤-1/2

Now solve the right inequality

3x+1 ≥4

Subtract 1 from each side

3x+1-1 ≥4-1

 3x ≥3

Divide by 3

3x/3 ≥3/3

x≥1

So we have

x  ≤-1/2 or x≥1

see attached

Notice closed circles where there is a greater than equal to  or less than equal to

Answer:

Explanation:

The type of model that best fits the situation of a $500 raise in a salary each year is a linear model.

In a linear model, the dependent variable changes a constant amount for constant increments of the independent variable.

In the given case, the dependent variable is the salary and the independent variable is the year.

You may build a table to show that for increments of 1 year the increments of the salary is $500:

Year         Salary        Change in year         Change in salary

2010           A                           -                                       -

2011           A + 500     2011 - 2010 = 1          A + 500 - 500 = 500

2012          A + 1,000   2012 - 2011 = 1          A + 1,000 - (A + 500) = 500

So, you can see that every year the salary increases the same amount ($500).

In general, a linear model is represented by the general equation y = mx + b, where x is the change of y per unit change of x, and b is the initial value (y-intercept).              

In this case m = $500 and b is the starting salary: y = 500x + b.

Answer:

Choice 4.

Step-by-step explanation:

f(g(x))

Replace g(x) with x^2+8 since g(x)=x^2+8.

f(g(x))

f(x^2+8)

Replace old input,x, in f with new input, (x^2+8).

f(g(x))

f(x^2+8)

2(x^2+8)+5

Distribute:

f(g(x))

f(x^2+8)

2(x^2+8)+5

2x^2+16+5

Combine like terms:

f(g(x))

f(x^2+8)

2(x^2+8)+5

2x^2+16+5

2x^2+21

Answer:

D

Step-by-step explanation:

Took the test

To solve (sin(x))^2 = 1/36, we find the arcsine of ±1/6. The solutions are sin⁻¹(1/6), π - sin⁻¹(1/6), 2π - sin⁻¹(1/6), and π + sin⁻¹(1/6) within the interval [0,2π].

To solve the equation (sin(x))^2 = 1/36 in the interval [0,2π], we first take the square root of both sides to get sin(x) = ±1/6. The sine function oscillates between -1 and 1 every 2π radians, which means that we are looking for angles where the sine value is ±1/6.

To find the specific angles, we use the arcsine function or inverse sine function. The principal value of sin⁻¹(1/6) gives us one of the solutions, and considering the symmetry of the sine function, the other solutions can be found in the second and fourth quadrants, where the sine function is positive and negative, respectively.

The solutions to sin(x) = 1/6 in the interval [0,2π] are x = sin⁻¹(1/6) and x = π - sin⁻¹(1/6). For sin(x) = -1/6, the solutions are x = 2π - sin⁻¹(1/6) and x = π + sin⁻¹(1/6). Thus, the solutions to the original equation (sin(x))^2 = 1/36 within [0,2π] are sin⁻¹(1/6), π - sin⁻¹(1/6), 2π - sin⁻¹(1/6), and π + sin⁻¹(1/6), all of which can be calculated to find the exact values.

Answer:

Option 2 - 3.2 inches.                      

Step-by-step explanation:

Given : The lengths of two sides of a right triangle are 5 inches and 8 inches.

To find : What is the difference between the two possible  lengths of the third side of the triangle?

Solution :

According to question, it is a right angle triangle

Applying Pythagoras theorem,

[tex]H^2=P^2+B^2[/tex]

Where, H is the hypotenuse the longer side of the triangle

P is the perpendicular

B is the base

Assume that H=8 inches and B = 5 inches

Substitute the value in the formula,

[tex]8^2=P^2+5^2[/tex]

[tex]64=P^2+25[/tex]

[tex]P^2=64-25[/tex]

[tex]P^2=39[/tex]

[tex]P=\sqrt{39}[/tex]

[tex]P=6.24[/tex]

Assume that P=8 inches and B = 5 inches

Substitute the value in the formula,

[tex]H^2=8^2+5^2[/tex]

[tex]H^2=64+25[/tex]

[tex]H^2=89[/tex]

[tex]H=\sqrt{89}[/tex]

[tex]H=9.43[/tex]

Therefore, The possible length of the third side of the triangle is

[tex]L=H-P[/tex]

[tex]L=9.43-6.24[/tex]

[tex]L=3.19[/tex]

Therefore, The difference between the two possible  lengths of the third side of the triangle is 3.2 inches.

So, Option 2 is correct.

The difference between the two possible lengths of the third side, rounded to the nearest tenth, is:

B. 3.2 inches

To determine the difference between the two possible lengths of the third side of a right triangle with given side lengths of 5 inches and 8 inches, we need to consider both cases where the unknown side could be the hypotenuse or one of the legs. We use the Pythagorean theorem, [tex]\(a^2 + b^2 = c^2\)[/tex].

Case 1: The unknown side is the hypotenuse [tex](\(c\))[/tex]

[tex]\[ c = \sqrt{5^2 + 8^2} = \sqrt{25 + 64} = \sqrt{89} \approx 9.4 \, \text{inches} \][/tex]

Case 2: The unknown side is one of the legs [tex](\(a\) or \(b\))[/tex]

Assume the known hypotenuse is 8 inches. Using the Pythagorean theorem, we solve for the other leg.

[tex]\[ 8^2 = 5^2 + x^2 \][/tex]

[tex]\[ 64 = 25 + x^2 \][/tex]

[tex]\[ x^2 = 64 - 25 \][/tex]

[tex]\[ x^2 = 39 \][/tex]

[tex]\[ x = \sqrt{39} \approx 6.2 \, \text{inches} \][/tex]

Difference between the two possible lengths

The two possible lengths of the third side are approximately 9.4 inches and 6.2 inches. The difference between these lengths is:

[tex]\[ 9.4 - 6.2 = 3.2 \][/tex]

Therefore, the difference between the two possible lengths of the third side, rounded to the nearest tenth, is:

B. 3.2 inches

The correct question is:

The lengths of two sides of a right triangle are 5 inches and 8 inches. What is the difference between the two possible

lengths of the third side of the triangle? Round your answer to the nearest tenth.

A. 3.1 inches

B. 3.2 inches

C. 10.0 inches

D. 15.7 inches

Answer:

Step-by-step explanation:

Rewrite 7(x+5)/(x-3)(x+5) for greater clarity:

7(x+5)

--------------

(x-3)(x+5)

Now cancel the x + 5 terms.  We get:  

     7

-----------

   x - 3

Please note:  the problem statement mentions "which of the following ..."

This implies that there were answer choices.  Please, next time, share those answer choices.  Thank you.

[tex]7(x+5)/(x-3)(x+5) for greater clarity:7(x+5)--------------(x-3)(x+5)Now cancel the x + 5 terms. We get: 7----------- x - 3[/tex]

No. Yes. A rational algebraic expression (or rational expression) is an algebraic expression that can be written as a quotient of polynomials, such as x2 + 4x + 4. An irrational algebraic expression is one that is not rational, such as √x + 4.

The steps to solve a rational equation are:

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Answer:

The statements which accurately describe f(x) are

The domain is all real numbers ⇒ 1st answer

The initial value of 3 ⇒ 3rd answer

The simplified base is 3√2 ⇒ last answer

Step-by-step explanation:

* Lets explain how to solve the problem

- The form of the exponential function is f(x) = a(b)^x, where a is the

  initial value , b is the base and x is the exponent

- The values of a and b are constant

- The domain of the function is the values of x which make the function

  defined

- The range of the function is the set of values of y that correspond

  with the domain

* Lets solve the problem

∵ [tex]f(x)=3(\sqrt{18}) ^{x}[/tex]

- The simplest form of is :

∵ √18 = √(9 × 2) = √9 × √2

∵ √9 = 3

∴ √18 = 3√2

∴ [tex]f(x)=3(3\sqrt{2})^{x}[/tex]

∵ [tex]f(x)=a(b)^{x}[/tex]

∴ a = 3 , b = 3√2

∴ The initial value is 3

∴ The simplified base is 3√2

- The exponent x can be any number

∴ The domain of the function is x = (-∞ , ∞) or {x : x ∈ R}

- There is no value of x makes y = 0 or negative number

∴ The range is y = (0 , ∞) or {y : y > 0}

* Lets find the statements which accurately describe f(x)

# The domain is all real numbers

∵ The domain is {x : x ∈ R}

∴ The domain is all real numbers

# The initial value is 3

∵ a = 3

∵ a is the initial value

∴ The initial value of 3

# The simplified base is 3√2

∵ b = √18

∵ b is the base

∵ The simplified of √18 is 3√2

∴ The simplified base is 3√2

- For more understand look to the attached graph

Answer:

The answer is -3.

Step-by-step explanation:

-3(-3+2)-6

First solve the parenthesis. -3+2= -1.

-3(-1)-6

-3 times -1 is 3. Two negatives always equal a positive.

3-6 = -3.

Answer:

(-2.3,0.5)

Step-by-step explanation:

Take the second equation we have y= (-16-9X)/10

Then, we will subtitute the value of y on the first equation

8X - 10(-16-9X/10)=-23

Because 10 is denominator, it will delate with the numeber 10 that is multipling the -16-9X. Then the equation is

8X-(-16-9X)=-23  

then: 8x+16+9X=-23  So, 17x=-23-16   => X=-2.3

Then we put the value of X in the first equation so

9 (-2.3) -10y=-16 =>  10y=-16+20.7

So, y=0.5

Answer:

A (-2.3,0.5)

Step-by-step explanation:

Answer:

25

Step-by-step explanation:

If the mean of the temperatures in the chart is 24° with standard deviation of 4º, there has been 25 years within one  standard deviation of the mean.

27° is the temperature value that is within one standard deviation of mean.

Mean is the average of the given numbers and is calculated by dividing the sum of given numbers by the total number of numbers.

Given

Mean of the temperatures in the chart [tex]\mu[/tex] [tex]\mew[/tex]= 24°

Standard deviation [tex]\sigma[/tex] = 4º

The lower and upper bound for temperature within one standard deviation of the mean is given as:

Lower bound =  [tex]\mu[/tex] -  [tex]\sigma[/tex] = 24° - 4º =  20°

Thus, the lower bound is  = 20°

Upper bound =  [tex]\mu[/tex] + [tex]\sigma[/tex] = 24° + 4º =  28°

Thus, the upper bound is  = 28°

Now, the temperature value between (Lower bound, Upper bound) that is (20°, 28°) is said to be within one standard deviation of the mean.

Hence, 27° is the temperature value that is within one standard deviation of mean.

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Answer:

Option A is the correct answer.

Step-by-step explanation:

Period of simple pendulum is given by the expression,

            [tex]T=2\pi \sqrt{\frac{l}{g}}[/tex]

Where l is the length of pendulum, g is acceleration due to gravity.

Here given for Venus

         Period, T = 2.11 s

        Length of pendulum, l = 1 m

     We need to find acceleration due to gravity, g

Substituting

            [tex]2.11=2\pi \sqrt{\frac{1}{g}}\\\\\sqrt{g}=\frac{2\pi}{2.11}\\\\g=8.87m/s^2[/tex]

Acceleration due to gravity of Venus = 8.9 m/s²

Option A is the correct answer.

Answer: B. DE

Since DE spans the entire circle it is the diameter :)

Answer:

B) DE is the diameter.

Step-by-step explanation:

Given  : A circle with center O.

To find : Which of the following segments is a diameter of O.

Solution : We have given circle with center O.

Diameter : A straight line passing from side to side through the center of a circle .

So, segment CF and DE are diameter of a circle which are passes through the center O.

Therefore, B) DE is the diameter.

For this case we have the following polynomial:

[tex]-24a ^ 6b ^ 4-40a ^ 3[/tex]

We must find the greatest common factor of the terms of the polynomial.

The GCF of the coefficients is given by:

[tex]24 = 3 * 8\\40 = 5 * 8[/tex]

Then we look for the GFC of the variables:

We have then:

[tex]a ^ 6 = a ^ 3a ^ 3\\a ^ 3 = a ^ 3[/tex]

Finally rewriting we have: [tex]-24a ^ 6b ^ 4-40a ^ 3 = -8a ^ 3 (3a ^ 3b ^ 4 + 5)[/tex]

Answer:

the complete factored form of the polynomial is:

[tex]-8a ^ 3 (3a ^ 3b ^ 4 + 5)[/tex]

Answer: The first option. (2,2)(3,1)(4,2)

Step-by-step explanation:

[tex]\bf 3y+x=9\implies 3y=-x+9\implies y=\cfrac{-x+9}{3}\implies y=\cfrac{-x}{3}+\cfrac{9}{3} \\\\\\ y=\stackrel{\stackrel{m}{~\hfill \downarrow }}{-\cfrac{1}{3}} x+3\qquad \impliedby \begin{array}{|c|ll} \cline{1-1} slope-intercept~form\\ \cline{1-1} \\ y=\underset{y-intercept}{\stackrel{slope\qquad }{\stackrel{\downarrow }{m}x+\underset{\uparrow }{b}}} \\\\ \cline{1-1} \end{array}[/tex]

Answer:

yes

Step-by-step explanation:

The difference of squares is x^2 - y^2 = (x-y) (x+y)

36a^2 = (6a)^2

9 = (3)^2

(6a -3) (6a+3)

This is the difference of squares

The correct answer is a. Yes, 36a² - 9 is a difference of squares

The given expression is [tex]36a^2 - 9[/tex].

To determine if it is a difference of squares we need to identify if it can be written in the form of a² - b², which factorizes to (a + b)(a - b).

We can see that

36a² is a perfect square because it can be written as (6a)² and 9 is also a perfect square because it can be written as 3². Therefore, we can rewrite the expression as:

[tex]36a^2 - 9 = (6a)^2 - 3^2[/tex]

Thus, we can see that the expression 36a² - 9 is a difference of 6a square and 3 square. So, it is indeed a difference of squares.

Answer: a. Yes, 36a² - 9 is a difference of squares

Answer: 2 gallons

Step-by-step explanation:

1 gallon = 4 quarts

8 divided by 4= 2

Answer:

left 29226 Right 134

Step-by-step explanation:

volume=base*height

(22*22+(11)^2 *3.14*3/4)*38=29226

((1.4+0.6)*2*0.7+1.4*2*1.4+0.6*1.4+0.6*1.4))* 16=134

Answer:

29,225.78 m^3 to the nearest hundredth.

134.4 m^3.

Step-by-step explanation:

The building:

The area of the floor = area of the square + 3/4 * area of the circle

= 22^2 + 3/4 π 11^2.

The volume of the building =

38 * (22^2 + 3/4 π 11^2)

= 29,225.78 m^3.

The greenhouse:

The sides consist of 2 pairs of trapezoids.

Area of a side  =  2 * (0.6/2)(0.7 + 2.1)  + 2 * (1.4/2)(2.1 + 2.7)

The length is 16 m so:

Volume =  16 * [  2 * (0.6/2)(0.7 + 2.1)  + 2 * (1.4/2)(2.1 + 2.7) ]

= 134.4 m^3.

Answer:

Answer is 0.3

Step-by-step explanation:

Let the probability that the train arrives on time. = p

The probability that the train leaves on time = 0.8

The probability that the train leaves on time and arrives on time = 0.24

Then the equation will be:

0.8 * p = 0.24

Move the constant value to the R.H.S

p = 0.24/0.8

p = 0.3

Thus the probability is 0.3....

Answer:

x=-1 y= -5

Step-by-step explanation:

y = – x – 6

y = x – 4

Substitute into y  = -x-6 into the second equation

y =x-4

-x-6 = x-4

Add x to each side

-x-6+x =x-4+x

-6 =2x-4

Add 4 to each side

-6+4 =2x-4+4

-2 = 2x

Divide by 2

-2/2 =2x/2

-1 = x

Now find y

y =-x-6

y = -(-1) -6

y =1-6

y = -5

Answer:

x = -1

y = -5

Step-by-step explanation:

Given:

We'd take one of the equations above and substitute it with the y variable:

x - 4 = -x - 6

-x is smaller, so we add x in both sides:

2x - 4 = -6

Add 4 in both sides:

2x = -2

Divide 2 in both sides:

x = -1

Solve for y

y -(-1) - 6 = -5

y = -5

Our answer is x = -1, y = -5

Answer:

Quadrant III

Step-by-step explanation:

(-4,5)  is in the II quadrant    because the x is negative and the y is positive

II                       I

(-,+)                     (+,+)            

-------------------------------------  X axis

III                         IV

(-,-)                         (+.-)

reflecting over the x axis means  it would be in the third quadrant