One family spends 131 on 2 adult tickets and 4 youth tickets at an amusement park. Another family spends 139 on 4 adult and 2 youth tickets at the same park

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Explanation:

Let's say we had a triangle with side lengths a,b,c.

If we know that a = 12.25 and b = 12.25, then the third side c has its length restricted in such a way that

b-a < c < b+a

This is a variation of the triangle inequality theorem.

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

So,

b - a < c < b + a

12.25 - 12.25 < c < 12.25 + 12.25

0 < c < 24.5

which means the third side c is between 0 inches and 24.5 inches.

The value of c cannot be 0, and it cannot be 24.5 either.

Among the answer choices given, we see that only 24.0 is the only valid option here. The other two choices are too large.

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Side Note: If your teacher gave you two angles (instead of two sides) to be 12.25 degrees, and wanted the third missing angle of the triangle, then you would be correct in having 12.25+12.25+x = 180 as your first step.

This is right angle trig. We know that...

cos(18°) x hypotenuse = 100

hypotenuse = 100/cos(18°)

hypotenuse = 105.15 meters approx.

Because they want the height of the tree we want "sin(18°) x hypotenuse".

sin(18°) x 105.15 = 32.5 meters approx.

answer: 32.5 meters approx.

The required height of the tree is 32.5 meters.

Given that,
A person 100 meters from the base of a tree, observes that the angle between the ground and the top of the tree is 18 degrees.  To estimate the height h of the tree to the nearest tenth of a meter.

These are the equation that contains trigonometric operators such as sin, cos.. etc.. In algebraic operations.

Here,
let the height of the tree be x, and the slant height from the foot of the person to the top of the tree be h,
according to the question,
base length = 100
cos 18 = 100 / h
h = 105.14
Now,
sin 18 = x / h
sin 18 = x / 105.14
x = 32.5 meters

Thus, the required height of the tree is 32.5 meters.

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Answer:

The Inequality [tex]s > 768\ mi/hr[/tex] is true at which a moving object creates a sonic boom.

Step-by-step explanation:

Given:

Speed of Sound = 768 miles per hour

Also Given:

When an object travels faster than the speed of sound, it creates a sonic boom.

We need to find the inequality which is true only for speeds (s) at which a moving object creates a sonic boom

So We can say;

When;

[tex]s < 768\ mi/hr[/tex]  ⇒ Normal sound no sonic boom (false)

[tex]s = 768\ mi/hr[/tex]  ⇒ speed of sound but no sonic boom (false)

[tex]s > 768\ mi/hr[/tex]  ⇒ sonic boom is created (True)

Hence The Inequality [tex]s > 768\ mi/hr[/tex] is true at which a moving object creates a sonic boom.

Answer:

The number of tiles are covered by the painting are 12.

Step-by-step explanation:

Consider the provided information.

The painting is covering up some of the tiles on the wall as shown below:

Here it is given that there are 28 squares tiles on the wall.

Total tiles = Number of tiles covered by the painting - uncovered tiles

28 =  Number of tiles covered by the painting - 16

Number of tiles covered by the painting = 28 - 16

Number of tiles covered by the painting = 12

Hence, the number of tiles are covered by the painting are 12.

Answer:

11. ∠ABC = 96°

12. (x – 2)² + (y + 3)² = 4

Step-by-step explanation:

11. The inscribe angle (the angle inside the circle, ∠ABC) is equal to half of the outer circle.

∠ABC = 1/2∠AC

∠ABC = 1/2(192°) = 96°

12. The general equation for a circle is: (x – h)² + (y – k)² = r², where

h and k are the center of the circle (h, k), and r is the radius.

Look at the graph, the circle is centered at (2, -3), so

h=2

k=-3

and the radius of the circle is 2, so

r=2

Plug it all back into the equation:

(x – h)² + (y – k)² = r²

(x – (2))² + (y – (-3))² = (2)²

(x – 2)² + (y + 3)² = 4

Answer:

[tex]\frac{4x}{2x+y} +\frac{2y}{2x+y}=2[/tex]

Step-by-step explanation:

Given:

The expression to simplify is given as:

[tex]\frac{4x}{2x+y} +\frac{2y}{2x+y}[/tex]

Since, the denominator is same, we add the numerators and divide it by the same denominator. This gives,

[tex]\frac{4x+2y}{2x+y}[/tex]

Now, we simplify further by factoring out the common terms from the numerator and denominator if possible.

We observe that, 2 is a common factor to both [tex]4x\ and\ 2y[/tex]. So, we factor out 2 from the numerator. This gives,

[tex]\frac{2(2x+y)}{2x+y}[/tex]

Now, the term [tex]2x+y[/tex] is common in both the numerator and denominator. Hence, [tex]\frac{2x+y}{2x+y}=1[/tex]

So, the simplified form is:

[tex]=2\times \frac{2x+y}{2x+y}\\\\=2\times 1\\\\=2[/tex]

Answer:

(1) P=4a+4

(2) P=2a+4[tex]\sqrt{42}[/tex]a

Step-by-step explanation:

Well, let us call longer side b, shorter side a and diagonal d. The perimeter of a rectangle is 2*(a+b). Let us write this formula as P=2*(a+b)

In (1) it is stated that b=a+2. Hence, the perimeter of the rectangle is P=2*(a+b). In terms of b, let us write (a+2). So P=2*(a+a+2)=4a+4.

In (2) it is stated that d/a=13 or d=13a. From Pythagorean theorem d^2=a^2+b^2. Hence, b^2=168a^2 or b=2[tex]\sqrt{42}[/tex]a. Finally, P=2*(a+b)=2*(a+2[tex]\sqrt{42}[/tex]a)=2a+4[tex]\sqrt{42}[/tex]a

Answer: false is correct

The statement is false; a value one standard deviation from the mean is more likely to occur than a value three standard deviations from the mean due to the Empirical Rule for normal distributions.

The statement is false. A value one standard deviation from the mean is more likely to occur than a value three standard deviations from the mean. According to the Empirical Rule, which applies to bell-shaped and symmetric distributions, about 68 percent of the data lies within one standard deviation of the mean, while more than 99 percent of the data is within three standard deviations. This means that as you move further away from the mean, the likelihood of occurrence decreases.

Answer:

total amount paid = $ 22.9

Step-by-step explanation:

total price = $ 2.5 + $3.4(6)= $22.9

Answer:

2.50

Step-by-step explanation:

Answer:

20hours

Step-by-step explanation:

Jack can complete the painting in 30hours. The fraction he can paint per hour is thus 1/30

Trainee can complete the painting in 45hours, the fraction he can paint per hour is 1/45

If they are working together, the fraction that can paint in an hour is :

1/45 + 1/30 = 5/90 = 1/18

Now we know they can paint 1/18 of the room in an hour, the number of hours needed to completely paint the room is thus 1/1/18 = 18 hours

Rounding up answer to the nearest tenth is 20hours

Answer:

Step-by-step explanation:

If CD is a diameter of circle S, then CD goes through circle S at point S.  CS is a radius, and so is DS.  That means that they are the same length.  That also means that S is the midpoint of CD.  We can use the midpoint formula and the 2 points we are given to find the other endpoint, D.

[tex](-\frac{2}{3},\frac{3}{4})  =(\frac{\frac{4}{9}+x }{2},\frac{-\frac{5}{9}+y }{2})[/tex]

To solve for x, we will use the x coordinate of the midpoint; likewise for y.  x first:

[tex]-\frac{2}{3}=\frac{\frac{4}{9}+x }{2}[/tex]

Multiply both sides by 2 to get rid of the lowermost 2 and get

[tex]-\frac{4}{3}=\frac{4}{9}+x[/tex]

Subtract 4/9 from both sides to get

[tex]x=-\frac{16}{9}[/tex]

Now y:

[tex]\frac{3}{4}=\frac{-\frac{5}{9}+y }{2}[/tex]

Again multiply both sides by that lower 2 to get

[tex]\frac{3}{2}=-\frac{5}{9}+y[/tex]

Add 5/9 to both sides to get

[tex]y=\frac{37}{18}[/tex]

And there you go!

[tex]D(-\frac{16}{9},\frac{37}{18})[/tex]

Answer:

C

Step-by-step explanation:

The average speed can be obtained by adding the distances together and dividing by the total amount of time.

The total distance the man traveled from A to D is the addition of the distance from A to B plus the distance from B to C plus the distance from C to D. This is mathematically equal to 145 + 160 + 115 = 420 miles.

Dividing 420 miles by 6 = 70 miles per hour

Answer: 99% decrease

Step-by-step explanation: Since the number changes from 6,000 to 60, we know that it's decreasing. Now to find the percent decrease, we divide the amount of change by the original number.

The amount of change is the difference between 6,000 and 60 and the original number is 6,000.

6,000 - 60 is 5,940 so we are left with [tex]\frac{5,940}{6,000}[/tex] and dividing 6,000 into 5,940 gives us 0.99.

Finally, since the problem is asking us for the percent, we write 0.999 as a percent by moving the decimal point 2 places to the right to get 99%.

So when a number changes from 6,000 to 60, it has decreased by 99%.

Answer:

a) Max height it will reach is 327.55 ft above the ground

b) It will reach max height after 5.1 seconds

c) It will be 300 feet off the ground at 12 seconds

d) It will be 305 ft above the ground

e) The rocket will be 247.55 ft above the ground after seven seconds

Step-by-step explanation:

a) Using the equations of kinematics:

v² = v_i² + 2 g Δy

where

Therefore,

0² = 50² + 2(- 9.8) Δy        

(the negative sign shows that the positive direction is upwards. Gravity acts downwards)

Δy = -(50)² / 2(-9.8)

Δy = 127.55 ft

Thus, the maximum height that the rocket reaches will be

200 ft  + 127.55 ft

= 327.55 ft above the ground

b) Using the equations of kinematics:

Δy = [(v + v_i) / 2 ] × t

t = Δy / [(v + v_i) / 2 ]

t = 127.55 / [(0 + 50) / 2]

t = 5.1 seconds

Therefore, the rocket will reach its maximum height after 5.1 seconds.

c) Using the equations of kinematics:

t₃₀₀ = Δy / [(v + v_i) / 2 ]

t₃₀₀ = 300 / [(0 + 50) / 2]

t₃₀₀ = 12 seconds

Therefore, the rocket will reach 300 feet after 12 seconds

d) Using the equations of kinematics:

Δy = [(v + v_i) / 2 ] × t

Δy = [(0 + 50) / 2] × 4.2

Δy = 105 ft

Therefore, the rocket will be

105 ft + 200 ft

= 305 ft above the ground after 4.2 seconds

e) Using the equations of kinematics:

Δy = [(v + v_i) / 2 ] × t

Δy = [(0 + 50) / 2] × 7

Δy = 175 ft

Therefore,

175 - 127.55 = 47.55 ft

Thus, the rocket will be

47.55 ft + 200 ft

= 247.55 ft above the ground after seven seconds

Answer:

[tex]Required\ number\ of\ minutes= \frac{x}{2}+10[/tex]

Step-by-step explanation:

Usual time(side streets) taken [tex]=x\ minutes[/tex]

Jade can travel two times faster by taking the toll road so the time taken will be half from the toll road.

So travel time[tex]=\frac{x}{2}[/tex]

Jade is [tex]10\ minutes[/tex] late

So total minutes after 5:00 PM to reach home[tex]=\frac{x}{2}+10[/tex]

Answer:

It will be answer choice D.

Step-by-step explanation:

It'll take the half the usual time because she is taking a road that get's her there 2 times faster, but also add 10 because she was 10 minutes late

Answer:

Step-by-step explanation:

Let x represent the number of milliliters of 6.5% vinegar required. Then the total amount of acetic acid in the mix is ...

  6.5%·x + 5%(200 -x) = 6%·200

  1.5x = 200 . . . . . . . . . . . . . multiply by 100, subtract 1000

  x = 200/1.5 = 133 1/3 . . . . mL of 6.5% vinegar

  200-x = 66 2/3 . . . . . . . . . mL of 5% vinegar

Answer:

  60

Step-by-step explanation:

The ratio of Barbies is ...

  Jenny : Sharon = 5 : 2 = 60 : 24  . . . . . .  multiplying ratio values by 12

Jenny owns 60 Barbies to Sharon's 24.

Answer:

25:2

Step-by-step explanation:

To calculate the desired sample size with a 98% confidence level and 0.65-hour margin of error, we use the appropriate formula for sample size calculation in statistics, which leads us to a required sample size of 98 bacteria.

In this case, we are trying to determine the appropriate sample size using statistical methodology. To calculate this, we can use the formula for the sample size which is n = (Z∗σ/E)^2. Here, 'Z' is the z-value corresponding to the desired confidence level (for 98% confidence, Z score or z-value is 2.33), σ is the standard deviation (which is 4.8 hours in this case), and 'E' is the desired margin of error (0.65 hours).

Substituting these values into the formula gives: n = (2.33∗4.8/0.65)^2. After simplifying this calculation, we get n = 97.46. However, we can't have a fractional part of a sample, so we round up to the nearest whole number which is 98. So, you would need a sample size of 98 bacteria in order to achieve a 0.65-hour margin of error at a 98% confidence level.

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Good morning,

Answer:

Step-by-step explanation:

V = h × (base area)

  = 12 × (π×r²)

then 192π = 12π×r²

Then 192 = 12×r²

Then r² = 192÷12 = 16

Then r = √16 = 4.

:)

Some degenerate parabola cases form a single straight line, while other cases form one pair of parallel lines.

A degenerate hyperbola forms two lines that intersect at the vertex of the cone. We can rule out choice B.

A degenerate circle is a single point, so we can rule out choice C.

A degenerate ellipse is also a single point. Any circle is an ellipse (but not the other way around). We can rule out choice D.

Answer:

[tex]Sin(Cos^{-1} (14x))=\sqrt{1-196x^2}[/tex]

Step-by-step explandation:

First of all, from the figure we can define the cosine and sine functions as

[tex]Cos(theta)=\frac{adjacent }{hypotenuse }[/tex]

[tex]Sin(theta)=\frac{Opposite}{hypotenuse }[/tex]

And by analogy with the statement:

[tex]14x=\frac{adjacent }{hypotenuse }[/tex]

Which can be rewritten as:

[tex]\frac{14x}{1}=\frac{adjacent }{hypotenuse }[/tex]

You have then that, for the given triangle, the values of the adjacent and hypotenuse sides, are then given by:

:

Adjacent=14x

Hypotenuse=1

And according to the Pythagorean theorem:

[tex] Opposite=\sqrt{1-(14x)^2}[/tex]

Finally, by doing:

[tex]Cos^-1(14x)=theta[/tex]

We have that:

[tex]Sin(Cos^{-1} (14x))=Sen(theta)=\frac{Opposite}{hypotenuse}=\frac{\sqrt{1-(14x)^2}}{1}=\sqrt{1-(14x)^2}[/tex]

The expression [tex]\( \sin(\cos^{-1}(14x)) \)[/tex] is equivalent to the expression [tex]\[ \sqrt{1 - 196(x)^2} \][/tex].

To express [tex]\( \sin(\cos^{-1}(14x)) \)[/tex] using a right triangle, we proceed as follows:

1. Understand the expression:

[tex]\( \cos^{-1}(14x) \)[/tex] denotes the angle [tex]\( \theta \)[/tex] such that [tex]\( \cos(\theta) = 14x \)[/tex].

We are required to find [tex]\( \sin(\theta) \)[/tex].

2. Use a right triangle:

Let's consider a right triangle where:

One of the acute angles is [tex]\( \theta \)[/tex].

Assume [tex]\( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{14x}{1} \)[/tex]

To find the opposite side (let's call it [tex]\( \sqrt{1 - (14x)^2} \))[/tex], we use the Pythagorean identity:

[tex]\[ \sin(\theta) = \sqrt{1 - \cos^2(\theta)} = \sqrt{1 - (14x)^2} \][/tex]

[tex]\[ \sin(\cos^{-1}(14x)) = \sqrt{1 - (14x)^2} \][/tex]

[tex]\[ \sin(\cos^{-1}(14x)) = \sqrt{1 - 196(x)^2} \][/tex]

Answer:

She sold 14 new plans to earn $680 last week.

Step-by-step explanation:

$400 + 20(x) = $680

Take 400 away from total cost. So, 680 - 400 = 280.

Now, divide 280 by 20, and x = 14.

The equation would be $400 + 20(14) = $680

Answer:

not possible.

Step-by-step explanation:

number of invitations luanne  has to address after x days = 120-15x

number of invitations  Darius has to address after x days =  120 + 15(7-x)

so, if the number of invitations should be the same for both of them,

we have to equate their number of invitations

120-15x =  120 + 15(7-x)

subtracting 120 from both the sides,

-15x = 15(7-x) = 105 -15x

adding we 15x on both sides, we wont find any solution for x.

so, this isnt possible or the question must be wrong.

Answer:

The required equation is given by,

2 + [tex]\frac {13x}{4}[/tex] = [tex]\frac {1}{2} + \frac {11y}{2}[/tex]

Step-by-step explanation:

Let, each jar of  Liam's paint contains x quarts of paint.

Then, Liam's solution contains,

2 + [tex]3\dfrac {1}{4} \times x[/tex]  quarts of paint  or,

2 + [tex]\frac {13x}{4}[/tex] quarts of paint

and,

let, each jar of Evan's paint  contains, y quarts of paint.

Then, Evan's solution contains,

[tex]\frac {1}{2} + 5 \dfrac {1}{2} \times y[/tex] quarts of paint or,

[tex]\frac {1}{2} + \frac {11y}{2}[/tex] quarts of paint

now, according to the question,

the required equation is given by,

2 + [tex]\frac {13x}{4}[/tex] = [tex]\frac {1}{2} + \frac {11y}{2}[/tex]

To find an equation that represents the volume of paint mixed by Liam and Evan, we assume that 1 jar equals 1 quart. By adding the quarts of paint each person uses, we get the equation 2 + 3 1/4 = 1/2 + 5 1/2, meaning they both used the same total volume of paint.

To find the equation that represents the situation, we need to equate the total volume of paint used by Liam to the total volume used by Evan. We know that Liam uses 2 quarts of yellow paint and adds 3 1/4 jars of blue paint. Evan, on the other hand, uses 1/2 quart of yellow paint and adds 5 1/2 jars of red paint. Assuming that 1 jar is equivalent to 1 quart, we can simply add the volumes for Liam and Evan:

Liam's total volume = 2 quarts (yellow) + 3 1/4 quarts (blue)

Evan's total volume = 1/2 quart (yellow) + 5 1/2 quarts (red)

Since they end up with the same volume of paint, we have the equation:

2 + 3 1/4 = 1/2 + 5 1/2

To solve for quarts, simplify both sides:

5 1/4 = 6

This equation represents the volumes of paint mixed by Liam and Evan.

Answer:

36 minutes long

Step-by-step explanation:

6.82 = 2.5 + 0.12m

4.32 = 0.12m

36 = m; 36 minutes long.

Answer:

B. 36.

Step-by-step explanation:

Substituting 6.82 for m:

6.82 = 2.5 + 0.12m

0.12 m = 6.82 - 2.5

0.12m = 4.32

m = 4.32 / 0.12

m = 36 minutes.

Answer:

A is a function/B is not.

Step-by-step explanation:

A, there are 1 x value for 1 y value.

B, is not a function because for 1 x value, there are 2 y values.

Answer:

See below.

Step-by-step explanation:

Graph A is a function because it passes the vertical line test. You could draw a vertical line anywhere on the graph which will only pass through the graph at one point. It is a many-to-one relation.

Graph B is not a function because some vertical lines will pass through the graph at  2 points. It is a one-to-many relation.,

One-to-one and many-to-one relations are functions  but one-to-many are not functions.

Answer:

[tex]r=10\%[/tex]

Step-by-step explanation:

The correct question is

A bank says you can double your money in 10 years if you put $1,000 in a simple interest account. What annual interest rate does the bank pay?

we know that

The simple interest formula is equal to

[tex]A=P(1+rt)[/tex]

where

A is the Final Investment Value

P is the Principal amount of money to be invested

r is the rate of interest  

t is Number of Time Periods

in this problem we have

[tex]t=10\ years\\ P=\$1,000\\ A=\$2,000\\r=?[/tex]

substitute in the formula above

[tex]2,000=1,000(1+10r)[/tex]

Solve for r

Divide by 1,000 both sides

[tex]2=1+10r[/tex]

Subtract 1 both sides

[tex]10r=2-1[/tex]

[tex]10r=1[/tex]

Divide by 10 both sides

[tex]r=1/10[/tex]

[tex]r=0.10[/tex]

Convert to percentage (multiply by 100)

[tex]r=0.10*100=10\%[/tex]

Answer:

Right sum < trapezoidal rule value < Left sum

Non of the completed options are valid. Option c is incomplete but it starts correctly. If there are no more options, the answer is (c).

Step-by-step explanation:

Lets call a = x0, x1, x2, x3 and x4 = b the endpoints of the intervals in increasing order. Lets also call L the length of each subinterval. Note that b-a = 4L. Lets denote yi = f(xi) for i in {0,1,2,3,4}.

Lets compute each sum:

Left sum = x0*L + x1*L + x2*L + x3*L = (y0+y1+y2+y3)L

Similarly

Right sum = (y1+y2+y3+y4)L

and trapezoidal rule value = L((y0)/2 + y1+ y2 + y3+ (y4)/2)

Since f is decreasing, we have that y0 > y1 > y2 > y3 > y4. Therefore, y0 > y4 and, as a result, Left sum > Right sum, because the Right sum has y4 instead of y0 multiplying by L.

On the other hand, multiplying L we have (y0)/2 + y1+ y2 + y3+ (y4)/2 on the trapezoidal rule value, thus it has half of y0 and half of y4, and as a consecuence, the trapezoidal sum is between the left sum and the right sum.

The correct answer is

Right sum < trapezoidal rule value < Left sum

Answer:

answer is c, i took the test

Step-by-step explanation:

Answer:

equation : 2x = 8

solution : x = 4

Step-by-step explanation:

let the number be x

"the sum of a number and itself "

= sum of x and x

= x + x

= 2x

"the sum of a number and itself  is 8"

2x = 8

solving the equation, divide both sides by 2

2x = 8

x = 8/2

x = 4

To find the length of the curve defined by $y=\dfrac{1}{3}(x^2+2)^{3/2}$ over the interval $1\leq x\leq 4$, we must integrate the square root of the sum of 1 and the square of the derivative of our function from 1 to 4.

To begin with, we want to find the derivative of our function. In other words, we need to compute $dy/dx$.

The derivative, with the Chain Rule gives us:
$y' = \dfrac{1}{3} \cdot \dfrac{3}{2} \cdot 2x \cdot (x^2+2)^{1/2}$
Simplifying gives:
$y' = x \cdot (x^2+2)^{1/2}$

Next, we substitute $y'$ into the formula for finding the length of a curve:

$L = \int_{a}^{b}\sqrt{1+(y')^2}dx$

We should note that $a = 1$ and $b = 4$ here. We substitute $y'=x \cdot (x^2+2)^{1/2}$ and obtain:

$L = \int_{1}^{4}\sqrt{1+(x \cdot (x^2+2)^{1/2})^2}dx$

We can now evaluate the integral, where we will square the entire derivative and add 1 as being under the square root.

So, finally, evaluating this integral gives us the length of the curve, which in this case is 24.

Therefore, the length of the curve $y=\dfrac{1}{3}(x^2+2)^{3/2}$ over the interval $1\leq x\leq 4$ is 24.

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