a fish tank is in the shape of a rectangular prism with dimensions 30 in by 12in by 15. The tan is 90% filled with water. You mistakenly place a rock, roughly the shape of a shape with 10in radius, in the tank. How much water will spill out of the tank? Round your answer to the nearest cubic inch.

Answer:

[tex]t\le 4[/tex]

Step-by-step explanation:

Let t hours be the number of hours needed to rent the room. If one hour costs $8.30, then t hours cost $8.30t. The reservation fee is $34, so the total cost is

[tex]\$(34+8.30t).[/tex]

The history club wants to spend $67.20, thus

[tex]34+8.30t\le67.20\\ \\8.30t\le 67.20-34\\ \\8.30t\le 33.20\\ \\t\le \dfrac{33.20}{8.30}=\dfrac{332}{83}=4[/tex]

The maximum possible whole number of hours is 4 hours.

Answer:

  a = 20; b = 2√101

Step-by-step explanation:

The triangle at upper left and the one at lower right are similar. Ratios of corresponding sides are equal, so ...

  200/a = a/2

  400 = a^2

  √400 = 20 = a

Then, by the Pythagorean theorem, the value of b can be found:

  b = √(a^2 + 2^2) = √(400 +4) = √4·√101

  b = 2√101

ANSWER

[tex] \sum_{n=1} ^{32} (4n + 1) = 2144[/tex]

EXPLANATION

The given series is

[tex] \sum_{n=1} ^{32} (4n + 1) [/tex]

The first term in this series is

[tex]a_1=4(1) + 1 = 5[/tex]

The last term is

[tex]l = 4(32) + 1 = 129[/tex]

The sum of the first n terms is

[tex]S_n= \frac{n}{2} (a + l)[/tex]

The sum of the first 32 terms is

[tex]S_ {32} = \frac{32}{2} (5 + 129)[/tex]

[tex]S_ {32} =16 \times 134[/tex]

[tex]S_ {32} =2144[/tex]

Therefore,

[tex] \sum_{n=1} ^{32} (4n + 1) = 2144[/tex]

Answer:

The simplest form is x/(x + 3)

Step-by-step explanation:

* To simplify the rational Expression lets revise the factorization

  of the quadratic expression

*  To factor a quadratic in the form x² ± bx ± c:

- First look at the c term  

# If the c term is a positive number, and its factors are r and s they

  will have the same sign and their sum is b.

#  If the c term is a negative number, then either r or s will be negative

   but not both and their difference is b.

- Second look at the b term.  

# If the c term is positive and the b term is positive, then both r and

  s are positive.  

Ex: x² + 5x + 6 = (x + 3)(x + 2)  

# If the c term is positive and the b term is negative, then both r and s

  are negative.  

Ex:  x² - 5x + 6 = (x -3)(x - 2)

# If the c term is negative and the b term is positive, then the factor

  that is positive will have the greater absolute value. That is, if

  |r| > |s|, then r is positive and s is negative.  

Ex: x² + 5x - 6 = (x + 6)(x - 1)

# If the c term is negative and the b term is negative, then the factor

  that is negative will have the greater absolute value. That is, if

  |r| > |s|, then r is negative and s is positive.

Ex: x² - 5x - 6 = (x - 6)(x + 1)

* Now lets solve the problem

- We have two fractions over each other

- Lets simplify the numerator

∵ The numerator is [tex]\frac{x+2}{x^{2}+2x-3}[/tex]

- Factorize its denominator

∵  The denominator = x² + 2x - 3

- The last term is negative then the two brackets have different signs

∵ 3 = 3 × 1

∵ 3 - 1 = 2

∵ The middle term is +ve

∴ -3 = 3 × -1 ⇒ the greatest is +ve

∴ x² + 2x - 3 = (x + 3)(x - 1)

∴ The numerator = [tex]\frac{(x+2)}{(x+3)(x-2)}[/tex]

- Lets simplify the denominator

∵ The denominator is [tex]\frac{x+2}{x^{2}-x}[/tex]

- Factorize its denominator

∵  The denominator = x² - 2x

- Take x as a common factor and divide each term by x

∵ x² ÷ x = x

∵ -x ÷ x = -1

∴ x² - 2x = x(x - 1)

∴ The denominator = [tex]\frac{(x+2)}{x(x-1)}[/tex]

* Now lets write the fraction as a division

∴ The fraction = [tex]\frac{x+2}{(x+3)(x-1)}[/tex] ÷ [tex]\frac{x+2}{x(x-1)}[/tex]

- Change the sign of division and reverse the fraction after it

∴ The fraction = [tex]\frac{(x+2)}{(x+3)(x-1)}*\frac{x(x-1)}{(x+2)}[/tex]

* Now we can cancel the bracket (x + 2) up with same bracket down

 and cancel bracket (x - 1) up with same bracket down

∴ The simplest form = [tex]\frac{x}{x+3}[/tex]

ANSWER

[tex]\frac{x}{x + 3}[/tex]

EXPLANATION

We want to simplify:

[tex] \frac{x +2 }{ {x}^{2} + 2x - 3} \div \frac{x + 2}{ {x}^{2}- x} [/tex]

Multiply by the reciprocal of the second fraction:

[tex] \frac{x +2 }{ {x}^{2} + 2x - 3} \times \frac{{x}^{2}- x}{ x + 2} [/tex]

Factor;

[tex] \frac{x +2 }{ (x + 3)(x - 1)} \times \frac{x(x - 1)}{ x + 2} [/tex]

We cancel out the common factors to get:

[tex] \frac{x}{x + 3} [/tex]

Answer:

288

Step -by-step explanation:I'm not explaining :D

Answer:

It took him 7.5 hrs to drive home.

Step-by-step explanation:

The distance is the same, going or returning.

Recall that distance = rate times time.

Here, distance from Andrew's to his sister's is (60 mph)(8 hr) = 480 mi

If distance = rate times time, then time = distance / rate.

Here we have time = 480 mi / 64 mph = 7.5 hrs

It took him 7.5 hrs to drive home.

ANSWER

[tex]\sin(40) \degree = 0.6428[/tex]

EXPLANATION

The sine and cosine ratios are complementary.

This means that:

[tex] \cos(x \degree) = \sin(90 - x) \degree[/tex]

We were given that,

cos(50°)=0.6428

We want to find the sine of the complement of this angle.

[tex] \cos(50\degree) = \sin(90 -5 0) \degree[/tex]

[tex]\cos(50\degree) = \sin(40) \degree[/tex]

This implies that,

[tex] \sin(40) \degree = 0.6428[/tex]

The answer is:

The first triangle is:

[tex]A=59.6\°\\C=71.4\°\\c=17.6units[/tex]

The second triangle is:

[tex]A=120.4\°\\C=10.6\°\\c=3.41units[/tex]

To solve the triangles, we must remember the Law of Sines form.

Law of Sines can be expressed by the following relationship:

[tex]\frac{a}{Sin(A)}=\frac{b}{Sin(B)}=\frac{c}{Sin(C)}[/tex]

Where,

a, b, and c are sides of the triangle

A, B, and C are angles of the triangle.

We are given,

[tex]B=49\°\\a=16\\b=14[/tex]

So, solving the triangles, we have:

Finding A, we have:

[tex]\frac{a}{Sin(A)}=\frac{b}{Sin(B)}\\\\Sin(A)=a*\frac{Sin(B)}{b}=16*\frac{Sin(49\°)}{14}\\\\Sin^{-1}(Sin(A)=Sin^{-1}(16*\frac{Sin(49\°)}{14})\\\\A=59.6\°[/tex]

Finding C, we have:

Now, if the sum of all the interior angles of a triangle is equal to 180°, we have:

[tex]A+B+C=180\°\\\\C=180-A-B\\\\C=180\°-59.6\°-49\°=71.4\°[/tex]

Finding c, we have:

Then, now that we know C, we need to look for "c":

[tex]\frac{14}{Sin(49\°)}=\frac{c}{Sin(71.4\°)}\\\\c=\frac{14}{Sin(49\°)}*Sin(71.4\°)=17.58=17.6units[/tex]

So, the first triangle is:

[tex]A=59.6\°\\C=71.4\°\\c=17.6units[/tex]

Finding A, we have:

[tex]\frac{a}{Sin(A)}=\frac{b}{Sin(B)}\\\\Sin(A)=a*\frac{Sin(B)}{b}=16*\frac{Sin(49\°)}{14}\\\\Sin^{-1}(Sin(A)=Sin^{-1}(16*\frac{Sin(49\°)}{14})\\\\A=59.6\°[/tex]

Now, since that there are two triangles that can be formed, (angle and its suplementary angle) there are two possible values for A, and we have:

[tex]A=180\°-59.6\°=120.4\°[/tex]

Finding C, we have:

Then, if the sum of all the interior angles of a triangle is equal to 180°, we have:

[tex]A+B+C=180\°\\\\C=180\°-A-B\\\\C=180\°-120.4\°-49\°=10.6\°[/tex]

Then, now that we know C, we need to look for "c".

Finding c, we have:

[tex]\frac{14}{Sin(49\°)}=\frac{c}{Sin(10.6)}\\\\c=\frac{14}{Sin(49)\°}*Sin(10.6\°)=3.41units[/tex]

so, The second triangle is:

[tex]A=120.4\°\\C=10.6\°\\c=3.41units[/tex]

Have a nice day!

Answer:

D

Step-by-step explanation:

The equation given gives y in terms of x, where y is Shaun's savings and x is the number of months.

We want to know the amount he will save in 25 months, so we can plug in x = 25 into the equation and get a value of y, that is the amount he will save. So:

[tex]y = 6x^2 + 75x+200\\y = 6(25)^2 + 75(25)+200\\y=5825[/tex]

D is the correct answer.

Final answer:

Adjacent sides in a kite are congruent, which can be proven using the definition of a kite, the Reflexive Property of Equality, and the SAS Postulate to show that the two triangles formed by the diagonal are congruent.

Explanation:

To prove that adjacent sides are congruent in a kite, recall the definition of a kite: a quadrilateral with two distinct pairs of adjacent sides that are congruent. Let's name the kite ABCD where AB and AD are one pair of congruent sides, and BC and DC are the other pair. The longer diagonal, which we'll call AC, bisects the kite into two congruent triangles, ∆ABC and ∆ADC. By the definition of a kite, we know that AB = AD and BC = DC.

Now, since AC is the common side in both ∆ABC and ∆ADC, by the Reflexive Property of Equality, AC = AC. With two sides and the included angle (BAC and DAC are congruent as they are both angles cut by diagonal AC) congruent in both triangles, we apply the SAS Postulate (Side-Angle-Side) to prove that the triangles are congruent. Once the triangles are proven congruent, all their corresponding parts are congruent. Therefore, AB = AD and BC = CD, which means the adjacent sides in the kite are indeed congruent.

ANSWER

[tex]W''(2, - 3)[/tex]

EXPLANATION

The given point bas coordinates

W(-2,3).

The mapping for reflection over the x-axis is

[tex](x,y)\to(x, - y)[/tex]

[tex]W(-2,3)\to \: W'(-2, - 3)[/tex]

This point W'(-2,-3) is then reflected over the y-axis.

The mapping for reflection over the y-axis is

[tex](x,y)\to (-x, y)[/tex]

[tex]W'(-2,-3)\to \: W''(2, - 3)[/tex]

Point W is initially transformed by reflecting it over the x-axis, which inverses the sign of the y-coordinate, followed by a reflection over the y-axis, which inverses the sign of the x-coordinate. The final position of point W" on the coordinate plane is therefore (2, -3).

To answer this mathematics question involving coordinate geometry and reflections, one must understand how reflections in the x-axis and y-axis affect the coordinates of a point.

A reflection of a point across the x-axis changes the sign of the y-coordinate, and a reflection across the y-axis changes the sign of the x-coordinate. Thus, if we start with point W at (-2,3), a reflection over the x-axis to create point W' changes the y-coordinate's sign, yielding point W' at (-2,-3). Then, reflecting W' over the y-axis changes the x-coordinate's sign, yielding our final point W" at (2,-3).

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Answer:

Answer choice B

Step-by-step explanation:

There are two options of x. You have both of them in picture 1.

Answer:

Step-by-step explanation:

x² - 4x +2=0

(x² - 4x +4) - 4+2=0

(x - 2)²= 2

x - 2 = √2   or x- 2 = -√2

x=2+√2 or x=2-√2

note : same solution :Magdaloskotova Beginner

(4+2√2)/2   = 2(2+√2)/2 = 2+√2

(4-2√2)/2   = 2(2-√2)/2 = 2-√2

The Box 3 on an electronic W-2 form represents the total wages subject to Social Security tax. To calculate it, look for the 'Social Security wages' value on the form and determine if it exceeds the annual limit set by the Social Security tax.

The Box 3 on an electronic W-2 form represents the total wages that are subject to the Social Security tax. To calculate the amount in Box 3, you can look for the value labeled 'Social Security wages' or 'SS wages' on the form. This value includes all taxable wages and tips that are subject to Social Security taxes, up to a certain limit.

For the year 2020, the Social Security tax is only applied to the first $137,700 of wages. If the 'Social Security wages' value provided on the form is higher than this limit, you should use the limit as the amount in Box 3. If the value is lower than the limit, then the 'Social Security wages' value itself should be used as the amount in Box 3.

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To create an equation for a graph where y depends on x, identify the type of relationship presented by the graph. For a linear relationship, use the formula y = mx + b, where 'm' is the slope and 'b' is the y-intercept. Determine the slope using two points on the line and then solve for 'b' using the slope and one of the points.

Creating an equation for a graph where y depends on x is a common task in algebra. It's essential to identify the type of relation presented by the graph, which may include linear, quadratic, exponential, or logarithmic relationships. Let's assume, for example, that the graph represents a simple linear relationship.

In this case, you would use the formula for a line, which is y = mx + b. 'm' represents the slope of the line, and 'b' stands for the y-intercept.

To find the slope 'm', you can choose two points on the line and calculate the change in y divided by the change in x. Once the slope is determined, plug one of the points' coordinates into the equation and solve for 'b', the y-intercept. This will give you the equation for the line.

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The equation for that graph, where y depends on x can be expressed in slope-intercept form as: y = -2x + 6.

To find the equation of the line, you can use the slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. First, find the slope (m) using any two given points on the graph, i.e. (0, 6) and (4, -2):

[tex]\[ m = \frac{{\text{{change in }} y}}{{\text{{change in }} x}} = \frac{{-2 - 6}}{{4 - 0}} = \frac{{-8}}{{4}} = -2 \][/tex]

Now, substitute one of the points (e.g., (0, 6)) and the slope into the equation:

y = mx + b

6 = -2(0) + b

b = 6

So, the equation of the line is y = -2x + 6.

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Answer:

12 possible outcomes.

Sample space:

[tex]\begin{array}{cccc}(F,1)&(A,1)&(I,1)&(R,1)\\(F,2)&(A,2)&(I,2)&(R,2)\\(F,3&(A,3)&(I,3)&(R,13)\end{array}[/tex]

Step-by-step explanation:

The collection of all possible outcomes of a probability experiment forms a set that is known as the sample space.

1. There are four possible outcomes for the first wheel: F, A, I and R

2. There are three possible outcomes for the second wheel: 1, 2 and 3

So, the sample space is

[tex]\begin{array}{cccc}(F,1)&(A,1)&(I,1)&(R,1)\\(F,2)&(A,2)&(I,2)&(R,2)\\(F,3&(A,3)&(I,3)&(R,13)\end{array}[/tex]

Answer:

C) r3 < r2 < r1

Step-by-step explanation:

Answer:

c

Step-by-step explanation:

ANSWER

The correct choice is D.

EXPLANATION

See the attachment for explanation;

Answer:

2

Step-by-step explanation:

Dividing 18 yds into 34 yds yields 2; Adrianna will have 2 pieces 18 yds long, with 8 yds left over.

Answer:

I would say D.) is the best choice in this situation, The amount of female children you have does not determine the Gender that your child will be, The kids have nothing to do with the situation, The Father and Mother are what determine the Gender.

Answer:

Option A is right

Step-by-step explanation:

Given that Amy has 3 children, and she is expecting another baby soon. Her first three children are girls.

The fourth baby is independent of first three babies

As regards child birth, each trial is independent of the other with probability for a girl or boy equally likely with p = 0.5 = q

Hence option A is right answer.

Option B is wrong because the previous 3 girls have nothing to affect the sex of 4th child

Option C is wrong, since each trial is independent

D) is wrong since even mom and dad cannot determine the sex of that baby, it is nature and pure chance.

the probability is 1/18

[tex]S=15\times15+4(7.5\times15)=\boxed{675}[/tex]

Answer:

Option C is correct.

Step-by-step explanation:

Given:

Length of the base rectangle = 15 unit

Width of the  base rectangle = 15 unit

Height of triangle on the side of pyramid = 15 unit

Length of the base of the triangle = 15 unit

Surface are of the rectangular pyramid = Area of Base + 4 × Area of triangle on side.

Area of base = length × width = 15 × 15 = 225 unit²

Area of the triangle = 1/2 × base  × height = 1/2 × 15 × 15 = 112.5 unit²

Surface Area = 225 + 4 × 112.5  = 225 + 450 = 675 unit²

Therefore, Option C is correct.

Answer:

(D) 2

Step-by-step explanation:

f(x) = 5x - 1

y = 5x - 1

5x = y + 1

x = (y + 1)/5

f⁻¹ (x) = (x + 1)/5

f(x) = 5x - 1

f(2) = 5(2) - 1

f(2) = 9

f⁻¹ (x) = (x + 1)/5

f⁻¹ (f(2)) = (9 + 1)/5

f⁻¹ (f(2)) = 2

The correct options for the Cavalieri's Principle are A and B, where the cone is paired with either a cylinder or a pyramid, both having equal base areas and heights.

Given Cavalieri's Principle which states that if two solids have the same height and cross-sectional area at every level (parallel cross-sections), then they have equal volumes.

To determine two solids would have congruent volumes,

A. A cone and a cylinder with equal base areas and heights would satisfy Cavalieri's Principle because they would have equal cross-sectional areas at every level.

B. A cone and a pyramid with equal base areas and heights would also satisfy Cavalieri's Principle.

A cone and a rectangular prism with equal base areas and heights would NOT satisfy Cavalieri's Principle because their cross-sectional areas would be different at different levels.

Also, cylinder and a sphere with equal radii would NOT satisfy Cavalieri's Principle because their cross-sectional areas would be different at different levels.

So, the correct answer is either A (cone and cylinder) or B (cone and pyramid).

ANSWER

6 inches

EXPLANATION

The given function is

[tex]h(t) = - {t}^{2} + 12t + 10[/tex]

The average rate of change from t=1 to t=5 is given by:

[tex] = \frac{h(5) - h(1)}{5 - 1} [/tex]

[tex]h(5) = - {(5)}^{2} + 12(5) + 10[/tex]

[tex]h(5) = - 25 + 60+ 10[/tex]

[tex]h(5) = 45[/tex]

Also,

[tex]h(1) = - {(1)}^{2} + 12(1) + 10[/tex]

[tex]h(1) = - 1+ 12+ 10[/tex]

[tex]h(1) = 21[/tex]

The average rate of change is now

[tex] = \frac{45 - 21}{4} [/tex]

[tex] = \frac{24}{4} [/tex]

[tex] = 6[/tex]

Answer:

Step-by-step explanation:

Left

When a square = a linear, always expand the squared expression.

x^2 - 2x + 1 = 3x - 5                Subtract 3x from both sides

x^2 - 2x - 3x + 1 = -5

x^2 - 5x +1 = - 5                      Add 5 to both sides

x^2 - 5x + 1 + 5 = -5 + 5

x^2 - 5x + 6 = 0

This factors

(x - 2)(x - 3)

So one solution is x = 2 and the other is x = 3

Second from the Left

i = sqrt(-1)

i^2 = - 1

i^4 = (i^2)(i^2)

i^4 = - 1 * -1

i^4 = 1

16(i^4) - 8(i^2) + 4

16(1) - 8(-1) + 4

16 + 8 + 4

28

Second from the Right

This one is rather long. I'll get you the equations, you can solve for a and b. Maybe not as long as I think.

12 = 8a + b

17 = 12a + b         Subtract

-5 = - 4a

a = - 5/-4 = 1.25

12 = 8*1.25 + b

12 = 10 + b

b = 12 - 10

b = 2

Now they want a + b

a + b = 1.25 + 2 = 3.25

Right

One of the ways to do this is to take out the common factor. You could also expand the square and remove the brackets of (2x - 2). Both will give you the same answer. I think expansion might be easier for you to understand, but the common factor method is shorter.

(2x - 2)^2 = 4x^2 - 8x + 4

4x^2 - 8x + 4 - 2x + 2

4x^2 - 10x + 6    The problem is factoring since neither of the first two equations work.

(2x - 2)(2x - 3)     This is correct.

So the answer is D

Graphing is overkill... Let [tex]x=0[/tex]. Then [tex]\sec0=1[/tex], while [tex]\tan0=0[/tex]. But [tex]1+1=2\neq0[/tex], so this is not an identity.

Answer:

It is not an identity

Step-by-step explanation:

If you graphic the two equations (left and right) separately, if they are an identity, they will be the same graphic, which is not true in this case.

Another way in order to know if an equation is an identity, you can replace some values ​​at x, for example 2 values:

X=30

X=60

And now we substitute in the equation, like this:

[tex]1+sec^2(30)=tan^2(30)[/tex]

2,33=0,33 this is not equal on both sides

[tex]1+sec^2(60)=tan^2(60)[/tex]

5=3 this is not equal on both sides

And if the results for each number x are the same on both sides of the equation, it is an identity. In this case they are different.

Answer:

The height of the trough is about 1.5 ft

Step-by-step explanation:

If each cubic foot of water contains about 7.5 gallons.

Then; 315 gallons is about [tex]\frac{315}{7.5}=42ft^3[/tex]

Let h be the height of the trough, then

[tex]7\times 4\times h=42[/tex]

This implies that;

[tex]28h=42[/tex]

Divide both sides by 28 to get:

[tex]h=\frac{42}{28}[/tex]

[tex]\therefore h=1.5[/tex]

The height of the trough is about 1.5 ft

the height of the trough is 1.5 feet.

To solve for the height of the trough that holds 315 gallons, we need to use the information that 1 cubic foot of water contains about 7.5 gallons. Since the trough is 7 feet long and 4 feet wide, we can first calculate the volume in cubic feet by dividing the volume in gallons by the gallon-to-cubic-feet conversion factor (315 gallons \/ 7.5 gallons/cubic foot). Then, we use this volume to find the height by dividing the volume by the product of the length and the width of the trough.

Here is a step-by-step approach:

Now, let's calculate:

Therefore, the height of the trough is 1.5 feet.

6x8= 48 inches

3 doubled=6

4 doubled=8

Answer:

D

Step-by-step explanation:

The following steps describes the correct order of steps for constructinf an angle bisector of angle DEF.

(1). Draw a circle with a radius less then the arms of angle.

(2). Draw a line from point of intersections of arc (circle) and arms of the angle.

(3). Draw two circles with radius = distance between point of intersection of circle and arms of angle, center taken as point of intersection of circle and arms. than draw an equilateral triangle.

(4). Use a straightedge to connect the vertex of angle and the right  most vertex of the equilateral triangle.

Hence option D is correct.

Answer:

Option C.

Step-by-step explanation:

Option C. A parabola is a set of all points that are the same distance from a point called focus and a line called vertex.

The correct option is option C.

A parabola is a set of all points that are the same distance from a point called focus and a line called vertex.

The parabola is symmetric about its axis.

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