Kayla needs to have $560.00 to buy a new laptop. She already has $200.00 in the bank. Each month she will add $45.00 to her bank account. How many more months until she has enough money to buy her laptop?

Answer: 28.1ft^2 so it’s C

Answer: C

Step-by-step explanation:

The volume of the solid formed by revolving the region bounded by the graphs of the given functions is (25π) / 2.

Integration is a method of adding or summing up the parts to find the whole. It is a reverse process of differentiation, where we reduce the functions into parts.

The region bound by all three functions is in the first quadrant. If this region were rotated about the y-axis, the solid formed would have washer-shaped cross-sections.

Therefore, we can use the Washer method: V = π∫ (ROuter)² - (RInner)² dy

R(outer) = 3 and R(inner) = √y  (Note: the inner radius is formed by the function y = x2 which we solve in terms of y since we are rotating about the y-axis).

V = π∫ (3)2 - (√y)2 dy

(the limits of integration are from y = 4 to y = 9, so a = 4 and b = 9)

= π∫  9 - y dy where a = 4 and b = 9

Integrate this definite integral to find the volume,

=  (25π) / 2.

Hence, the volume of the solid formed by revolving the region bounded by the graphs of the given functions is (25π) / 2.

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Answer:

[tex]\textsf{A)} \quad \dfrac{25\pi}{2}[/tex]

Step-by-step explanation:

To find the volume of the solid of revolution created by rotating the region bounded by the curve y = x², x = 3, and y = 4 about the y-axis, we can use the method of cylindrical shells.

As the region of revolution is bounded by the graphs of two functions, y = x² and y = 4, the volume of a solid of revolution using cylindrical shells is given by the integral:

[tex]\displaystyle V=2\pi\int^b_ax\left(f(x)-g(x)\right)\;\text{d}x[/tex]

where:

The height of the shell, f(x) - g(x), is determined by the vertical distance between the curve y = x² and the line y = 4. Therefore, the height is:

[tex]f(x)-g(x)=x^2-4[/tex]

The upper bound of the integral is b = 3, since the region is bounded by x = 3.

The lower bound is the x-value of the point of intersection of the two functions y = x² and y = 4. These two functions intersect when x = 2. Therefore, the lower bound of the integral is a = 2.

Substitute these values into the integral:

[tex]\displaystyle V=2\pi\int^3_2x\left(x^2-4\right)\;\text{d}x[/tex]

Expand the brackets:

[tex]\displaystyle V=2\pi\int^3_2\left(x^3-4x\right)\;\text{d}x[/tex]

Evaluate the integral by using the power rule:

[tex]\boxed{\begin{minipage}{4 cm}\underline{Integrating $x^n$}\\\\$\displaystyle \int x^n\:\text{d}x=\dfrac{x^{n+1}}{n+1}+\text{C}$\end{minipage}}[/tex]

Therefore:

[tex]V=2\pi\left[\dfrac{x^4}{4}-\dfrac{4x^2}{2}\right]^3_2[/tex]

[tex]V=2\pi\left[\dfrac{x^4}{4}-2x^2\right]^3_2[/tex]

[tex]V=2\pi\left[\left(\dfrac{(3)^4}{4}-2(3)^2\right)-\left(\dfrac{(2)^4}{4}-2(2)^2\right)\right][/tex]

[tex]V=2\pi\left[\left(\dfrac{81}{4}-18\right)-\left(4-8\right)\right][/tex]

[tex]V=2\pi\left[\left(\dfrac{9}{4}\right)-\left(-4\right)\right][/tex]

[tex]V=2\pi\left(\dfrac{25}{4}\right)[/tex]

[tex]V=\dfrac{25\pi}{2}[/tex]

Therefore, the volume of the solid formed by revolving the region bounded by the graphs of y = x², x = 3, and y = 4 about the y-axis is:

[tex]\Large\boxed{\boxed{\dfrac{25\pi}{2}}}[/tex]

If there are 812 students in your school and there are 36 more girls than boys. Then there are 414 girls in the school.

In this problem, let's denote the number of boys as B and the number of girls as G. We are given two pieces of information: there are 812 students in total, and there are 36 more girls than boys.

So, we can set up two equations based on these statements:

1. Total number of students: (B + G = 812)

2. The number of girls is 36 more than boys: (G = B + 36)

Now, we can substitute the expression for G from the second equation into the first equation:

[B + (B + 36) = 812]

Combine like terms:

[2B + 36 = 812]

Subtract 36 from both sides:

[2B = 776]

Now, divide by 2 to solve for B:

[B = 388]

Now that we know the number of boys, we can find the number of girls using the second equation:

[G = 388 + 36]

[G = 424]

So, there are 388 boys and 424 girls in the school. To find the number of girls, we add the number of boys (388) to the given difference (36):

[424 = 388 + 36]

Therefore, the final answer is that there are 424 girls in the school.

The answer is x=1. Just did the assignment

The equation is a complex exponentiation problem that can be solved by simplifying and rearranging the equation. By plugging in the given potential solutions, it can be determined that x = -1 is the correct answer.

This equation is an exponential equation. You can start solving by simplifying the bases, as both bases are actually the 2, but to different powers (512 = 2^9 and 1/64 = 2^-6). The equation can be rewritten as (2^9)^x - 2 = (2^-18)x - 512. Simplifying this gives 2^(9x) - 2 = 2^(-18x) - 512. From the options given (x = -1, 0, or 1), we can plug these into the simplified equation to see which one gives a valid solution. Plugging these values in reveals that x = –1 is the valid solution to the equation. None of the other choices work out correctly.

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The property that justifies the step is the Commutative Property of Addition, which states that the order of addition does not affect the result.

The property that justifies the illustrated step in solving the equation 3x + 4 - 2x + 5 = 6x + 2 as 3x - 2x + 4 + 5 = 6x + 2 is the Commutative Property of Addition.

This property states that the order in which numbers are added does not affect the result. So, we can rearrange the terms in any order without changing their sum.

In this step, we are rearranging the terms -2x and 5 so that like terms, namely 3x and -2x, and 4 and 5, are grouped together.

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The equation which has the steepest graph is:

             Option: A

     A.   [tex]y=11x-144[/tex]

We know that the slope-intercept form of a line is given by:

                 [tex]y=mx+c[/tex]

where m is the slope of the line and c is the y-intercept of a line.

A)

[tex]y=11x-144[/tex]

The slope is: 11

B)

[tex]y=\dfrac{1}{2}x+3[/tex]

The slope is: [tex]\dfrac{1}{2}[/tex]

C)

[tex]y=7x+16[/tex]

The slope is: 7

D)

[tex]y=3x+144[/tex]

The slope of this line is: 3

The line which has the largest absolute value of slope is:

               [tex]y=11x-144[/tex]

Hence, the graph of this equation will be steepest.

Final answer:

Two equivalent ratios for 9:20 are 18:40 and 27:60, found by multiplying both the numerator and denominator of the ratio by 2 and 3, respectively.

Explanation:

To find two equivalent ratios for 9:20, you can multiply or divide both terms of the ratio by the same non-zero number. Remember, an equivalent ratio has the same proportional relationship as the original one.

Here's how we can calculate two equivalent ratios:

Multiply both terms by 2: 9 × 2 = 18 and 20 × 2 = 40, so one equivalent ratio is 18:40.

Multiply both terms by 3: 9 × 3 = 27 and 20 × 3 = 60, so another equivalent ratio is 27:60.

These both represent the same ratio as 9:20 when simplified.

Answer:

#1) d. ΔJKL is not a right triangle because no two of its sides are perpendicular; #2) -1/3, 3, -7, is, two of these slopes have a product of -1; #3) a. Quadrilateral DEFG is a rhombus because opposite sides are parallel and all four sides have the same length; #4) 1, -1/6, 1, -2/5, is not, only one pair of opposite sides is parallel; #5) c. Quadrilateral PQRS is not a rectangle because it has only one right angle.

Step-by-step explanation:

#1) The slope of any line segment is found using the formula

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

For JK, this gives us (1-1)/(-5-0) = 0/-5 = 0.  For KL this gives us (1--5)/(0-2) = 6/-2 = -3.  For LJ this gives us (-5-1)/(2--5) = -6/7.  None of these slopes are negative reciprocals, so none of the angles are right angles and this is not a right triangle.

#2) The slope of JK is (2-1)/(0-3) = 1/-3 = -1/3.  The slope of KL is (1--5)/(3-1) = 6/2 = 3.  The slope of LJ is (2--5)/(0-1) = 7/-1 = -7.  Two of these slopes have a product of -1, 3 and -1/3.  This means they are negative reciprocals so this has a right angle; this means JKL is a right triangle.

#3) The slope of DE is (5-4)/(-2-2) = 1/-4 = -1/4.  The slope of EF is (4-0)/(2-0) = 4/2 = 2.  The slope of FG is (0-1)/(0--4) = -1/4.  The slope  of GD is (1-5)/(-4--2) = -4/-2 = 2.  Opposite sides have the same slope so they are parallel.

Next we use the distance formula to find the length of each side:

[tex]d=\sqrt{(y_2-y_1)^2+(x_2-x_1)^2}[/tex]

Using our points, the length of DE is

[tex]\sqrt{(5-4)^2+(-2-2)^2}=\sqrt{1^2+(-4)^2}=\sqrt{1+16}=\sqrt{17}[/tex]

The length of EF is

[tex]d=\sqrt{(4-0)^2+(2-0)^2}=\sqrt{4^2+2^2}=\sqrt{16+4}=\sqrt{20}[/tex]

The length of FG is

[tex]d=\sqrt{(0-1)^2+(0--4)^2}=\sqrt{(-1)^2+(4)^2}=\sqrt{1+16}=\sqrt{17}[/tex]

The length of GD is

[tex]d=\sqrt{(1-5)^2+(-4--2)^2}=\sqrt{(-4)^2+(-2)^2}=\sqrt{16+4}=\sqrt{20}[/tex]

Opposite sides have the same length and are parallel, so this is a parallelogram.

#4) The slope of AB is (-1-2)/(-4--1) = -3/-3 = 1.  The slope of BC is (2-1)/(-1-5) = 1/-6 = -1/6.  The slope of CD is (1--3)/(5-1) = 4/4 = 1.  The slope of DA is (-3--1)/(1--4) = -2/5.  Only one pair of opposite sides is parallel, so this is not a parallelogram.

#5) The slope of PQ is (2-4)/(-4-3) = -2/-7 = 2/7.  The slope of QR is (4-0)/(3-5) = 4/-2 = -2.  The slope of RS is (0--2)/(5--3) = 2/8 = 1/4.  The slope of SP is (-2-2)/(-3--4) = -4/1 = -4.  Only one pair of sides has slopes that are negative reciprocals; this means this figure only has 1 right angle, so it is not a rectangle.

The answers provided for ΔJKL, ΔJKL as a right triangle, and quadrilateral ABCD are correct. For quadrilateral DEFG, the answer may be correct depending on side lengths, and for quadrilateral PQRS, slopes must be calculated to determine if it's a rectangle.

To determine whether the given shapes are right triangles, rhombuses, rectangles, or parallelograms, we use properties such as the slopes of lines to check for perpendicularity, parallelism, and equal lengths.

For ΔJKL with vertices J(-5, -1), K(0, 1), L(2, -5), you calculated it's not a right triangle because none of the slopes of the sides are negative reciprocals of each other (indicating perpendicular sides). This is correct; no slopes have a product of -1, so answer (d) is correct.

In the second case, ΔJKL with vertices J(0, 2), K(3, 1), L(1, -5) has slopes -1/3, 3, and -7 for sides JK, KL, and JL respectively. Since -1/3 and 3 are negative reciprocals, ΔJKL is a right triangle due to the perpendicular sides JK and KL. So, the filled answers are correct.

Regarding quadrilateral DEFG, you might want to reconsider option (b), as you cannot determine whether it's a rhombus solely based on one pair of opposite sides being parallel. Instead, check if all sides have the same length using the distance formula, and if the opposite sides are parallel by comparing slopes. Since you didn't provide the side lengths, I will only address the slopes here, which you've noted. Answer (b) can be correct if the side lengths are not equal, but if they are, it should be (c).

For quadrilateral ABCD, your assessment of the slopes leads to the conclusion that it is not a parallelogram, which is correct as only one pair of opposite sides (AB and CD having a slope of 1) is parallel. The filled answers are appropriate.

Last, for quadrilateral PQRS, you must calculate the slopes of the sides to determine if the sides are perpendicular to each other by checking if the products of the corresponding slopes are -1. If the slopes of adjacent sides are negative reciprocals of each other, then PQRS has right angles, and it could be a rectangle. Calculate the slopes, and if none are negative reciprocals, then answer (d) is correct stating that PQRS is not a rectangle because it has no right angles.

Answer:

22sqrt(3)/3

You are correct.

Step-by-step explanation:

You have a 30-60-90 triangle.

In a 30-60-90 triangle, the hypotenuse is twice the length of the short leg. The long leg is sqrt(3) times the length of the short leg.

You are given the length of the long leg.

1) Find the length of the short leg.

HG = GI/sqrt(3) = 11/sqrt(3)

2) Find the length of the hypotenuse, HI.

HI = 2HG = 2(11/sqrt(3)) = 2(11/sqrt(3))*(sqrt(3))/(sqrt(3)) = 22sqrt(3)/3

Low taxation would lead to slower economic growth because it encourages citizens and businesses to consume more and invest less is false.

We have,

Low taxation can actually lead to faster economic growth if it encourages citizens and businesses to consume and invest more.

With more money in their pockets, consumers may have more disposable income to spend on goods and services, which can stimulate demand and create more jobs.

Additionally, with lower taxes, businesses may have more money to invest in capital equipment, research and development, or expansion, which can lead to increased productivity and economic growth.

It's important to note that the relationship between taxation and economic growth is complex and multifaceted.

While lower taxes can be beneficial for growth, they may also lead to reduced government revenues, which can limit the government's ability to fund important public services and investments.

Moreover, the impact of tax policy on economic growth can vary depending on a range of factors, such as the size and structure of the economy, the nature of the tax system, and broader economic conditions.

Thus,

Low taxation would lead to slower economic growth because it encourages citizens and businesses to consume more and invest less is false.

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Answer:

The correct option is A.

Step-by-step explanation:

Let number of sold wristbands and headbands be x and y respectively.

It is given that the jack earns $2 for each wristband and $3 for each headband. So, total revenue is

[tex]T=2x+3y[/tex]

He wants to earn at least $50.

[tex]2x+3y\geq 50[/tex]

He needs to sell at least 5 wristbands.

[tex]x\geq 5[/tex]

The number of sold wristbands and headbands can not be negative. so,

[tex]x\geq 0,y\geq 0[/tex]

The system of inequalities is

[tex]2x+3y\geq 50[/tex]

[tex]x\geq 5[/tex]

[tex]x\geq 0,y\geq 0[/tex]

Therefore, correct option is A.

Answer:

The second choice is correct.

Step-by-step explanation:

Reflecting ABC across the y-axis will negate the x-coordinates, which will place it directly above triangle DEF.  Counting down from A to D, B to E and C to F, we see that each is 5 units above the image; this means a translation 5 units down will finish mapping the pre-image to the image.  Since they are mapped on top of each other, they are congruent.

Answer:

the answer is (x - 1)(6 x-1) on edge, option D.

The product is (x - 1) (6x - 1).

Multiplication of two numbers is defined as the addition of one of the number repeatedly until the times of the other number.

a × b means that a is added to itself b times or b is added to itself a times.

The given expression is [6(x² - 1) (6x - 1)] / [6 (x + 1)]

6 is in the numerator as well as denominator. Cancel it.

[6(x² - 1) (6x - 1)] / [6 (x + 1)] =  [(x² - 1) (6x - 1)] / [(x + 1)]

We have to use one of the algebraic identity here.

a² - b² = (a + b) (a - b)

Using this,

x² - 1 = x² - 1² = (x - 1)(x + 1)

[6(x² - 1) (6x - 1)] / [6 (x + 1)] =  [(x² - 1) (6x - 1)] / [(x + 1)]

                                            = [(x - 1)(x + 1)(6x - 1)] / [(x + 1)]

Again, x + 1 is both in the numerator as well as denominator. Cancel it.

[6(x² - 1) (6x - 1)] / [6 (x + 1)] = (x – 1)(6x – 1)

Hence the product is (x – 1)(6x – 1).

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After stretching for 8 minutes, Leticia had 30 minutes left for running. Given that she takes 6 minutes to complete one round, she completed 5 rounds in the remaining time.

The question asks how many times Leticia jogged around her block after stretching for 8 minutes within her total exercise time of 38 minutes. Let's deduct the time Leticia spent stretching from the total time to find out how much time she spent running. That is 38 - 8 = 30 minutes. Since we know that running around the block once takes Leticia 6 minutes, we then divide her running time by this to discover how many laps she was able to make.

We do this by 30 / 6 = 5. Therefore, Leticia jogged around her block 5 times.

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