What times what equals 18

Final answer:

To find the floor dimensions of a building that is 14 feet longer than it is wide with an area of 1715 sq ft, we solve the quadratic equation w * (w + 14) = 1715, resulting in w=100 feet and length l=114 feet.

Explanation:

The student is asked to find the length (l) and width (w) of a one-story building floor which is 14 feet longer than it is wide, with a total area of 1715 square feet. To solve this, we set up an equation where w represents the width and w + 14 feet represents the length. Since the area of a rectangle is length times width, the equation is w * (w + 14) = 1715.

After solving this quadratic equation, we find that w=100 feet which makes the length l = w + 14 = 114 feet. Therefore, the actual dimensions of the building are 114 ft by 100 feet.

There are 12 possible choices for the player.

"It is a way of selecting items from a collection "

"[tex]^nC_r=\frac{n!}{r!(n-r)!}[/tex]"

n! = n × (n - 1) × . . . × 2 × 1

For given question,

There are 3 doors and 4 curtains behind each door.

The number of possible ways of selecting a door from the 3 doors.

Using combination formula,

[tex]^3C_1=\frac{3!}{1!(3-1)!}\\\\ ^3C_1=\frac{3\times 2!}{2!}\\\\ ^3C_1=3[/tex]

The number of possible ways of selecting a curtain from the 4 curtains.

Using combination formula,

[tex]^4C_1=\frac{4!}{1!(4-1)!}\\\\ ^4C_1=\frac{4\times 3!}{3!}\\\\ ^3C_1=4[/tex]

So, the number of choices for the player to pick a door and then pick a curtain behind the door :

[tex]^3C_1\times ^4C_1\\\\=3\times 4\\\\=12[/tex]

Therefore, there are 12 possible choices for the player.

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The question is about using the Monte Carlo method to estimate the value of π, by generating random points in a square, counting how many fall inside a unit circle, and finally using this count to calculate π.

This question involves the implementation of a famous problem in computational mathematics known as the Monte Carlo method for estimating the value of π:

This process is based on the mathematical fact that the ratio of the area of a unit circle to the square it is inscribed in is π/4.

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1+3^2 = 1+9 = 10

10/5 = 2

2-6+2 = -4+2

 answer is -2

The equation is [tex]\(x^2 + 12y^2 - 3072 = 0\),[/tex] obtained from integrating [tex]\(12y \, dy = -x \, dx\)[/tex] with the initial condition.

To find the equation of the curve passing through the point (0, 16) with a slope given by [tex]\( \frac{{dy}}{{dx}} = -\frac{x}{12y} \),[/tex] you can use separation of variables.

Given that [tex]\( \frac{{dy}}{{dx}} = -\frac{x}{12y} \)[/tex], we can rewrite it as:

[tex]\[ 12y \, dy = -x \, dx \][/tex]

Now, integrate both sides:

[tex]\[ \int 12y \, dy = \int -x \, dx \]\[ 6y^2 = -\frac{1}{2}x^2 + C \][/tex]

Given that the curve passes through the point (0, 16), substitute these values into the equation:

[tex]\[ 6(16)^2 = -\frac{1}{2}(0)^2 + C \]\[ 6(256) = C \]\[ C = 1536 \][/tex]

Therefore, the equation of the curve is:

[tex]\[ 6y^2 = -\frac{1}{2}x^2 + 1536 \][/tex]

This can also be written in standard form as:

[tex]\[ x^2 + 12y^2 - 3072 = 0 \][/tex]

So, the equation of the curve is [tex]\( x^2 + 12y^2 - 3072 = 0 \).[/tex]

The number of attendance on Thursday would be 9,315.

Given that,

During a 5-day training at your department, the number of visitors tripled each day.

Here, the training started on a Monday with 345 attendees.

Hence, the number of visitors on Tuesday is 3 times with 345 attendees.

That is, On Tuesday;

Number of attendees = 345 x 3

                                     = 1,035

On Wednesday;

Number of attendees = 1,035 x 3

                                     = 3,105

On Thursday;

Number of attendees = 3,105 x 3

                                     = 9,315

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Shantel will need to study for 60 minutes more to complete her history review.

Shantel is studying history and after 20 minutes, she is [tex]\frac{1}{4}[/tex]  of the way done.

To determine how much longer she will study, we need to calculate how much time is needed to complete the remaining [tex]\frac{3}{4}[/tex] of her study time.

[tex]\frac{1}{4}[/tex] of the study done in 20 minutes means she will study for 4 * 20 = 80 minutes in total.

To find out how much longer she needs to study, subtract the time she has already studied (20 minutes) from the total study time (80 minutes): 80 - 20 = 60 minutes.

Therefore, Shantel will need to study for 60 minutes more to complete her history review.

The value of y'' will be (- 5x/y² - 25x⁴/2y⁵).

What is implicit differentiation?

In implicit differentiation, we differentiate each side of an equation with two variables by treating one of the variable as a function of other.

Given that;

The equation is;

5x³ + 2y³ = 1

Now, Apply implicit differentiation rule we get;

5x³ + 2y³ = 1

After differentiation;

15x² + 6y² dy/dx = 0

dy/dx = - 15x² / 6y²

dy/dx = -5x²/2y²

Again differentiation;

d²y/dx² = -5/2 (y² × 2x - x² × 2y dy/dx) / y⁴

d²y/dx² = -5/2 (2xy² - 2x²y dy/dx) / y⁴

Substitute value of dy/dx;

d²y/dx² = - 5/2 (2xy² - 2x²y × -5x²/2y²) / y⁴

d²y/dx² = - 5/2 (2xy² + 5x⁴/y) / y⁴

d²y/dx² = - 5x/y² - 25x⁴/2y⁵

y'' = - 5x/y² - 25x⁴/2y⁵

Thus, The value of y'' will be (- 5x/y² - 25x⁴/2y⁵).

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The number 720,808 in expanded form with exponents is written as:

[tex]\[ 7 \times 10^5 + 2 \times 10^4 + 8 \times 10^2 + 8 \times 10^0 \][/tex]

700,000  : [tex]7 \times 10^5 \\[/tex]

20,000  : [tex]2 \times 10^4 \\[/tex]

800  : [tex]8 \times 10^2 \\[/tex]

8  : [tex]8 \times 10^0[/tex]

[tex]\[ 7 \times 10^5 + 2 \times 10^4 + 8 \times 10^2 + 8 \times 10^0 \][/tex]

This representation helps in understanding the value of each digit based on its position in the number.