A scientist illegally introduces new species of honey bees in their labs. The growth of the population of this new species is modeled by the equation below, where x is the time in weeks following the introduction of the species. What is happening to the illegal honey bee population each week? B\left(x\right)=a_0\left(3\right)^xB(x)=a 0 ​ (3) x

Question:

Find the point (,) on the curve [tex]y = \sqrt x[/tex] that is closest to the point (3,0).

[To do this, first find the distance function between (,) and (3,0) and minimize it.]

Answer:

[tex](x,y) = (\frac{5}{2},\frac{\sqrt{10}}{2}})[/tex]

Step-by-step explanation:

[tex]y = \sqrt x[/tex] can be represented as: [tex](x,y)[/tex]

Substitute [tex]\sqrt x[/tex] for [tex]y[/tex]

[tex](x,y) = (x,\sqrt x)[/tex]

So, next:

Calculate the distance between [tex](x,\sqrt x)[/tex] and [tex](3,0)[/tex]

Distance is calculated as:

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

So:

[tex]d = \sqrt{(x-3)^2 + (\sqrt x - 0)^2}[/tex]

[tex]d = \sqrt{(x-3)^2 + (\sqrt x)^2}[/tex]

Evaluate all exponents

[tex]d = \sqrt{x^2 - 6x +9 + x}[/tex]

Rewrite as:

[tex]d = \sqrt{x^2 + x- 6x +9 }[/tex]

[tex]d = \sqrt{x^2 - 5x +9 }[/tex]

Differentiate using chain rule:

Let

[tex]u = x^2 - 5x +9[/tex]

[tex]\frac{du}{dx} = 2x - 5[/tex]

So:

[tex]d = \sqrt u[/tex]

[tex]d = u^\frac{1}{2}[/tex]

[tex]\frac{dd}{du} = \frac{1}{2}u^{-\frac{1}{2}}[/tex]

Chain Rule:

[tex]d' = \frac{du}{dx} * \frac{dd}{du}[/tex]

[tex]d' = (2x-5) * \frac{1}{2}u^{-\frac{1}{2}}[/tex]

[tex]d' = (2x - 5) * \frac{1}{2u^{\frac{1}{2}}}[/tex]

[tex]d' = \frac{2x - 5}{2\sqrt u}[/tex]

Substitute: [tex]u = x^2 - 5x +9[/tex]

[tex]d' = \frac{2x - 5}{2\sqrt{x^2 - 5x + 9}}[/tex]

Next, is to minimize (by equating d' to 0)

[tex]\frac{2x - 5}{2\sqrt{x^2 - 5x + 9}} = 0[/tex]

Cross Multiply

[tex]2x - 5 = 0[/tex]

Solve for x

[tex]2x =5[/tex]

[tex]x = \frac{5}{2}[/tex]

Substitute [tex]x = \frac{5}{2}[/tex] in [tex]y = \sqrt x[/tex]

[tex]y = \sqrt{\frac{5}{2}}[/tex]

Split

[tex]y = \frac{\sqrt 5}{\sqrt 2}[/tex]

Rationalize

[tex]y = \frac{\sqrt 5}{\sqrt 2} * \frac{\sqrt 2}{\sqrt 2}[/tex]

[tex]y = \frac{\sqrt {10}}{\sqrt 4}[/tex]

[tex]y = \frac{\sqrt {10}}{2}[/tex]

Hence:

[tex](x,y) = (\frac{5}{2},\frac{\sqrt{10}}{2}})[/tex]

The area of the rectangle is 128 square feet.

To find the area of a rectangle, you multiply the length by the width. In this case, the rectangle is 16 feet long and 8 feet wide.

Area = Length * Width

Substituting the given values:

Area = 16 feet * 8 feet

To perform the calculation, multiply the numbers:

Area = 128 square feet

Therefore, the area of the rectangle is 128 square feet.

The area of a rectangle represents the total amount of space enclosed within its boundaries. In this case, the rectangle measures 16 feet in length and 8 feet in width, so the area is calculated as 16 feet multiplied by 8 feet, resulting in 128 square feet.

The concept of area is fundamental in geometry and is used to measure the size of two-dimensional shapes. In the case of a rectangle, the area represents the product of its length and width, as every point within the rectangle contributes to its overall size.

Knowing the area of a rectangle can be useful in various situations. For example, if you want to determine how much flooring material is required to cover a rectangular room, you would need to know the area of the floor.

For more such questions on area visit:

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Answer:

B

Step-by-step explanation:

Answer:

5x - 9 = 3

Please consider giving me brainliest if this helped you! :)

answers:

4.C po

Step-by-step explanation:

i hope its help

Answer:

$25,377.11 or

$25,759.92

Step-by-step explanation:

hope that helps

Answer:

Triangle GHI is the right angled triangle.

Step-by-step explanation:

A right angled triangle follows the pythagoras theorem where,

h² = p² + b²

h is the longest side.

So,

15² = 12² + 9²

225 = 144 + 81

225 = 225

Hence this follow the pythagoras theorem and is a right angled triangle.

Answer:

66 ur welcome

Answer: x = -29/4

Step-by-step explanation: Let's solve for x.

4x = -29

4x/4 = -29/4

x = -29/4

Answer:

24×36×5=4,320

4,320÷6=720

Answer:

I don't understand your question.

Answer:

531 in^2

Step-by-step explanation:

21 x 18 = 378 in^2

17 x 9 = 153 in^2

378 + 153 = 531 in^2

Answer:

4x^4y^8

Step-by-step explanation:

Answer:

1200INCHES²

Step-by-step explanation:

the formula for the area of rectangle and rhombus is same so l*b will work....120*10=1200

Answer:

[tex] \frac{0.56}{w - 6} [/tex]

Step-by-step explanation:

1) Simplify 4.8 - 2 to 2.8.

[tex] \frac{2.8}{5(w - 6)} [/tex]

2) Simplify.

[tex] \frac{0.56}{w - 6} [/tex]

Therefor, the answer is 0.56/w-6.

Answer:

5/27

Step-by-step explanation:

P(9) = 2/9

P(not 6) = 5/6

2/9 x 5/6 = 10/54 or 5/27

Answer:

She can use any square object around her. Since squares are made up of right angles, any side she chooses can be used to compare the angle of the door. If the square corner fits perfectly, the angle is 90. If the square fits with space, obtuse. If the square doesn’t fit, acute

Step-by-step explanation:

Answer:

45.4

Step-by-step explanation:

Answer:

$17.29

Step-by-step explanation:

8% = 0.08

216.09(0.08) = 17.29

Answer:

$7.75

Step-by-step explanation:

$26.75 + $11.25=$38

$53.50 - $38=$15.5

$15.5/2= $7.75

Answer:

D

Step-by-step explanation:

Look at where the graph intercepts on the x-axis

If u can't see it's 2 and 10

Answer:

the answer is D

Step-by-step explanation:The answer is 2 and 10 because its directly on the line.

Hope I helped

Answer: 30 mins.

Step-by-step explanation:

The mode is the value that appears the most times.

In such a situation, first order the numbers in an ascending order:

15, 20, 25, 25, 30, 30, 30, 35, 35, 40, 45

As can be seen from the above, 30 is the number that appears the most so the mode is 30 mins.

Answer:

D

Step-by-step explanation:

Answer:

please be more specific

Step-by-step explanation:

This isn't a lot of info, we don't know how fast he is, so find the speed, and divide it by feet! easy! Hope this helps enough?

Answer:

[tex]let \: \alpha \: and \: \beta \: be \: two \: roots \: of \: above \: [/tex]

[tex]equation \: .[/tex]

[tex] \alpha = 2.52 \: and \: \beta = -0.52[/tex]

Answer:

9.95

Step-by-step explanation:

Answer:

9 19/20 = 9.95

Step-by-step explanation: