An animal trainer at the aquarium
trains bottlenose dolphins to jump
for performances. One dolphin
jumped out of the water at a height
of 15 feet and then dove back under
the water to a depth of 20 feet. Write
an expression to represent the vertical
distance the dolphin traveled. What is
the vertical distance?

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]

Answer:

$17.29

Step-by-step explanation:

8% = 0.08

216.09(0.08) = 17.29

Answer:

look at the answer below

Step-by-step explanation:

this is xy + 25 = 0 if you wanted it. hope it helps

Answer:

(Rational Functions Practice)

3. Graph the function. (1 point)

xy+25=0

Step-by-step explanation:

Graph B. is the correct answer

Answer:

1) b - correct statement: The interquartile range of the number of volunteers is 16

2) c - correct statement: About 50% of the students have between a 75 and a 90

Step-by-step explanation:

1B --> 66 - 50

2C --> About 50% of the students have between a 75 and a 90

Answer:

9.549296

Step-by-step explanation:

30/pi = 9.549296 for diameter

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:

1 1/2 feet

Step-by-step explanation:

Martin is making a model of a Native American canoe. He has 5 1/2 feet of wood he uses 2 3/4 feet for the whole and 1 1/4 feet for a paddle how much does he have left

This is calculated as:

5 1/2 feet - ( 2 3/4 feet + 1 1/4 feet)

Lowest Common Denominator 4

5 1/2 feet - (2 + 1 ( 3 + 1/4)

5 1/2 feet - 3 + 1

5 1/2 feet - 4 feet

= 1 1/2 feet of wood left

Answer:

24×36×5=4,320

4,320÷6=720

Answer:

I don't understand your question.

Answer: y = 4

Step-by-step explanation:

Answer:

11 = (15 -6r) + 2r

11= 15-6r+2r

11=15-4r

4=4r

r=1

Answer:

Trudy will have 4 pieces of the fabric.

Step-by-step explanation:

Since, 1 foot of the fabric = 12 inches

If Trudy cuts the fabric into pieces of 3 inch.

Number of pieces she will have = [tex]\frac{\text{Total length of the fabric}}{\text{Length of one piece}}[/tex]

                                                     = [tex]\frac{12}{3}[/tex]

                                                     = 4 pieces

Therefore, she will have 4 pieces of the fabric.

Answer:

Step-by-step explanation:

Through is upholstering a rectangular ottoman that measures 21 cm x 18 cm x 15 cm. What will be the total square centimeters of fabric that Sue must use to cover all faces of the Ottoman

Answer:

The volume of the snow globe is 78 cubic inches.

Step-by-step explanation:

Given:

A snow globe consists of a pedestal in the shape of a cube with a sphere mounted on top of the pedestal. The pedestal has an edge length of 4 inches and the sphere has a diameter of 3 inches.

Now, to get the volume of the snow globe.

So, we get the volume of pedestal which is in the shape of cube by putting formula:

Edge = 4 inches.

Volume of pedestal = 64 cubic inches.

Now, to get the volume of sphere by putting formula:

Diameter = 3 inches.

Radius () =

Volume of sphere = 14.131 cubic inches.

Now, to get the volume of snow globe by adding both the volumes, the volume of sphere and the volume of pedestal:

Rounding to the nearest cubic inch = 78 cubic inches.

Therefore, the volume of the snow globe is 78 cubic inches.

Answer:

9.95

Step-by-step explanation:

Answer:

9 19/20 = 9.95

Step-by-step explanation:

Answer:

What you did is correct!

Answer:

378x³⁰²

Step-by-step explanation:

378x378x378x378x378x378x378x378x378

=378*x^378*x^378*x^378*x^378*x^378*x^378*x^378*x^378

=378*x3024

=378x³⁰²

Hope this helps

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:

75 square feet

Step-by-step explanation:

Break this apart and turn it into a rectangle

You can the use 15 times 5 to find your answer

Answer:

B.) y - 6 = -8/5(x + 2)

Step-by-step explanation:

Remember in order to find the slope intercept form on 2 points, you need these equations: [tex]\frac{x_2 -x_1}{y_2 -y_1}[/tex] and [tex]y-y_1=m(x-x_1)[/tex]

To find the slope, you will need to use the first equation. In this case the equation will look like this: [tex]\frac{(-2)-6}{3-(-2)}=-\frac{8}{5}[/tex]

Then use the second equation, plug in the numbers, and you will get B as your answer

Answer:

First one true, second false, third true

Step-by-step explanation:

Answer:

D.  y = -2/3 x + 4/3

Step-by-step explanation:

When you have a perpendicular line, everything except the y-intercept becomes opposite. So 3/2x + 6 will become -2/3x.

To find the slope, you have to plug the coordinates into the equation so far which is y = -2/3x + b

y = -2/3x + b

2 = -2/3(-1) + b

2 = 2/3 + b

4/3 = b

So the equation is y = -2/3 x + 4/3.

Answer:

the graph is the one with h(x)=2x-6

Step-by-step explanation:

it will hit the y-axis at (0,-6) and it will hit the x-axis at (3,0)

The graph of h(x)=p(x)-6 is shown below.

All the points (and only those points) which lie on the graph of the function satisfy its equation.

Thus, if a point lies on the graph of a function, then it must also satisfy the function.

We are given that;

p(x)=2x

Now,

To find the graph of h(x) = p(x) - 6, we need to use the concept of transformations of functions. A transformation is a change in the position, shape, or size of a function’s graph.

To graph h(x), we can start by graphing p(x) = 2x and then shifting it down by 6 units. The graph of p(x) is a straight line that passes through the origin and has a slope of 2. To shift it down by 6 units, we can subtract 6 from the y-coordinate of every point on the line. For example, the point (1, 2) on p(x) becomes (1, -4) on h(x).

The graph of h(x) is also a straight line that has the same slope as p(x), but has a y-intercept of -6. The equation of h(x) is y = 2x - 6.

Therefore, the graph of the function will be shown below.

Learn more about points lying on graph of a function here:

brainly.com/question/1979522

#SPJ2

Answer:

12×5+45-36=60+9=69 is your answer

Answer:

12×5=60+45=105-36=69

Answer:

$25,377.11 or

$25,759.92

Step-by-step explanation:

hope that helps

Answer: 2

2x-7=11  X=9  set up the up the negative portion of solutions Set up the negative portion of the 2x - 7= - 11   x= -2  and then x = 9,-2

Step-by-step explanation:

Answer:

Step-by-step explanation:

-2x+7≥-11

-2x≥-18

2x≤18

x≤9