[tex]F(x)=(6 \sqrt{x} -2)(5 \sqrt{x} +7)[/tex]

The height of the air balloon is 63.4 yards.

Trigonometry is a branch of mathematics that uses variables to determine heights and distances. It is the study of the properties of right angled triangles and trigonometric functions and of their applications.

For the given situation,

The diagram below shows the situation described above.

The distance between Caleb and Emily = 100 yards

Let the distance from Caleb to the air balloon be 'x'

Let the distance from Emily to the air balloon be 'y'

Let height of the air balloon be 'h'

⇒ [tex]x+y=100[/tex]

⇒ [tex]x=100-y[/tex]

We know that, [tex]tan[/tex] θ = [tex]\frac{perpendicular }{base}[/tex]

Now consider the triangle with angle 45°,

[tex]tan 45[/tex]° = [tex]\frac{h}{x}[/tex]

We know that tan 45° = 1 and substitute x = 100-y,

⇒ [tex]1 = \frac{h}{100-y}[/tex]

⇒ [tex]h=100-y[/tex]

This is equation 1.

Now consider the triangle with angle 60°,

[tex]tan60[/tex]° = [tex]\frac{h}{y}[/tex]

We know that, tan 60° = √3

⇒ [tex]\sqrt{3} = \frac{h}{y}\\[/tex]

⇒ [tex]h=y\sqrt{3}[/tex]

This is the equation 2.

On equating equation 1 and 2,

⇒ [tex]100-y=y\sqrt{3}[/tex]

⇒ [tex]y+y\sqrt{3}=100[/tex]

⇒ [tex]y(\sqrt{3}+1 )=100[/tex]

⇒ [tex]y=\frac{100}{\sqrt{3}+1 }[/tex]

Thus height, [tex]h=y\sqrt{3}[/tex]

Substitute the value of y in h,

⇒ [tex]h=\sqrt{3}(\frac{100}{\sqrt{3} +1} )[/tex]

⇒ [tex]h=\frac{100\sqrt{3} }{\sqrt{3}+1 }[/tex]

⇒ [tex]h=\frac{173.205}{2.732}[/tex]

⇒ [tex]h=63.397[/tex]

⇒ [tex]h=63.4[/tex]

Hence we can conclude that the height of the air balloon is 63.4 yards.

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Answer:

Step-by-step explanation:

(a) The attachment shows the graph of the parabola in blue. It also shows an inscribed rectangle in black.

(b) The upper right point of the rectangle is shown in the attachment as (x, y). The dimension y is the height of the rectangle. The x-dimension is half the width of the rectangle, which is symmetrical about the y-axis. Hence the width is 2x.

(c) As with any rectangle, the area is the product of length and width:

... A = (2x)(9 -x²) . . . . . the attachment shows a graph of this

... A = -2x³ +18x . . . . . expanded form suitable for differentiation

A suitable domain for A is where both x and A are non-negative: 0 ≤ x ≤ 3.

(d) The derivative of A with respect to x is ...

... A' = -6x² +18

This is defined everywhere, so the critical values will be where A' = 0.

... 0 = -6x² +18

... 3 = x² . . . . . . . divide by -6, add 3

... √3 = x . . . . . . . -√3 is also a solution, but is not in the domain of A

(e) The rectangle will have its largest area where x=√3. That area is ...

... A = 2x(9 -x²) = 2√3(9 -(√3)²) = 2√3(6)

... A = 12√3 . . . . square units . . . . ≈ 20.785 units²

The parabola y = 9 - x^2 is graphed, and a rectangle's vertices and area calculations are superimposed on it. The derivative of the area function gives critical values, determining the maximum area is 18 square units.

To solve this problem, we must utilize quadratic equations, Two-Dimensional (x-y) Graphing, and calculus.

(a) To begin, we need to draw a graph of the parabolic equation y = 9 - x^2. The vertex of the parabola is at the point (0,9) since the x-coordinate of the vertex is -b/2a in a general quadratic equation of the form y = ax^2 + bx + c, and here a = -1 and b = 0.

(b) For the rectangle, the upper right vertex is at point (x, y=(9 - x^2)), with base length 2x and height (9 - x^2).

(c) The area (A=base * height) of the rectangle is then given by A = 2x * (9 - x^2). The domain of A is specified by the values of x for which A is defined. Here A is defined for all x ∈ (-∞, ∞).

(d) The derivative of A with respect to x is A’ = 2(9 - 3x^2), which can be set equal to zero, the solution of which gives critical values x = ±√3.

(e) By the second derivative test or comparing values at the endpoints and critical points, you can determine that the largest area is 18 square units when x = √3.

Therefore,

b) The upper right vertex is at point (x, y=(9 - x^2)), with base length 2x and height (9 - x^2).

c) Here, A is defined for all x ∈ (-∞, ∞).

d) The solution of which gives critical values x = ±√3.

e) The largest area is 18 square units.

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The correct answer is:

Total capacity of tank in quarts:

20 gallons = 20 × 4 = 80 quarts

Gasoline used in quarts:

4 gallons 1 quart = (4 × 4) + 1 = 17 quarts

Gasoline left in the tank in quarts:

80 − 17 = 63 quarts

Gasoline left in the tank in gallons and quarts:

63 quarts = 63 ÷ 4 = 15 remainder 3 = 15 gallons 3 quarts

x = 0

y = 1 + 2x Substitute x = 0

y = 1 + 2(0) Any number times 0 is 0

y = 1 + 0 Simplify

y = 1

I hope this could help.

Answer:

<1, 3>, square root of ten

Step-by-step explanation:

perimeter = 2W+2L

Length = 4W

160 = 2W +2(4W)

160 = 10W

w = 160/10 = 16 yards

length = 16*4 = 64 yards

Check:

64*2 = 128, 16*2 = 32

128+32 = 160

Length = 64 yards, Width = 16 yards

Casey earned $780.25 in tips last week. This was calculated by first finding out her earnings from her hourly wage, and then subtracting this from her total earnings.

First, we need to calculate how much Casey earns from the hourly rate alone. Given that Casey's hourly rate is $4.55 per hour and she worked for 26 hours last week, we multiply the two to get her earnings from the hourly wage, which is $4.55 * 26 = $118.30.

Then we need to find out how much Casey made in tips. Casey's total earnings last week were $898.55. We subtract the earnings from her hourly rate from her total earnings to find out how much she made in tips. So, $898.55 - $118.30 = $780.25.

Therefore, Casey earned $780.25 in tips last week, which corresponds to option D.

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a. The commission from the sale of $ 44,000 worth of​ furnaces is $1,800

b. Commission would double as well.

A percentage is a minimum number or ratio that is measured by a fraction of 100.

We are given that salesperson's commission rate is 5 ​%

Then Commission paid on $36,000 worth of furnaces with a rate of 5% is:

= Amount of sales x commission rate

= 36,000 x 5%

= $1,800

If the sales were to double, the commission would be based on double the amount so it would double;

= 36,000 x 2

= $72,000

Commission = 72,000 x 5%

= $3,600

therefore Commission is doubled.

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Answer:

Standard form: 600

Unit form: 6 hundreds.

Step-by-step explanation:

We have been given a number [tex]10\times 6[/tex]-tens. We are asked to write our given number in unit form and standard form.  

To write our given number in standard form, we will expand our given number as shown below:

6-tens [tex]6\times 10=60[/tex]

[tex]10\times 6[/tex]-tens would be  [tex]10\times 60=600[/tex]

Therefore, our given number in standard form would be 600.

We know that unit form is writing a number using place value units.

We can see that our given number has 0 ones, 0 tens and 6 hundreds.

Therefore, our given number in unit form would be 6 hundreds.

We have that  the Tens unit form of 10 x 6 and  Standard form is mathematically

[tex]6*10^1[/tex]

6tens

From the question we are told that

10 x 6

10 x 6=60

Standard form is a number, between 1 and 10 multiplied by a power of 10

Generally the equation for the Standard form   is mathematically given as

x*10^y

Therefore

the Standard form  of 10 x 6  is mathematically given as

[tex]6*10^1[/tex]

And

Generally the equation for the Tens unit form of 10 x 6  is mathematically given as

x*tens(10)

Tens-unit form is mathematically given as

6tens

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The height of the tree when it was transplanted was:

                              4 feet

As we could observe that the given table represents a linear function.

The table is given as follows:

Year since it was      4        4.5        5      

transplanted

Height  (Feet)            12       13          14

We will find the linear function.

Let y denotes height and x denote year since it was transplanted.

We know that any linear function passing through two points (a,b) and (c,d) is given by:

[tex]y-b=\dfrac{d-b}{c-a}\times (x-a)[/tex]

Here let (a,b)=(4,12) and (c,d)=(5,14)

Hence, the linear function is calculated as follows:

[tex]y-12=\dfrac{14-12}{5-4}\times (x-4)\\\\\\y-12=2(x-4)\\\\\\y-12=2x-8\\\\\\y=2x-8+12\\\\\\y=2x+4[/tex]

Now the height of tree when it was transplanted is the value of y when x=0

Hence, when x=0 we have:

y=4

    Hence, the height was:

               4 feet

The formula for finding the slope of a line is expressed as:

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

Given the coordinate points (2, 1) and (2, 0). Get the slope of the line passing through the coordinate points

[tex]m=\frac{0-1}{2-2}\\m=\frac{-1}{0}\\m=\infty[/tex]

The equation of the line in point-slope form is [tex]y-y_0=m(x-x_0)[/tex]

Since the slope of a vertical line does not exist, hence the equation of the line cannot be written in point-slope form

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Answer: No

Step-by-step explanation: It does not go through the origin (0,0) in a straight line. Also, it has a y-intercept, meaning it doesn't go through the origin (0,0) and goes through the line that is up and down; vertically.

The number of people George can feed is 6.

The unitary method is a technique for solving a problem by first finding the value of a single unit, and then finding the necessary value by multiplying the single unit value.

Given that, George bought 4 submarine sandwiches for a birthday party.

Each person will eat 2/3 of a sandwich

Number of people George can feed

= 4÷2/3

= 4×3/2

= 6

Therefore, the number of people George can feed is 6.

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The perimeter of the rectangle with four times longer in length than its width and an area of 400 cm2 is 100 cm.

To solve this, let's assume the width of the rectangle is w centimeters. This would make the length 4w centimeters. Since the area of a rectangle is given by the product of its length and width, we have:

w x 4w = 400

Solving for w, we find that w2 = 100, so w = 10 cm.

Hence, the length is 4 x 10 = 40 cm.

The perimeter of a rectangle is given by the formula 2l + 2w, so substituting in our values:

Perimeter = 2(40) + 2(10) = 80 + 20 = 100 cm.

Therefore, the perimeter of the rectangle is 100 cm.

Answer with explanation:

Side of cube = S

Surface Area of Cube = S.A

Relation between Side of a cube and surface area

   [tex]S=\frac{S.A}{6}[/tex]

→If surface area of cube =180 Square meters

Side of cube (S)

                     [tex]S_{1}=\frac{180}{6}\\\\=30[/tex] meters

→  If surface area of cube =120 Square meters

Side of cube (S)

                     [tex]S_{2}=\frac{120}{6}\\\\=20[/tex] meters

[tex]S_{1}-S_{2}=30 -20=10\\\\S_{1}=S_{2}+10[/tex]

Side of cubic having surface area 180 square meters is greater by 10 meters, than a cube with the surface area of 120 square meters.