Find the 5th term in the expansion of (3x + 1)7 in simplest form.

The values of a and b are a = -3 and b = 42. The solutions for x can be written as:

x = -3 - √42

x = -3 + √42

To solve the quadratic equation 2x² + 12x = 66 by completing the square, we need to follow these steps:

Step 1: Move the constant term to the right side:

2x² + 12x - 66 = 0

Step 2: Divide the equation by the leading coefficient (2):

x² + 6x - 33 = 0

Step 3: To complete the square, we take half of the coefficient of x, square it, and add it to both sides of the equation:

x² + 6x + (6/2)² = 33 + (6/2)²

x² + 6x + 9 = 33 + 9

x² + 6x + 9 = 42

Step 4: Rewrite the left side as a perfect square:

(x + 3)² = 42

Step 5: Take the square root of both sides:

√(x + 3)² = ±√42

x + 3 = ±√42

Step 6: Solve for x:

x = -3 ± √42.

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We have two explicit functions of x for y:

[tex]y = -1/2 + \sqrt{(x^2 - x + 1/4)} \\or\\y = -1/2 - \sqrt{(x^2 - x + 1/4)}[/tex]

To define y as an explicit function of x, we need to solve for y in terms of x in the given equation:

[tex]x + y + y^2 = x^2[/tex]

First, let's simplify the equation by moving all the terms to one side:

[tex]y^2 + y + (x - x^2) = 0[/tex]

Now, we can use the quadratic formula to solve for y:

[tex]y = (-b + \sqrt{(b^2 - 4ac)} ) / 2a[/tex]

where a = 1, b = 1, and [tex]c = x - x^2.[/tex]Substituting these values, we get:

[tex]y = (-1 + \sqrt{(1 - 4(x - x^2)} )) / 2[/tex]

Simplifying further:

[tex]y = (-1 + \sqrt{(1 - 4x + 4x^2)} ) / 2\\y = (-1 + \sqrt{(4x^2 - 4x + 1)} ) / 2\\y = (-1 + 2\sqrt{(x^2 - x + 1/4)} ) / 2\\y = -1/2 + \sqrt{(x^2 - x + 1/4)}[/tex]

Therefore, we have two explicit functions of x for y:

[tex]y = -1/2 + \sqrt{(x^2 - x + 1/4)} \\or\\y = -1/2 - \sqrt{(x^2 - x + 1/4)}[/tex]

Either of these expressions represents y as an explicit function of x.

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The prove of the statement is described below.

First of all ,

Labeling the given figure

And draw  a diagonal,

In the quadrilateral PQRS,

The side PQ is equal to the side RS.

Therefore,

PQ = RS

And also, the side PQ is parallel to the side RS.

Then,

PQ || RS.

Now, draw the diagonal PR.

We know that a parallelogram's diagonal divides the parallelogram into two congruent triangles.

By the rule of  Side-Angle-Side,

we can show that ∆ PQR ≅ ∆ RSP.

Therefore,

The side QR is parallel to PS

Then,

QR || PS.

Hence,

PQRS is a parallelogram.

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Answer:

A = (1/4)√(4 + 11 + 14)√(-4 + 11 + 14)√(4 - 11 + 14)√(4 + 11 - 14)

A = (1/4)√29√21√7

= about 16.32 mm²

Answer:

  16.27 mm²  (see comment)

Step-by-step explanation:

You want the area of a triangle with side lengths 11 mm and 14 mm, and the angle between them 12.2°.

The area is given by the formula ...

  A = 1/2ab·sin(C)

  A = 1/2(11 mm)(14 mm)·sin(12.2°) ≈ 16.27 mm²

The area of the triangle is about 16.27 square millimeters.

__

Additional comment

If you use Heron's formula for the area from the three side lengths, you find it is about 16.32 mm². That's the trouble with over-specified geometrical figures. The result you get depends on which of the given values you use. (To get the area accurate to 4 sf, the angle needs to be specified to 4 sf: 12.24°.)

  s = (4 +11 +14)/2 = 14.5

  A = √(s(s -a)(s -b)(s -c))

  A = √(14.5(14.5 -4)(14.5 -11)(14.5 -14)) = √(14.5·10.5·3.5·0.5) = √266.4375

  A ≈ 16.32

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Answer:

mean, 25.8 median 23

Step-by-step explanation:

Arrange the data in an ascending order and the median is the middle value. If the number of values is an even number, the median will be the average of the two middle numbers.

The mean of a set of numbers is the sum divided by the number of terms.

The area of the triangle is 6x³y⁵

The space enclosed by the boundary of a plane figure is called its area.

A triangle is a polygon with three sides having three vertices.

There are different types of triangle, scalene triangle, equailteral triangle, isosceles triangle, right triangle e.t.c

The area of a triangle is expressed as ;

A = 1/2 bh

where b is the base and h is the height of the of the triangle.

Base = 3x²y²

height = 4x4y³

A = 1/2 × 3x²y² × 4x4y³

A = 1/2 × 12x³y⁵

A = 6x³y⁵

Therefore the area of the triangle is 6x³y⁵

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The area of the Triangle is [tex]48x^{3} y^{4}[/tex]

Triangle is a two-dimensional  three-sided polygon, which has three vertices, three sides and three angles. It is a shape formed when three straight lines meet.

How to determine this

Area of triangle = 1/2 base * height as given

Where area of triangle = ?

Base = [tex]3x^{2} y2[/tex]

i.e 3 * 2 [tex]x^{2} y[/tex]

Base, b = [tex]6x^{2} y[/tex]

Height = [tex]4x4y^{3}[/tex]

i.e [tex]4x[/tex] * [tex]4y^{3}[/tex]

Height,b = [tex]16xy^{3}[/tex]

Area of triangle = 1/2 * [tex]6x^{2} y[/tex] * [tex]16xy^{3}[/tex]

Area = 1/2 * 96* [tex]x^{2+1}[/tex] * [tex]y^{1+3}[/tex]

Area = 1/2 * 96 * [tex]x^{3}[/tex] * [tex]y^{4}[/tex]

Area = 48 * [tex]x^{3} y^{4}[/tex]

Area of the triangle = [tex]48x^{3} y^{4}[/tex]

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The surface areas and volumes of the solids are listed below:

Case 1: A = 118 cm², V = 70 cm³

Case 2: A = 121.5 in², V = 91.125 in³

Case 3: A = 192 m², V = 144 m²

Case 4: A = 480 ft², V = 576 ft³

Case 5: A = 87.965 yd², V = 62.832 yd³

In this problem we need to determine the surface area and the volume of each solid. The surface area is the sum of the areas of all faces. The area and volume formulas required to answer this question are shown below:

Area

Rectangle

A = w · l

Triangle

A = 0.5 · w · l

Circle

A = π · r²

Lateral face of a cylinder

A = 2π · r · h

Volumes

Prism

V = A · h

Where:

Now we proceed to determine the surface areas and volumes of the solids:

Case 1

A = 2 · [(5 cm) · (7 cm) + (5 cm) · (2 cm) + (7 cm) · (2 cm)]

A = 2 · (35 cm² + 10 cm² + 14 cm²)

A = 2 · 59 cm²

A = 118 cm²

V = (5 cm) · (7 cm) · (2 cm)

V = 70 cm³

Case 2

A = 6 · (4.5 in)²

A = 121.5 in²

V = (4.5 in)³

V = 91.125 in³

Case 3

A = 2 · 0.5 · (8 m) · (6 m) + (6 m) · (10 m) + (6 m)² + (8 m) · (6 m)

A = 192 m²

V = 0.5 · (8 m) · (6 m)²

V = 144 m²

Case 4

A = 2 · 0.5 · (12 ft) · (8 ft) + 2 · (10 ft) · (12 ft) + (12 ft)²

A = 480 ft²

V = 0.5 · (12 ft) · (8 ft) · (12 ft)

V = 576 ft³

Case 5

A = 2π · (2 yd)² + 2π · (2 yd) · (5 yd)

A = 87.965 yd²

V = π · (2 yd)² · (5 yd)

V = 62.832 yd³

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The measure of m∠A in the triangle is:

m∠A = 38.44°

The cosine rule is used for solving triangles which are not right-angled in which two sides and the included angle are given. The following are cosine rule formula for angles:

cos(A) = (b² + c² − a²)/2bc

cos(B) = (c² + a² − b²)/ 2ac

cos(C) = (a² + b² − c²)/2ab

Where a, b and c are the length of the sides, and A, B, and C are the measures of the angles

We have:

a = BC = 20

b = AC = 23

c = AB = 32

Substituting into:

cos(A) = (b² + c² − a²)/2bc

cos(A) = (23² + 32² − 20²)/(2*23*32)

cos(A) = 0.7833

A = cos⁻¹(0.7833)

A = 38.44°

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This is a linear equation problem with two variables. One way to solve it is by using the substitution method. Let x be the price of the tie and y be the price of the pair of pants. Then we have:

$$

\begin{aligned}

x + y &= 87.18 \\

x &= y - 2.12

\end{aligned}

$$

Substituting x into the first equation, we get:

$$

\begin{aligned}

(y - 2.12) + y &= 87.18 \\

2y - 2.12 &= 87.18 \\

2y &= 89.30 \\

y &= 44.65

\end{aligned}

$$

Therefore, the price of the pair of pants is $44.65. To find the price of the tie, we plug y into the second equation:

$$

\begin{aligned}

x &= y - 2.12 \\

x &= 44.65 - 2.12 \\

x &= 42.53

\end{aligned}

$$

Therefore, the price of the tie is $42.53.

Answer:

42.53

Step-by-step explanation:

write equation

tie + pants = 87.18

since ties cost 2.12 less than pants,

tie = pants -2.12

now substitute back into first equation

pants - 2.12 + pants = 87.18

solve for pants by adding 2.12 to the other side

divide by 2 since there are 2 pants on one side

find pants and then substitute value into the tie equation to find the price of the tie

The equation of the parabola is: y = (1/4)(x-3)²+ 2

In Parabola mathematics, it is defined as a set of points that are equidistant from a fixed point called the focus and a fixed line called the directrix. In this case, we are given the focus (3,5) and the directrix y=1, y=1, and we need to find the equation of the parabola.

To find the equation of the parabola, we first need to determine the vertex. The vertex is the midpoint between the focus and the directrix, which in this case is (3,3). Since the parabola is symmetric, we know that the axis of symmetry passes through the vertex and is perpendicular to the directrix. Therefore, the equation of the axis of symmetry is x=3.

Next, we need to find the distance between a point on the parabola and the focus, as well as the distance between that same point and the directrix. Let (x,y) be a point on the parabola. The distance between (x,y) and the focus is given by the distance formula: √((x-3)² + (y-5)²)

The distance between (x,y) and the directrix is simply the absolute value of the difference between y and 1: |y-1|

Since the point (x,y) is equidistant from the focus and the directrix, we have: √((x-3)²+ (y-5)²) = |y-1|

Squaring both sides and simplifying, we get: (x-3)²= 4(y-2)

Therefore, the equation of the parabola is: y = (1/4)(x-3)²+ 2

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NOTE : I HAVE ANSWERED THE QUESTION IN GENERAL AS GIVEN QUESTION IS INCOMPLETE.

Complete Question : Find the Parabola with Focus (3,5) and Directrix y=1 (3,5) , y=1

Answer:

Step-by-step explanation:

The converse of the conditional statement "If a polygon has three sides, then it is a triangle" is:

D. If a polygon is a triangle, then it has three sides.

In the original conditional statement, the "if" part is "a polygon has three sides," and the "then" part is "it is a triangle." The converse switches the positions of the "if" and "then" parts, resulting in the statement "If a polygon is a triangle, then it has three sides."

The value of x from the given triangle with angle 55° and side 27 units is 15.5 units.

A) By using Pythagoras theorem,

x²=16²+6²

x²=292

x=17.1 units

B) By using Pythagoras theorem,

23²=x²+9²

x²=529-81

x²=448

x=21.2 units

C) By using Pythagoras theorem,

10²=x²+8.5²

x²=100-72.25

x²=27.75

x=5.3 units

D) By using Pythagoras theorem,

x²=12²+15²

x²=369

x=19.2 units

E) Here, cos60°=6√3/x

1/2=6√3/x

x=12√3 units

F) Here, cos45°=√10/x

1/√2=√10/x

x=10 units

G) Here, sin60°=30/x

2/√3=30/x

x=15√3 units

H) Here, cos30°=x/16

2/√3=x/16

x=32/√3 units

I) Here, sinx=25/26

x=74°

J) Here, sin45°=x/16√2

1/√2=x/16√2

x=16 units

K) tan34°=x/28

0.6745=x/28

x=18.886

L) tanx°=17/18.5

tanx°=0.9189

x=42.5°

M) Here, cosx=9/12

x=41.4°

N) Here, sin16°=4/x

0.2756=4/x

x=4/0.2756 units

x=14.5

O) Here, cos55°=x/27

0.5735=x/27

x=15.4845

Therefore, the value of x from the given triangle with angle 55° and side 27 units is 15.5 units.

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The value of the parameter p that satisfies the given conditions is 36.

Let the roots of the quadratic equation [tex]3x^2 - 24x + p = 0[/tex] be denoted by α and β, where α is the root that is triple the value of β.

Then we have:

α = 3β

The sum and product of the roots of the quadratic equation are given by:

α + β = 8 (from the coefficient of x in the linear term)

αβ = p/3 (from the constant term)

Substituting α = 3β in the first equation gives:

3β + β = 8

4β = 8

β = 2

Therefore, α = 6.

So the roots of the quadratic equation are α = 6 and β = 2.

The product of the roots is:

αβ = 6 × 2 = 12

From the equation αβ = p/3, we have:

p/3 = 12

p = 36

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The probability of drawing two yellow marbles is 0.015 to the nearest thousandth.

The probability of drawing two yellow marbles after drawing a yellow marble on the first draw and without replacement drawing another yellow marble is determined as follows:

Total number of marbles in the bag = 4 + 2 + 6

Total number of marbles in the bag = 12 marbles in the bag.

P(Yellow on first draw) = 2/12 or 1/6

P(Yellow on second draw | Yellow on first draw) = 1/11

The probability of both marbles being yellow will be:

P(Both marbles yellow) = (1/6) * (1/11)

P(Both marbles yellow)  = 1/66 or 0.015

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Answer:

They are both positive on (2, ∞)

Step-by-step explanation:

g(x) > 0

log(2x + 1) > 0

2x + 1 > 0

x > –1/2

(-1/2, ∞)

Explanation:

The error is 42-40 = 2 meters

Divide this over the actual width

2/42 = 0.0476 = 4.76% approximately

This rounds to 5%

Answer:

D. 5%

Step-by-step explanation:

Percent Error = (|40 - 42| / 42) × 100

Percent Error = (2 / 42) × 100

Percent Error ≈ 4.76%

Answer:

$6.06

Step-by-step explanation:

$1.29 x 4.7 = 6.063

Round: 6.06

The 9th term of the geometric sequence 4, -16, 64, ... is 262144.

To find the 9th term of the geometric sequence 4, -16, 64, ... , we need to determine the common ratio (r) of the sequence.

To do this, we can divide any term by its preceding term:

-16 / 4 = -4

64 / -16 = -4

We see that the common ratio (r) is -4.

To find the 9th term, we can use the formula for the nth term of a geometric sequence:

Tn = a * r^(n-1)

Where Tn is the nth term, a is the first term, r is the common ratio, and n is the term number.

In this case, the first term a is 4, the common ratio r is -4, and we want to find the 9th term.

T9 = 4 * (-4)^(9-1)

T9 = 4 * (-4)^8

T9 = 4 * 65536

T9 = 262144

Therefore, the 9th term of the geometric sequence 4, -16, 64, ... is 262144.

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Mrs. Garcia invested $2240 in the 8.5% account and $4091 in the 8% account.

Let x be the amount invested in the 8.5% account, and y be the amount invested in the 8% account. Since the total investment is $6331, we have x + y = 6331.

The total interest received is $517.68, which can be expressed as 0.085x + 0.08y = 517.68, where 0.085 and 0.08 are the decimal equivalents of the interest rates.

We can now solve this system of equations to find x and y. One possible method is to use substitution, where we solve for one variable in terms of the other from one of the equations, and substitute it into the other equation. From x + y = 6331, we have y = 6331 - x. Substituting this into the second equation, we get:

0.085x + 0.08(6331 - x) = 517.68

Simplifying and solving for x, we get:

0.005x + 506.48 = 517.68

0.005x = 11.2

x = 2240

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53 revolutions will be needed by the wheels to make it travel 300 feet.

Option D is the correct answer.

We have,

To solve this problem, we can use the formula:

Number of revolutions = Distance / Circumference of the wheel

First, let's convert the distance from feet to inches:

300 feet = 300 x 12 = 3600 inches

Next, we need to calculate the circumference of the wheel using the given diameter:

Circumference = π x diameter

Circumference = 3.14 x 22 inches

Circumference ≈ 69.08 inches

(rounded to two decimal places)

Now we can calculate the number of revolutions:

Number of revolutions = 3600 inches / 69.08 inches

= 52.14 revolutions

Rounding to the nearest whole number.

= 52 revolutions.

Therefore,

53 revolutions will be needed by the wheels to make it travel 300 feet.

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The bicycle wheels will need 53 full revolutions to travel 300 feet.

The circumference of a wheel can be calculated using the formula:

circumference = π x diameter

So, circumference = 3.14 x 22

= 69.08 inches

Now, converting feet to inches

300 feet = 300 x 12 inches

300 feet = 3600 inches

So, number of revolutions:

revolutions = distance / circumference

= 3600 / 69.08

≈ 52.11

Therefore, the bicycle wheels will need 53 full revolutions to travel 300 feet.

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Answer: C

Step-by-step explanation:

Answer:

D.  (-2, 10)

Step-by-step explanation:

Given inequalities:

    [tex]y > -\dfrac{5}{3}x+2[/tex]

    [tex]y > 3x+2[/tex]

Both inequalities have been given in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept.

The first equation has a negative slope, which means that as x increases, y decreases. Therefore, it is the dashed line that begins in quadrant II and ends in quadrant IV.

The second equation has a positive slope, which means that as x increases, y increases. Therefore, it is the dashed line that begins in quadrant III and ends in quadrant I.

When graphing inequalities, the inequality sign ">" indicates a dashed line and shading above the line.

As both inequalities should be shaded above the lines, the region where the shaded regions overlap is the upper "V" of the graph (see attachment).

A point that satisfies both inequalities is any point that is located in the overlapping shaded region, but not found on the dashed lines.

Therefore, the point that satisfies both inequalities is (-2, 10).

The model that represents the expression 87 - 42 is (d)

From the question, we have the following parameters that can be used in our computation:

87 - 42

Using their place values, we have

87 = 8 tens 7 units

42 = 4 tens 2 units

This means that

87 - 42 = 8 tens 7 units - 4 tens 2 units

Subtract the tens

87 - 42 = 4 tens 7 units - 2 units

Subtract the units

87 - 42 = 4 tens 5 units

The model that represents the expression 87 - 42 is 4 tens 5 units

This is represented by model (d) bottom right

Hence, the model that represents the expression 87 - 42 is (d)

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Answer:

It's B

Step-by-step explanation:

Look at the sides with the number compare them with each other and you will find they are the most similar ones with each other.

The percentage of buyers is approximately 68.26% of buyers of new houses paid between [tex]$149,000[/tex] and [tex]$151,000[/tex] .

We are given that the prices of the new homes are normally distributed with a mean of [tex]$150,000[/tex] and a standard deviation of $1000.

Using the 68-95-99.7 rule, we know that: approximately 68% of the data falls within one standard deviation of the mean approximately 95% of the data falls within two standard deviations of the mean, approximately 99.7% of the data falls within three standard deviations of the mean.

In order to determine the proportion of customers who spent between $149,000 and , we must first determine the z-scores for these values:

z1 = (149,000 - 150,000) / 1000 = -1 z2 = (151,000 - 150,000) / 1000 = 1

Now, we can determine the proportion of data that falls between z1 and z2 using the z-table or a calculator. The region to the left of z1 is 0.1587, and the area to the left of z2 is 0.8413, according to the z-table. Thus, the region bounded by z1 and z2 is:

0.8413 - 0.1587 = 0.6826

We can get the percentage of consumers who spent between by multiplying this by 100% is  [tex]$149,000[/tex]  and [tex]$151,000[/tex]:

0.6826 x 100% = 68.26%

Therefore, the standard deviation of customers who paid between is [tex]$149,000[/tex] and [tex]$151,000[/tex]  for this model of new homes.

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The number of pages that each book contained is given as follows:

465 pages.

The number of pages that each book contained is obtained applying the proportions in the context of the problem.

A head teacher shared 27 notebooks among 9 students, hence the number of the notebooks per student is given as follows:

27/9 = 3 notebooks per student.

One of them students found that each of his books combined 155 pages, hence the number of pages on the book is given as follows:

155 x 3 = 465 pages.

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The detergent with the coupon is the better buy since it has a lower price per ounce.

To determine which option is the better buy, we need to compare the prices per ounce for each detergent brand.

First, let's calculate the price per ounce for the 32-ounce bottle of detergent after applying the coupon:

Price per ounce = (Price - Coupon) / Ounces

Price per ounce = ($1.39 - $0.35) / 32

Price per ounce = $1.04 / 32

Price per ounce ≈ $0.0325

Next, let's calculate the price per ounce for the 20-ounce bottle of the other brand:

Price per ounce = Price / Ounces

Price per ounce = $0.79 / 20

Price per ounce = $0.0395

Comparing the two price per ounce values, we can see that the price per ounce for the detergent with the coupon is approximately $0.0325, while the price per ounce for the other brand is $0.0395.

Therefore, the detergent with the coupon is the better buy since it has a lower price per ounce.

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The first equation is a quadratic equation, and the second equation is a linear equation. The solution to the system is x = -2 and y = 7, which satisfies both equations.

A linear equation is an equation in which the highest power of the variable is always 1. It is also known as a one-degree equation. The standard form of a linear equation in one variable is of the form Ax + B = 0. Here, x is a variable, A is a coefficient and B is constant.

The system that is equivalent to:

O

O

O

[5-x=9x²

y = 5-x

is: x = -2, y = 7

In the given system, the first equation is a quadratic equation, and the second equation is a linear equation. The solution to the system is x = -2 and y = 7, which satisfies both equations.

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Answer: sin(0) = 40.804/41

Step-by-step explanation:

Answer: $71,400

Step-by-step explanation:

We will set up a proportion with income in the numerator and pounds in the denominator.

       [tex]\displaystyle \frac{\$12}{4.2\;lbs} =\frac{\$x}{24,990\;lbs}[/tex]

Then we will cross-multiply.

       4.2 * x = 12 * 24,990

       4.2x = 299,880

Lastly, we will divide both sides of the equation by 4.2.

       x = $71,400

The income from 24,990 pounds of strawberries would be approximately $71,690.14.

To find the income from 24,990 pounds of strawberries, we need to use a proportion:

4.2 pounds of strawberries sells for $12, so 1 pound of strawberries sells for $12/4.2 = $2.86 (rounded to two decimal places).

Therefore, 24,990 pounds of strawberries would sell for:

$2.86 x 24,990 = $71,690.14 (rounded to two decimal places).

The height of the plane is 5 miles.

To solve this problem, we can visualize it as a right triangle. The horizontal distance traveled by the plane forms the base of the triangle, which is 12 miles. The total distance traveled by the plane forms the hypotenuse of the triangle, which is 13 miles. We need to find the height, which corresponds to the vertical side of the triangle.

Using the Pythagorean theorem, we can calculate the height as follows:

height^2 + 12^2 = 13^2

height^2 + 144 = 169

height^2 = 169 - 144

height^2 = 25

Taking the square root of both sides, we get:

height = √25

height = 5

Therefore, the height of the plane is 5 miles.

To learn more about height :

https://brainly.com/question/28990670

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