In 8 years, Claire will be three times her current age. In how many years will she be 20 years old?

The value of y is -3. The equation formed from the question is 4x + y = 65 and 5x -3y = 94.

First, we will solve for x

Multiplying the 4x + y = 65 equation by 3 we get

12x +3y =195

Now solving both the equation

The variable y gets cancelled because of having the same value

∴ 17x = 289

x= 17

Now putting the value of x in equation 4x + y = 65 to get the value of y

4*17 +y =65

68 +y = 65

y= -3

Now putting the value of x in equation 5x -3y = 94  to get the value of y

5*17 -3y =94

85 -3y =94

-3y = 94-85

y= -3

Therefore, the value of y is -3

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[tex] \mathbb{ANSWER:}[/tex]

The slope is -2 while the y-intercept is -1.

[tex] \mathbb{EXPLANATION:}[/tex]

The slope and the intersection with the y-axis of the graph or the y-intercept is already given if you will just simply look at the equation. An equation written in y = mx + b is in slope-intercept form, where m is the slope and b is the y-intercept. So, the number being multiplied to x or the coefficient of x is the slope while the constant is the y-intercept.

The comparison between the distance he biked and swam is 3 : 1.

A numerical expression is a mathematical statement written in the form of numbers and unknown variables. We can form numerical expressions from statements.

From the given information the comparison of the distance he swam to the distance he biked can be represented in the form of ratios.

Given, He rode his bike 3/4 miles and swam for 1/4 mile.

Therefore, The ratio of biking : swimming = 3/4 : 1/4.

Now, Multiplying by 4 to get rid of the denominator we have,

Biking : Swimming = 3 : 1.

As the days in both comparisons increase in the same manner the ratio will remain the same.

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Answer:

Step-by-step explanation:

d^2

When dividing subtract indices OR

dxdxdxdxd /dxdxd

d's cancel out leaving dxd=d^s

The logarithmic equation that is equivalent to the given exponential equation

The logarithmic equation that is equivalent to the exponential equation [tex]((1)/(3))^-4=81 (is)  log(1/3) 81 = -4[/tex]

To rewrite an exponential equation as a logarithmic equation,

we can use the following formula:

bx = y → logb y = x

In this case, b = (1/3), x = -4, and y = 81. So, we can plug these values into the formula to get the logarithmic equation:

[tex]log(1/3) 81 = -4[/tex]

This is the logarithmic equation that is equivalent to the given exponential equation.

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If 2x-1 is a factor of 24x⁴ - 2x² +2x +f , the value of f is  1 - 2x.

Synthetic division is a method of polynomial division used to divide a polynomial by a linear expression of the form x - a, where a is a constant. It is a shorthand way of performing long division for polynomials and is particularly useful when the divisor is of the form x - a, as it eliminates the need for writing out the full divisor in each step of the division process.

Since 2x-1 is a factor of the given polynomial, we know that it will divide exactly into the polynomial, leaving no remainder. Therefore, we can use polynomial long division to find the quotient when 24x⁴ - 2x² + 2x + f is divided by 2x-1:

           12x³ + 5x² - 3x - f/2 + 1/4

      ____________________________________

2x - 1 |   24x⁴ + 0x³ - 2x² + 2x + f

         - (24x⁴- 12x³)

----------------------

12x^3

- (12x³ - 6x^2)

----------------------

5x^2

- (5x² - 2.5x)

----------------------

2.5x

- (2.5x - f/4)

-----------------------

f/4 + 2x/4 - 1/4

Since there is no remainder, the last term in the quotient must be zero. Therefore:

f/4 + 2x/4 - 1/4 = 0

Simplifying and solving for f, we get:

f/4 = 1/4 - 2x/4

f = 1 - 2x

Therefore, f = 1 - 2x.

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Pythagorean theorem: Show that ⟨x, y⟩ = 0

Let V be an Euclidean space and let x, y ∈ V.(a) If ∥x + y∥ = ∥x − y∥ then determine ⟨x, y⟩.We know that ∥x + y∥² = ⟨x + y, x + y⟩ = ⟨x, x⟩ + ⟨x, y⟩ + ⟨y, x⟩ + ⟨y, y⟩ = ∥x∥² + 2⟨x, y⟩ + ∥y∥²And ∥x − y∥² = ⟨x − y, x − y⟩ = ⟨x, x⟩ − ⟨x, y⟩ − ⟨y, x⟩ + ⟨y, y⟩ = ∥x∥² − 2⟨x, y⟩ + ∥y∥²Since ∥x + y∥ = ∥x − y∥, we have ∥x∥² + 2⟨x, y⟩ + ∥y∥² = ∥x∥² − 2⟨x, y⟩ + ∥y∥²⟨x, y⟩ = 0(b) Show that 2⟨x, y⟩ ≤ ∥x∥² + ∥y∥².We know that ∥x + y∥² = ∥x∥² + 2⟨x, y⟩ + ∥y∥²And ∥x + y∥² ≥ 0So ∥x∥² + 2⟨x, y⟩ + ∥y∥² ≥ 0⟨x, y⟩ ≤ (∥x∥² + ∥y∥²)/2(c) If (z, y) = 0 for all z ∈ V then show that y = 0.Let z = y, then (y, y) = 0⟨y, y⟩ = ∥y∥² = 0∥y∥ = 0So y = 0(d) If ∥x∥ = ∥y∥ for some x, y ∈ V then ⟨x + y, x − y⟩ = 0.We know that ∥x + y∥² = ∥x∥² + 2⟨x, y⟩ + ∥y∥²And ∥x − y∥² = ∥x∥² − 2⟨x, y⟩ + ∥y∥²Since ∥x∥ = ∥y∥, we have ∥x∥² = ∥y∥²So ∥x + y∥² − ∥x − y∥² = 4⟨x, y⟩ = 0⟨x, y⟩ = 0(e) Pythagorean theorem: Show that ⟨x, y⟩ = 0 if and only if ∥x + y∥² = ∥x∥² + ∥y∥².We know that ∥x + y∥² = ∥x∥² + 2⟨x, y⟩ + ∥y∥²If ⟨x, y⟩ = 0, then ∥x + y∥² = ∥x∥² + ∥y∥²If ∥x + y∥² = ∥x∥² + ∥y∥², then 2⟨x, y⟩ = 0⟨x, y⟩ = 0

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Answer:

Degree: The degree of the function is 3.

X-Intercepts with multiplicity greater than 1: x = -2 and x = 0

How many distinct x-intercepts?: There are two distinct x-intercepts.

How many zeros are there?: There are three zeros, one at x = -2 and two at x = 0 (with a multiplicity of 2).

The solution is: population in 2015 will be 40,115,896

The exponential is an example of a mathematical function that is useful in determining if something is increasing or decreasing exponentially is the exponential function. As implied by its name, an exponential function uses exponents.

But take note that an exponential function does not have a variable as its exponent and a constant as its base (if a function has a variable as the base and a constant as the exponent then it is a power function but not an exponential function).

calculate the average percent change in population in California from 2000-2009.

Add all the percentage than divide by 9 since there is 9 year

1.97+1.71+1.65+1.42+1.22+1.02+1.07+1.22+0.93/9 = 1.3567%

The number of periods from 2009 to 2015 is 6:

Future: 37,000,000•(1+0.013567)^6 = 40,115,896.06

Hence, the solution is: population in 2015 will be 40,115,896

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The European company will earn 128,750 Euros in profit for selling 1,500 headphones.

To find the profit for selling 1,500 headphones, we need to plug in the value of h into the profit function E(h) and simplify.

Since h is measured in thousands of headphones, we need to divide 1,500 by 1,000 to get h = 1.5.

Now we can plug in h = 1.5 into the profit function and simplify:

E(1.5) = -4(1.5)^(4) + 7(1.5)^(3) - 7(1.5) + 20

E(1.5) = -4(5.0625) + 7(3.375) - 10.5 + 20

E(1.5) = -20.25 + 23.625 - 10.5 + 20

E(1.5) = 12.875

So the profit for selling 1,500 headphones is 12.875 ten thousands of Euros, or 128,750 Euros.

Therefore, the European company will earn 128,750 Euros in profit for selling 1,500 headphones.

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The surface area of the cone is 578.1 square feet.

We know that for a cone of radius R and slant height H the surface area is given by the formula:

S = pi*R^2 + pi*R*(H)

with pi = 3.14

On the diagram we can see that H = 19.3ft, and the diameter of the base is 14 ft, then the radius is:

R = 14ft/2 = 7ft

REplacing these values in the formula above we will get:

S = 3.14*(7ft)^2 + 3.14*7ft*(19.3ft) = 578.1 ft²

So the correct option is the second one.

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So the slope-intercept form of the equation of the line passing through the two given points is:

y = -5x + 26

If an equation is expressed in the form y=mx+b, where m denotes the slope and b the y-intercept, it is said to be linear.

The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept.

To find the slope of the line passing through the two given points (5,1) and (2,16), we can use the formula:

m = (y2 - y1)/(x2 - x1)

where (x1, y1) = (5,1) and (x2, y2) = (2,16)

m = (16 - 1)/(2 - 5) = -5

So the slope of the line is -5.

To find the y-intercept b, we can use the point-slope form of the equation of a line:

y - y1 = m(x - x1)

Using the point (5,1) and the slope m = -5, we get:

y - 1 = -5(x - 5)

Simplifying this equation, we get:

y = -5x + 26

So the slope-intercept form of the equation of the line passing through the two given points is:

y = -5x + 26

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Option D i.e. number line with closed circle on point negative 7 over 22 and arrow shaded to the left represents the solution to the inequality.

A comparison between two or more mathematical expressions is known as an inequality.

We are given an inequality as one seventh m is less than or equal to negative one twenty second.

So, from the above information, we get the inequality as

m/7 ≤ −1/22

Now, on solving the above inequality, we get the following

⇒m/7 ≤ −1/22

⇒m ≤ −7/22

Hence, Option D i.e. number line with closed circle on point negative 7 over 22 and arrow shaded to the left represents the solution to the inequality.

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The process to find the scale factor of a shape and the concept of sequence of transformations is explained below.

In order to find the scale factor of a shape, we need to compare the corresponding side lengths of the original shape to the corresponding side lengths of the new shape after a dilation.

The scale factor is the ratio of any two corresponding side lengths.

The sequence of transformations is defined as the order in which different transformations are applied to a shape.

For example, if we have a triangle and we first translate it, then rotate it, and then dilate it, the sequence of transformations would be : translate → rotate → dilate.

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The given question is incomplete, the complete question is

How do you find the scale factor of a shape? And What is the sequence of transformations?

Debra's two area baking plates are 17 square inches different in size.

The area surrounding a shape forms its perimeter. Area is a unit of measurement for interior space. Surface area is a measurement of a solid shape's exposed surface, whereas area is a two-dimensional measurement of the size of a flat surface (three-dimensional).

The square baking dish has the following surface area with an 8-inch side length:

Area of first square = (side length)² = 8² = 64 square inches

Similarly, the surface area of a square baking dish with 9-inch sides is:

Area of second square = (side length)² = 9² = 81 square inches

Difference in area = |Area of second square - Area of first square|

Difference in area = |81 - 64|

Difference in area = 17 square inches

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Answer: 372 units²

Step-by-step explanation:

     We can use this formula to solve for the surface area:

Surface Area = 2lw + 2lh + 2hw

     We will substitute the known values and solve:

Surface Area = 2lw + 2lh + 2hw

Surface Area = 2(12)(3) + 2(12)(10) + 2(10)(3)

Surface Area = 372 units²

the actual distance the Patel family will travel by the time they reach Clarksville is 72 miles.

What is the distance in math?

As its name implies, any distance formula outputs the distance (the length of the line segment). In coordinate geometry, there is a number of formulas for finding distances, such as the separation between two points, the separation between two parallel lines, the separation between two parallel planes, etc.

Smithville to Jonesville distance in map = 3(3/4) inches = (15/4) inches .

given that,

(1/2) inch = 4 miles .

1 inch = 8 miles .

Distance from Smithville to Jonesville = (15/4) * 8 = 30 miles .

similarly,

Jonesville to Clarksville distance in map = 5(1/4) = (21/4) inches .

Distance from Jonesville to Clarksville = (21/4) * 8 = 42 miles .

The actual distance the Patel family will travel by the time they reach Clarksville = 30 + 42 = 72 miles

Hence, the actual distance the Patel family will travel by the time they reach Clarksville is 72 miles.

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Shaun ordered 7 tacos and 8 burritos, for a total of 160160 calories from tacos and 320320 calories from burritos.

To solve this problem, we need to use a system of equations. Let's call the number of tacos Shaun ordered x and the number of burritos he ordered y. We can then write the following equations:

160x + 320y = 3680 (the total number of calories consumed)
x + y = 15 (the total number of items ordered)

We can rearrange the second equation to get y = 15 - x and then substitute that into the first equation to solve for x:
160x + 320(15 - x) = 3680
160x + 4800 - 320x = 3680
-160x = -1120
x = 7

Now that we know x, we can plug it back into the second equation to solve for y:
7 + y = 15
y = 8

So, Shaun ordered and ate 7 tacos and 8 burritos.

Tacos eaten (x) : 7
Burritos eaten (y) : 8

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Note that in the above sketch, ΔABC ≇ ΔDCA. That is they are NOT congruent. The proof is that ∠ADC which is supposed to be = ∠BAC are not equal. Thus, line AC which is supposed to be equal to Line BC are not equal.

When two triangles are congruent, it means they are exactly the same in terms of their shape and size, so all corresponding sides and angles are equal.

In ΔABC,

Line AB = 8ft
Line BC = 11ft

Line AC = 13.6ft

∠ABC = 90°

∠BAC = 53.97°
∠ACB = 36.03°


As you can see,

Where as, ∠ADC = 62° - given

∠BAC = 53.97°

Also

Where as:

AC (The Hypotenuse ΔABC which is also the Adjacent Side of ΔACD) = 13.6ft

BC (The adjacent side of ΔABC) = 11ft.

Note that in order to prove congruence, at least two angles and one side from both Triangles must be equal (Angle Angle Side Theorem). Or

Two sides and one angle from both Triangles must be equal (Side - Angle - Side).

Or All three angles (Angle - Angle - Angle);
Or All three Sides (Side - Side Side).

In this case, only one side from both Triangles AC is common to both Triangles.

On the basis of the above, therefore, ΔABC ≇ ΔDCA.


The calculations showing how we arrived at the missing sides and angles are given below:


ΔABC:

AC = √(AB² + BC²)
AC = √(8² + 11²)
AC = √(64 + 121)
AC = √(185)
AC = 13.6014705087

AC [tex]\approx[/tex] 13.60

∠ ACB = arcsin (AB/AC)
∠ ACB = arcsin (8/13.6014705087)
= arcsin(0.58817169767505)
∠ ACB = 0.6288 rad converted to degrees

∠ ACB  = 36.03°

Thus, since the sum of Angles in a Triangle is 180°
∠BAC = 180-36.03° -90°
∠BAC = 53.97°


ΔACD


y (AD) = AC / sin(β)
y (AD) =  13.6/ sin(62°)
y (AD)  = 13.6/0.88294759285

AD = 15.40°

CD = √(AD² - AC²)

CD = √(15.4029526893712 - 13.62)

CD = √52.290951550999

CD = 7.23

The outstanding angle in ΔACD is ∠CAD

Since the sum of Angles in a Triangle is 180°

∠CAD = 180 - 90 - 62
∠CAD = 28°

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Full Question:

Although part of your question is missing, you might be referring to this full question: See attached Image.

The interval giving the middle 68% of the lifetime of bulbs in hours is given as follows:

(1350, 1450).

The Empirical Rule states that, for a normally distributed random variable, the symmetric distribution of scores is presented as follows:

The middle 68% of scores is within one standard deviation of the mean, hence the bounds of the interval are given as follows:

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Answer:

First, we need to find the radius of the pipe by dividing the diameter by 2:

radius = 3ft / 2 = 1.5ft

Next, we can use the formula for the volume of a cylinder:

V = πr²h

where V is volume, π (pi) is approximately 3.14, r is the radius, and h is the height (or length) of the cylinder.

Plugging in the values we have:

V = 3.14 x (1.5ft)² x 17ft

V = 3.14 x 2.25ft² x 17ft

V = 3.14 x 38.25ft³

V ≈ 120ft³

Rounding to the nearest whole number and including the unit:

The volume of the pipe is about 120 cubic feet.

[tex]\bold{Work \ Shown:}[/tex]

[tex]d = 3 = diameter[/tex]

[tex]r = d\div2 = 3\div2 = 1.5 = radius[/tex]

[tex]h = 17 = height \ or\ length \ of \ the\ pipe[/tex]

[tex]\pi = 3.14\ approximately[/tex]

[tex]V = volume \ of \ the \ cylinder \ pipe[/tex]

[tex]V = \pi \times r^2 \times h[/tex]

[tex]V = 3.14 \times (1.5)^2\times 17[/tex]

[tex]V = 120.105[/tex]

[tex]V=120[/tex]

The cylindrical pipe's volume is about 120 cubic feet.

As a result, Jamie could rewrite the equation as (1+sin theta)/cos² theta by multiplying both the denominator and the numerator by (1+sin theta), and then replacing 1-sin² theta with cos² theta.

A ²is part of a whole. The number is expressed mathematically as a quotient, which divides the denominator and the numerator. Both are integers in a simple fraction. A complicated fraction contains a fraction in the numerator or denominator. The denominator of a correct fraction is greater than the numerator

Jamie can double the numerator and write 1/(1-sin theta) to ensure that the fraction's denominator is cos² theta. Neither 1-cos theta nor any other option, but the numerator by (1+sin theta):

1/(1-sin theta) × (1+sin theta)/(1+sin theta) = (1+sin theta)/(1-sin² theta)

We may replace cos² theta for 1-sin² theta using the identity sin² theta + cos² theta = 1:

(1+sin theta)/cos² theta

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(a) Revenue function: R = 13x
Cost function: C = 2x + 1540
(b) Break-even point: 2x + 1540 = 13x
x = 1540/11
x = 140 units

(c) Break-even point as a percent of capacity: 140/2500 x 100% = 5.6%
(d) Break-even point in sales dollars: 13 x 140 = $1820
(a) The revenue function is R(x) = 13x, where x is the number of units sold. The cost function is C(x) = 2x + 1540, where x is the number of units sold.
(b) To find the break-even point, we set R(x) = C(x):
13x = 2x + 1540
11x = 1540
x = 140
So the firm needs to sell 140 units to break even.
(c) The break-even point as a percent of capacity is 140/2500 = 0.056 or 5.6%.
(d) The break-even point in sales dollars is 140 units * $13/unit = $1820.

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The demand for nurses in the country can be expressed as the function D(x) = 2.8 + 0.044x. The year in which the demand for nurses in the country will reach 3.5 million is in 2027.

a. The demand for nurses in the country can be expressed as a function of the number of years after 2011, x, by using the equation D(x) = 2.8 + 0.044x. This equation represents the initial demand in 2011 (2.8 million) plus the expected increase in demand each year (0.044 million) multiplied by the number of years after 2011 (x).

b. To find the year in which the demand for nurses will reach 3.5 million, we can set D(x) equal to 3.5 and solve for x:

3.5 = 2.8 + 0.044x

0.7 = 0.044x

x = 0.7 / 0.044

x = 15.9

Since x represents the number of years after 2011, we can add 15.9 to 2011 to find the year in which the demand will reach 3.5 million:

2011 + 15.9 = 2026.9

Therefore, the demand for nurses in the country will reach 3.5 million during the year 2027.

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We can start by converting the mass of the bag of dog food from kilograms to grams:

8 kilograms = 8,000 grams

Then, we can divide the total mass of the bag of food by the daily amount given to the dog:

8,000 grams ÷ 100 grams/day = 80 days

Therefore, the bag of dog food will last for 80 days.

The measure of angle 1 would be equal to 59 degrees.

That quadrilateral in which opposite sides are parallel is called a parallelogram.

Thus, a parallelogram is always a quadrilateral but a quadrilateral can or cannot be a parallelogram.

We have been given the angles as 31 degrees and 90 degree

We know that the sum of angle of triangle is 180 then the angle 1 will be;

31 + 90 + m∠1 = 180

m∠1 = 180 - 90 - 31

m∠1 = 59

Hence, m∠1 = 59

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The volume of the cone is  1004.8 cm³.

Let's calculate the volume of the cone with radius 8 cm and the slant height is 17 cm. Therefore, let's find the volume of the cone as follows:

volume of a cone = 1 / 3 πr²h

where

Therefore,

h² = 17² - 8²

h = √289 - 64

h = √225

h = 15 cm

Therefore,

volume of a cone = 1 / 3 × 3.14 × 8² × 15

volume of a cone = 1 / 3 × 3.14 × 64 × 15

volume of a cone = 3014.4 / 3

volume of a cone = 1004.8 cm³

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According to the information we can infer that the volume of the prism would be 63cm³

To find the volume of the prism we must perform the following mathematical operation:

First of all we must find the cube root of each of the options.
[tex]\sqrt[3]{84} = 4,3795191398878892657\\\sqrt[3]{63} = 3,9790572078963918596\\\sqrt[3]{42} = 3,4760266448864497867\\\sqrt[3]{7} = 1,9129311827723891012[/tex]

Once we find these values we must check them with the formula for the volume of a pyramid. In this case, it would be the area of the base times the height. Then we must take into account that the cube root of these numbers would be the equivalent to the side length of the figures.

In the case of 63, the procedure would be as follows:

Then we can establish that this value corresponds to the length of the sides and the height of the pyramid because as a result it gives us the volume of the pyramid. So to find the volume of the prism we must do the following operation:

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Answer:

Area = 15.6

Perimeter = 20.6

Step-by-step explanation:

Both smaller triangles are right triangles

The left triangle is isosceles right triangle since 2 angles are 45o so AB is hypotenuse and the other 2 sides are equal

so c^2 = a^2 + b^2

c^2 = 2a^2

(3√2)^2 = 2a^2

2a^2 = 18

a^2 = 9

a = 3

For the right triangle the angle at B is 60o, hypotenuse is BC

so cos(60) = adjacent/hypotenuse

1/2 = 3/BC

=> BC = 6

sin(60) = opposite/hypotenuse

√3/2 = opp/6

opp = 3√3

so AC = 3 + 3√3 = 6√3

Area of triangle = A = 1/2bh = 1/2(6√3)(3) = 9√3 = 15.6

Perimeter = 3√2 + 6√3 + 6 = 20.6