Find the area of this composite figure: *find the area of each figure, then add those areas together

The exponential function that relates the mass of the substance (y) to the number of years since 2004 (t) is: y = 450 * (0.923) ^ t

In this function, 0.923 is the decimal equivalent of 1 - 0.077, which represents the decay factor of 7.7%. The exponent t represents the number of years since 2004.

So, for example, if we want to know the mass of the substance in 2010 (6 years after 2004), we can plug t = 6 into the function:

y = 450 * (0.923) ^ 6 = 298.8 milligrams

Therefore, the mass of the substance in 2010 would be approximately 298.8 milligrams.

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Note that the exponential function that shows the relationship between the mass y of the substance and the number of years t since 2004 is:

y = 450 * 0.923^t

The general formula for exponential decay is:

y = a * (1 - r)^t

where:

y = final amount

a = initial amount

r = decay rate

t = time elapsed

In this problem, the initial amount is 450 milligrams, and the decay rate is 7.7% per year, or 0.077 as a decimal. We can use the formula above to write an exponential function for the relationship between y and t:

y = 450 * (1 - 0.077)^t

Simplifying this expression, we get:

y = 450 * 0.923^t

Therefore, the exponential function that shows the relationship between the mass y of the substance and the number of years t since 2004 is:

y = 450 * 0.923^t

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Full Question:

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

In 2004, a sample of a radioactive substance had a mass of 450 milligrams. Since then, the sample has decayed by 7.7% each year. Let t be the number of years since 2004. Let y be the mass of the substance in milligrams. Write an exponential function showing the relationship between y and t.

If the height of the triangular roof is half its base, then the height of the roof is:

h = (1/2) * 32 = 16 feet

The area of a triangle is given by the formula:

A = (1/2) * base * height

Plugging in the values we have:

A = (1/2) * 32 * 16

A = 256 square feet

Therefore, the area of the triangular roof is 256 square feet.

The given cylindrical wind chime of dimensions 16 inches long will not fit inside the box in the shape of rectangular-prism having dimensions 10 inches x 8 inches x 6 inches.

1)shorter than

2) shorter than

3)will not fit.

What is a rectangular-prism?

Rectangular prism also called as cuboid, is a closed three dimensional solid having six faces. This prism has eight vertices or corners and 12 edges. The perimeter of this shape is the total length of 12 edges. The area of this shape is the sum of area of for or six surfaces depending on the given situation.

Given dimensions of cuboid/rectangular prism:

Length=10 inches, height= 6 inches & Width=8 inches

Dimensions of cylindrical wind chime: Length of cylinder = 16 inches

Case I: comparision of diagonal of base of the box & length of wind chime.

Diagonal of base can be found using pythagorean theorem:

Diagonal=[tex]\sqrt{L^{2} +W^{2} }[/tex]

              =[tex]\sqrt{(10)^{2} +(8)^{2} }[/tex]

              =[tex]\sqrt{100+64\\}[/tex]

               =[tex]\sqrt{164}[/tex]

               =12.8 inches ≠ 16 inches

Diagonal of rectangular prism base shorter than length of wind chime.

Case II:comparision of interior diagonal of the box & length of wind chime.

Interior diagonal of box=[tex]\sqrt{L^{2} +W^{2} +H^{2} }[/tex]

                                       =[tex]\sqrt{(10)^{2} + (8)^{2} +(6)^{2} }[/tex]

                                       =[tex]\sqrt{100+64+36}[/tex]

                                        =[tex]\sqrt{200}[/tex]

                                        =14.14 inches ≠ 16 inches

Interior Diagonal of rectangular prism is shorter than length of wind chime.

As the wind chime is greater than both the diagonals of cuboid, the chime will not fit into the prism.

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Refer to the complete question attached below.

a) To graph y=2g(x)+3, we need to take the graph of y=g(x) and vertically stretch it by a factor of 2, then shift it upward by 3 units.

Start by identifying some key points on the graph of y=g(x). For example, if we take x=0, we see that y=g(0)=2, so the point (0,2) is on the graph. If we take x=1, we see that y=g(1)=4, so the point (1,4) is on the graph. If we take x=-1, we see that y=g(-1)=1, so the point (-1,1) is on the graph.

Now, to graph y=2g(x)+3, we take these key points and apply the transformation.

- Vertically stretch: Multiply the y-coordinates by 2. So (0,2) becomes (0,4), (1,4) becomes (1,8), and (-1,1) becomes (-1,2).
- Shift upward: Add 3 to each y-coordinate. So (0,4) becomes (0,7), (1,8) becomes (1,11), and (-1,2) becomes (-1,5).

These points give us enough information to sketch the graph of y=2g(x)+3. It will look like the original graph of y=g(x), but stretched vertically and shifted upward.

b) To graph y=-h(x-2), we need to take the graph of y=h(x) and reflect it across the y-axis, then shift it to the right by 2 units, and finally reflect it again across the x-axis.

Again, start by identifying some key points on the graph of y=h(x). For example, if we take x=0, we see that y=h(0)=1, so the point (0,1) is on the graph. If we take x=1, we see that y=h(1)=3, so the point (1,3) is on the graph. If we take x=-1, we see that y=h(-1)=2, so the point (-1,2) is on the graph.

Now, to graph y=-h(x-2), we take these key points and apply the transformation.

- Reflect across y-axis: Multiply the x-coordinates by -1. So (0,1) becomes (0,1), (1,3) becomes (-1,3), and (-1,2) becomes (1,2).
- Shift to the right: Add 2 to each x-coordinate. So (0,1) becomes (2,1), (-1,3) becomes (1,3), and (1,2) becomes (3,2).
- Reflect across x-axis: Multiply the y-coordinates by -1. So (2,1) becomes (2,-1), (1,3) becomes (1,-3), and (3,2) becomes (3,-2).

These points give us enough information to sketch the graph of y=-h(x-2). It will look like the original graph of y=h(x), but reflected across the y-axis, shifted to the right, and reflected across the x-axis.

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In the word problem , the length of wood Mr. DeWitt carved off is 3.6cm.

What is word problem?

Word problems are often described verbally as instances where a problem exists and one or more questions are posed, the solutions to which can be found by applying mathematical operations to the numerical information provided in the problem statement. Determining whether two provided statements are equal with respect to a collection of rewritings is known as a word problem in computational mathematics.

According to the given ,

Length of wood = 20.3 cm

The length of boat = 16.7 cm.

Now Length of wood he carved off is,

=> Length of wood - length of boat

=> 20.3 - 16.7

=> 3.6 cm.

Hence the length of wood Mr. DeWitt carved off is 3.6cm.

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The type of the function in the question is a linear function

The range of -∞ to +∞ means that the equation is valid for all real numbers.

This range alone provides enough information to determine whether the equation is linear, exponential, or quadratic.

Because only a linear function can have this range

A linear equation is of the form y = mx + b, where m and b are constants and x is the independent variable.

And it always have a range of -∞ to +∞ , otherwise stated

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Answer: In dilation, the parent function is scaled or stretched by a given value, and can be presented by the equation y = a*x^2 where "a" is the dilation factor and a ≠ 0.

When a > 1, the dilation stretches the graph vertically, making the parabola narrower and taller. When a < 1, the dilation compresses the graph vertically, making the parabola wider and shorter. When a is negative, the dilation reflects the graph across the x-axis.

Step-by-step explanation:

Based on the information, the ordinate to the origin is -3.

In mathematics, an "ordinate" refers to the vertical coordinate of a point on a graph, while the "origin" typically refers to the point (0, 0) where the x- and y-axes intersect. Therefore, the phrase "ordinate to the origin" means the vertical distance of a point from the origin, which is simply the value of its y-coordinate.

For example, if you have a point with coordinates (3, 5), the ordinate to the origin would be 5, because that is the distance from the point to the x-axis (which passes through the origin).

The first question is incomplete. Regarding the second question, the equation y = -6x - 3 is in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept (or the ordinate to the origin). Therefore, the slope is -6 and the y-intercept is -3. So the ordinate to the origin is -3.

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The required coordinate points are (3, 0) and (-4, 0)  

To find the exponential function from two points, you need to know the coordinates of two points on the graph of the function.

If you have access to the graph, you can try to estimate the coordinates of two points by using the grid lines on the axes.

Choose two points that lie on the curve and try to read their coordinates as accurately as possible from the gridlines.

Here we have an exponent graph

In the given picture,

The curve cuts the y-axis at 3 and the x-axis at - 4

Hence,

The required coordinate points are (3, 0) and (-4, 0)

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Answer:

x = 7, y = 5

Step-by-step explanation:

Given the 2 equations

x = 12 - y → (1)

2x + 3y = 29 → (2)

Substitute x = 12 - y into (2)

2(12 - y) + 3y = 29 ← distribute and simplify left side

24 - 2y + 3y = 29

24 + y = 29 ( subtract 24 from both sides )

y = 5

Substitute y = 5 into (1)

x = 12 - 5 = 7

Solution is x = 7, y = 5

Answer:

So the solutions for x and y are: (x, y) = (0, 1/5), (5, -39/5), (10, -79/5), (15, -119/5), ...

What are parallel lines?

Parallel lines are two lines in a plane that never intersect. This means that they maintain the same distance between each other at all points.

Subtracting the first equation from the second, we get:

-8x - 5y = -45 - (-1)

-8x - 5y = -44

Now we can see that the two equations are equivalent, which means that there are infinitely many solutions for x and y that satisfy both equations. Geometrically, the two equations represent two parallel lines in the x-y plane that never intersect, so there is no unique solution.

However, we can still find a solution for x and y in terms of one variable. Let's solve for y in terms of x:

-8x - 5y = -1

-5y = 8x - 1

y = (-8/5)x + 1/5

Now we can choose any value for x and plug it into this equation to get the corresponding value of y. For example, if we set x = 0, then y = 1/5. If we set x = 5, then y = -39/5.

So the solutions for x and y are:

(x, y) = (0, 1/5), (5, -39/5), (10, -79/5), (15, -119/5), ...

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The statements which are true are that the radius of the circle is 3 units, the center of the circle lies on the x-axis and the radius of this circle is the same as the radius of the circle whose equation is x² + y² = 9.

What is a circle?

A circle is formed by all points in a plane that are at a particular distance from the center. To put it another way, it is the curve that a moving point in a plane draws to maintain a constant distance from another point.

We are given a circle whose equation is [tex]x^{2}[/tex] + [tex]y^{2}[/tex] - 2x - 8 = 0.

We know that the general form of a circle is  [tex]x^{2}[/tex] + [tex]y^{2}[/tex] + 2gx + 2fy + C = 0, where (-g, -f) is the center and √[tex]g^{2}[/tex] + [tex]f^{2}[/tex] - C is the radius.

So, in the equation, g is -1 and f is 0.

So, the center is (1, 0) which represents that the center lies on the x - axis.

Now,

⇒ Radius = √[tex]g^{2}[/tex] + [tex]f^{2}[/tex] - C

⇒ Radius = √1 + 0 - (-8)

⇒ Radius = √1 + 8

⇒ Radius = √9

⇒ Radius = 3 units

So, the radius of the circle is 3 units.

Now, radius of circle x² + y² = 9 is

[tex]r^{2}[/tex] = 9

r = 3

So, the  radius of this circle is the same as the radius of the circle whose equation is x² + y² = 9.

Hence, the required solution has been obtained.

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The complete question has been attached below.

Answer:

To determine the number of different ways the leader can appoint 3 members to clean up camp from a troop of 18 members, we can use the combination formula: n C r = n! / r!(n - r)! where n is the total number of members in the troop (18), and r is the number of members to be chosen (3). Plugging in the values, we get: 18 C 3 = 18! / 3!(18 - 3)! = (18 x 17 x 16) / (3 x 2 x 1) = 816 Therefore, there are 816 different ways the leader can appoint 3 members to clean up camp from a troop of 18 members.

Step-by-step explanation:

pi ( x^2 - 36 )

pi ( x-6)(x+6)     Done.

The SUMIF function in excel combines a condition and a sum of the values which meet the stated condition.  The SUMIF statement required on Cell A17 would be =SUMIF(C1:C50, '>=40')

How to write functions in Excel?

In Microsoft Excel, formulas are usually used to generate results or values for a cell.

Now, there are different ways that a formula can be written but then the formula must start with an equal sign. The formula to be used for the cells given has the function "SUMIF" and this is used to provide a sum of numeric values based on a condition.

The SUMIF syntax goes thus ; SUMIF(row_range, condition)

Assume the total membership count is in cells A1 to A50.

Cells, where the values are greater than or equal to 40, should be summed.

Hence, the required formula would be =SUMIF(C1:C50, '>=40')

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The angles that have the given tangents, correct to 2 significant figures, are: 45°, 30° and 15°

To find the angles that have the given tangents, we can use the inverse tangent function (tan⁻¹) or the arctan function on a calculator. Remember that the arctan function gives an angle in radians, so we need to convert the result to degrees.

[tex]\tan \theta = 1.000\ \theta = \tan^{-1}(1.000)\\\\ \theta \simeq 45^{\circ}[/tex]

tan θ = 0.577

θ = tan⁻¹(0.577)

θ ≈ 30°

tan θ = 0.268

θ = tan⁻¹(0.268)

θ ≈ 15°

tan θ = 4

θ = tan⁻¹(4)

θ ≈ 75.96°

θ ≈ 76° (rounded to 3 significant figures)

tan θ = 2.747\\θ = tan⁻¹(2.747)

θ ≈ 70.53°

θ ≈ 71° (rounded to 3 significant figures)

tan θ = 3.732

θ = tan⁻¹(3.732)

θ ≈ 75.89°

θ ≈ 76° (rounded to 3 significant figures)

Therefore, the angles that have the given tangents, correct to 2 significant figures, are:

45°

30°

15°

And the angles that have the given tangents, correct to 3 significant figures, are:

76°

71°

76°

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By metric system it can be determined that Jasοn lives οn 0.075 hectare.

The metric system is a prοcess οf measuring things. The measurements are used wοrldwide in calculatiοns and research. Sοme examples οf measurements οf things using the metric system are given belοw:

Meter is the unit οf measuring distances and lengths.

Kilοgram οr Gram is used tο measure weight.

Jasοn lives οn 750m plοt .

The given prοblem can be sοlved by metric system.

By metric system

1 hectare =10000m

10000m = 1 hectare

1m= 1/10000 hectare

750m= 750/10000 hectare

Hence,

Jasοn lives οn 0.075 hectare.

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A sample size of at least 54,444 guests to ensure a 95% confidence level and a margin of error of 3% and a sample size of at least 217777 guests to ensure a 95% confidence level and a margin of error of ±1.5%.

A. To calculate the sample size needed for a 95% confidence level and a margin of error of 3%, we can use the following formula:

n = (Z^2 * p * q) / E^2

Where:

n = sample size

Z = Z-score (which corresponds to the level of confidence) = 1.96 (for a 95% confidence level)

p = proportion of the population that has the characteristic we are interested in (we don't know this value, so we use 0.5 for a conservative estimate)

q = 1 - p

E = margin of error

Plugging in the values, we get:

n = (1.96^2 * 0.5 * 0.5) / 0.03^2 = 54,444.44

Therefore, we need a sample size of at least 54,444 guests to ensure a 95% confidence level and a margin of error of 3%.

B. If Ouranos Resorts wants to reduce the margin of error to ±1.5%, the required sample size will increase. Using the same formula as above, we get:

n = (1.96^2 * 0.5 * 0.5) / 0.015^2 = 217777

Therefore, we need a sample size of at least 217777 guests to ensure a 95% confidence level and a margin of error of ±1.5%. As we can see, the required sample size has increased by a factor of 4, which means that the survey will be more expensive and time-consuming to conduct.

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Using the graph, the answers to the questions are

Using the graph f(1) is the value of y when x is zero.

This is solved by tracing point x = 1 to the graph then tracing the corresponding value of y

Doing this give f(1) to be equal to 2 and this is not negative

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The length of line EF is proved to be parallel to the length of line EF.

To prove that EF is parallel to BD, we need to show that the opposite interior angles are equal.

Let's denote the center of the circle as O, and the intersection point of BD and AE as X. Since ABCD is a square, we have:

From the above information, we can deduce that:

∠ABD = 45°, and

∠OBD = 90°.

Since line_AF = line_AE, we can also deduce that:

∠FAE = ∠FEA.

Now, let's consider the triangle AFE. We have:

∠AFE + ∠FAE + ∠FEA = 180° (sum of angles in a triangle)

∠AFE + 2∠FAE = 180° (substituting ∠FEA with ∠FAE)

∠AFE = 180° - 2∠FAE

Also, in triangle AXε, we have:

∠AXε + ∠AEX + ∠XAE = 180° (sum of angles in a triangle)

∠AXε + ∠AEX + ∠FAE = 180° (substituting ∠XAE with ∠FAE)

∠AXε + ∠FAE + ∠FAE = 180° (rearranging terms)

∠AXε + 2∠FAE = 180°

Now, let's consider the quadrilateral ABXE. We have:

∠ABX + ∠AXε + ∠AEB = 360° (sum of angles in a quadrilateral)

∠ABX + ∠AEX + ∠AEB = 360° (rearranging terms)

∠ABX + ∠FAE + ∠AEB = 360° (substituting ∠AEX with ∠FAE)

∠ABX + 2∠FAE = 360° (substituting ∠AEB with ∠FAE)

∠ABX = 360° - 2∠FAE

Finally, let's consider the triangle BDE. We have:

∠BDE + ∠BED + ∠EBD = 180° (sum of angles in a triangle)

∠BDE + ∠AEB + ∠EBD = 180° (substituting ∠BED with ∠AEB)

∠BDE + ∠FAE + ∠EBD = 180° (substituting ∠AEB with ∠FAE)

∠BDE + 2∠FAE = 180° (since ∠FAE = ∠FEA)

∠BDE = 180° - 2∠FAE

From the above equations, we can see that:

∠ABX = ∠BDE (since both are equal to 360° - 2∠FAE)

∠ABD = ∠EBD (since both are equal to 45°)

Therefore, by the angle-angle (AA) criterion for similarity, we have:

triangle ABD is similar to triangle EXD.

This implies that:

BD/AD = XD/ED (by the property of similar triangles)

Since AD = BD (since ABCD is a square), we have:

BD/BD = XD/ED

1 = XD/ED

XD = ED

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The equation in slope-intercept form y = x - 1

In general, to find the equation of a trend line in a scatter plot, you can use linear regression. Linear regression is a statistical method that finds the line of best fit through a set of data points. The equation of the trend line can then be written in the slope-intercept form, y = mx + b, where "m" is the slope of the line and "b" is the y-intercept.

The yellοw pοints are lοcated at the pοints (1, 0) and (8, 7)

Fοr a linear equatiοn that pases trοugh the pοints (x1, y1) and (x2, y2), the slοpe is:

s = (y₂ - y₁)/(x₂ - x₁)

Then, fοr οur equatiοn, the slοpe is

s = (7-0)/(8-1) = 1

Then οur linear equatiοn is

y = 1 × x + b

where b is the y-axis intercept.

Tο find b, we can use that when x = 1, y = 0

0 = 1 × 1 + b

b = -1

Then οur equatiοn is y = x - 1

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Complete question:

What is the equation of the trend line in the scatter plot?

Use the two yellow points to write the equation in slope-intercept form. Write any coefficients as integers, proper fractions, or improper fractions in simplest form.

The angle measures in the figure are 45°, 55° and 145° respectively.

Angles are measured in degrees (°) using a protractor, which is a measuring device used to calculate or draw angles in terms of degrees

From the figure,

7x + 5 + 35 = 180      (angle on a straight line)

7x = 180-35-5

7x = 180-40

7x = 140

Making x the subject of the relation by dividing both sides by 7

7x/7 = 140/7

x = 20

Now, 7(20) +5 = 140+5 = 145°

Also, 90 -35 = 55

Therefore the angles that are measures of angles are 45°, 55°,145°

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The equation Hats 2030=230.09+23x represents a linear relationship between the number of years since 2010 (x) and the projected number of hats sold in the year 2030.

The slope of the equation, 23, indicates that for each additional year since 2010, the projected number of hats sold in 2030 increases by 23. The y-intercept of the equation, 230.09, represents the projected number of hats sold in 2030 if x is equal to zero (i.e., in the year 2010). This equation can be used to forecast future hat sales based on past trends and can help businesses and organizations plan for inventory and production needs. However, it is important to note that unforeseen events, changes in consumer behavior, and other factors can impact actual hat sales and should be considered when making business decisions.

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The equation is 9x + 8y = 81.

The answer of the given question based on circle is we have shown that XY = WZ, since both line segments are equal to XM.

The Pythagorean theorem is fundamental concept in mathematics that relates to sides of right triangle. It states that square of the length of hypotenuse (the side opposite the right angle) is equal to sum of the squares of lengths of the other two sides (the legs).

To prove that XY = WZ, we will use the fact that the line segment connecting the center of a circle to the midpoint of a chord is perpendicular to the chord.

Let M be the midpoint of XY, and N be the midpoint of WZ. Since X = Z, we know that XM is parallel to ZN, and therefore MZNX is a parallelogram.

Let O be center of circle k(O). Then, by the perpendicularity property mentioned above, we know that OM is perpendicular to XY, and ON is perpendicular to WZ.

Since M and N are midpoints of XY and WZ respectively, we have:

XM = MY, and ZN = NW

Therefore, by the Pythagorean theorem, we have:

OX² = OM² + MX², and OZ² = ON² + NZ²

Adding these two equations gives:

OX² + OZ² = OM² + MX² + ON² + NZ²

Since MZNX is a parallelogram, we know that OM = ON, and MX = NZ. Substituting these values gives:

OX² + OZ² = 2OM² + 2MX²

But since OM is perpendicular to XY and ON is perpendicular to WZ, we have:

OM² + MX² = XM²/4, and ON² + NZ² = ZN²/4

Substituting these values gives:

OX² + OZ² = XM²/2 + ZN²/2

But since MZNX is a parallelogram, we know that XM = ZN. Therefore:

OX² + OZ² = XM²

Thus, we have shown that XY = WZ, since both line segments are equal to XM.

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Answer:

x = 4

Step-by-step explanation:

Use Pythagoras theorem

Answer:

To solve for Z in the equation X = Y + Z, we need to isolate Z on one side of the equation. We can do this by subtracting Y from both sides:

X - Y = Y + Z - Y

Simplifying the right-hand side, we get:

X - Y = Z

Therefore, the equation X = Y + Z solved for Z is:

Z = X - Y