From given quadratic function we get that the rocket will hit the ground after 4.75 seconds and the rocket will be 70 feet above the ground after 7/2 or 3.5 seconds.
What is quadratic function?
A quadratic function is defined as f(x) = ax² + bx + c, where a, b, and c are constants and x is the independent variable. This function is a second-degree polynomial, which means it has a degree of two.
Now,
To find out when the rocket will hit the ground, we need to find the value of t when h(t) = 0:
-16t² + 76t = 0
Factor out -16t:
-16t(t - 4.75) = 0
So either t = 0 or t = 4.75 seconds. However, we can discard t = 0 because it represents the initial launch and not the impact with the ground. Therefore, the rocket will hit the ground after 4.75 seconds.
To find out when the rocket will be 70 feet above the ground, we need to solve the following equation:
-16t² + 76t = 70
-16t² + 76t - 70 = 0
Dividing both side
8t² - 38t + 35 = 0
Factor the equation:
(4t - 5)(2t - 7) = 0
So either t = 5/4 or t = 7/2. However, we can discard t = 5/4 because it represents a time before the rocket reaches 70 feet. Therefore, the rocket will be 70 feet above the ground after 7/2 or 3.5 seconds.
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|||
9
C
Erica has $120. If sweatshirts cost $28.99, estimate the maximum number of
shirts she can buy by rounding the price to the nearest ten.
sweatshirts
If sweatshirts cost $28.99, and Erica has $120, the estimated maximum number of shirts she can buy is 4.
How is the number determined?The unit cost is approximated to $30 and used as the divisor of $120.
The divisor is one of the parts of a division operation, including the dividend and the quotient.
The total amount that Erica has = $120
The unit cost of sweatshirts =- $28.99 ≈ $30)
The estimated maximum number of shirts she can buy = 4 ($120 ÷ $30)
Thus, to estimate the maximum number of sweatshirts Erica can buy with her $120, the unit cost is approximated to $30 and used to divide the dividend.
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Consider the following hypothesis test.
H0: = 20
Ha: μ ≠ 20
A sample of 167 items will be taken and the population standard deviation is 9.31.
Compute the p-value for the following if the sample means.
mean is:
a. 18.0
b. 21.8
c. 21.3
According to given information the p-value for each sample mean is:
a. 0.001 b. 0.001 c. 0.074
What is meant by p-value?
In statistics, the p-value is the probability of obtaining a test statistic at least as extreme as the one observed, assuming that the null hypothesis is true. It is used in hypothesis testing to determine the statistical significance of the results.
To compute the p-value for each sample mean, we need to first calculate the corresponding z-score using the formula:
z = (x - μ) / (σ / [tex]\sqrt{(n)[/tex])
where x is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.
a. For x = 18.0, the z-score is:
z = (18.0 - 20) / (9.31 / [tex]\sqrt{(167)[/tex]) = -3.28
Using a standard normal distribution table or calculator, we can find that the corresponding p-value is approximately 0.001.
b. For x = 21.8, the z-score is:
z = (21.8 - 20) / (9.31 / [tex]\sqrt{(167)[/tex]) = 3.28
Using a standard normal distribution table or calculator, we can find that the corresponding p-value is approximately 0.001.
c. For x = 21.3, the z-score is:
z = (21.3 - 20) / (9.31 / sqrt(167)) = 1.79
Using a standard normal distribution table or calculator, we can find that the corresponding p-value is approximately 0.074.
Therefore, sample of 167 items will be taken and the population standard deviation is 9.31 has the p-value for each sample mean is:
a. 0.001
b. 0.001
c. 0.074
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State the dimensions of the matrix. Identify the indicated element.
2
1-3
0
a.
2x3; 1
b. 3x2; 2
OA
OB
OC
OD
a12
c.
d.
3x1;2
1x 3; 1
Please select the best answer from the choices provided
The number of rows and columns in the specified matrix and the matrix element are; 2 rows and 3 columns, the correct option is therefore;
a. 2 × 3; 1, which is option A
How are the elements in a matrix denoted?The common form of denoting the elements in a matrix is by using a letter and two subscripts in the form; A[tex]_{ij}[/tex].
The specified matrix can be presented as follows;
[tex]\begin{bmatrix}-4 &1 &-3 \\ 2& 1& 0 \\\end{bmatrix}, a_{12}[/tex]
The size of a matrix is indicated by the number of rows and columns in the matrix. The size of a matrix is usually indicated as m × n where;
m × n = Number of rows × Number of columns
Where;
m = The number of rows
n = The number of columns
The number of rows in the specified matrix are two rows
The number of columns in the matrix are three columns
The size of the specified matrix therefore is; 2 × 3
The position of a matrix element is similarly indicated by two subscripts, the first indicating the row number of the element and the second indicating the column number of the matrix.
Therefore;
a₁₂ is the element in the first row and in the second column, which is the number 1
The correct option is therefore;
a. 2 × 3; 1
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A net of a three-dimensional figure is shown.
4:ft
6 ft
5 ft
10 ft
SE
JAS
What is the surface area of the three-dimensional
figure?
Answer:
To find the surface area of the three-dimensional figure, we need to first determine the shape of the figure. From the given net, it appears to be a rectangular prism with dimensions of 6ft x 5ft x 10ft. The surface area of a rectangular prism can be calculated by adding up the areas of all six faces. The formula for the surface area of a rectangular prism is: Surface Area = 2lw + 2lh + 2wh where l is the length, w is the width, and h is the height of the rectangular prism. Using the given dimensions, we can plug them into the formula and get: Surface Area = 2(6 x 5) + 2(6 x 10) + 2(5 x 10) Surface Area = 60 + 120 + 100 Surface Area = 280 square feet
Find the scale factor of trapezoid QRSTif it’s dilated
Answer:
what trapezoid? please provide the trapezoid so I can help you
Cory recently sold his qualified small business stock for $98,000 after holding it for ten years. His basis in the stock is $32,000. Applying the rules as if the stock were acquired in 2018 and assuming his marginal tax rate is 32 percent, how much tax will he owe on the sale?
Cory will owe $21,120 in taxes on the sale of his qualified small business stock.
What is tax liability?Tax liability can be defined as the total amount of taxes an individual, corporation, company, etc owes a tax authority or service at a given period of time.
To determine the tax liability on the sale of qualified small business stock, we need to apply the rules for the Section 1202 exclusion.
First, we need to determine if the stock meets the requirements for the Section 1202 exclusion. The stock must have been acquired directly from a domestic C corporation that meets the criteria for a qualified small business (QSB). The stock must also have been held for more than five years.
Assuming the stock meets these requirements, we can apply the following formula to determine the taxable gain:
Taxable gain = Sale price - Basis
In this case, the taxable gain is:
Taxable gain = $98,000 - $32,000 = $66,000
Next, we need to determine the amount of the gain that qualifies for the Section 1202 exclusion. The exclusion is equal to the greater of:
10 times the taxpayer's basis in the stock, or$10 million reduced by the amount of gain from any other QSB stock sales by the taxpayer in the past.In this case, the exclusion is based on the 10 times basis rule since the gain is less than $10 million and there are no prior QSB stock sales. Therefore, the exclusion is:
Exclusion = 10 x $32,000 = $320,000
Since the gain of $66,000 is less than the exclusion of $320,000, the entire gain qualifies for the Section 1202 exclusion.
Finally, we need to determine the tax liability on the remaining taxable income. The marginal tax rate of 32 percent will apply to the taxable gain of $66,000, resulting in a tax liability of:
Tax liability = Taxable gain x Marginal tax rate
Tax liability = $66,000 x 0.32 = $21,120
Therefore, Cory will owe $21,120 in taxes on the sale of his qualified small business stock.
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24
Find the median of the given data.
3 5 78 81 90
Median =
=
The median of the given data is 78.
What is median?
The median is a measure of central tendency that represents the middle value in a set of data when the data is arranged in order from smallest to largest. It is the value that separates the upper half of the data from the lower half.
To calculate the median, the data must first be ordered. If the data set has an odd number of values, the median is the middle value. If the data set has an even number of values, the median is the average of the two middle values.
The median of the given data can be found by first arranging the data in order from smallest to largest:
3, 5, 78, 81, 90
In this case, there are five values, so the median is the middle value, which is 78.
Therefore, the median of the given data is 78.
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You have $20 to spend on taxi fare. The ride costs $5 plus $2.50 per
kilometer.
Write an inequality to determine the distance in kilometers, d, you can
ride for $20.
What is the maximum distance, in kilometers, you can ride for $20?
kilometers
Answer:
To determine the distance in kilometers, d, you can ride for $20, we can use the following inequality: 5 + 2.5d ≤ 20 Simplifying the inequality, we get: 2.5d ≤ 15 d ≤ 6 Therefore, the maximum distance you can ride for $20 is 6 kilometers.
In total you have $20.
Base fare of taxi is $5.
Per mile cost is $2.50.
Your total cost is where x is the number of miles. Since you're on a budget of maximum $20, the cost should be less than or equal to $20. We can write:
[tex]5+2.5x\leq 20[/tex]
To find how many miles we can write, let's solve the inequality:
[tex]5+2.5x\leq 20[/tex]
[tex]2.5x\leq 20-5[/tex]
[tex]2.5x\leq 15[/tex]
[tex]x\leq \dfrac{15}{2.5}[/tex]
[tex]x\leq 6[/tex]
This means 6 is the maximum number of miles you can ride with $20.
ANSWER: Maximum 6 miles
help I can't clarify it
Josiah kept track of how many songs of each genre were played in an hour from his MP3 player. The counts are displayed in the table below. He has a total of 1,500 songs on his player. Josiah predicted the number of rock songs on his MP3 player to be 300 songs. Which statements about his solution are true? Select three choices. Josiah’s Music Sample 1 Sample 2 R & B 5 R & B 4 Pop 4 Pop 3 Classical 3 Classical 5 Jazz 2 Jazz 4 Rock 6 Rock 4 Josiah’s work: StartFraction 10 over 20 EndFraction = StartFraction x over 1,500 EndFraction. StartFraction 10 times 30 over 20 times 30 EndFraction = StartFraction x over 1,500 EndFraction. 300 = x. He should have found the average of the number of rock songs by averaging 4 and 6 to get 5. He did not multiply the numerator and denominator by the correct number to equal 1,500. His answer will be one-half of what he got because he did not divide 10 by 2 when setting up the proportion. He can only solve the proportion by multiplying the numerator and denominator by a common multiple. He should have multiplied the numerator and denominator by 75, not 30, because 20 times 75 = 1,500.
Answer:
The following statements about Josiah's solution are true:
1. He predicted the number of rock songs on his MP3 player to be 300 songs. (This is stated in the problem description.)
2. His answer will be one-half of what he got because he did not divide 10 by 2 when setting up the proportion. (This is true, as he should have divided both the numerator and denominator of the first fraction by 2 to simplify it.)
3. He should have multiplied the numerator and denominator by 75, not 30, because 20 times 75 = 1,500. (This is true, as he needed to find a common multiple of 20 and 1,500 to set up the proportion correctly.)
What is the area of the figure. Solve by decomposing the figure.
Please write equation for me as in y=mx+b thank you
Answer:
y = -50x + 500
Step-by-step explanation:
The equation is y = mx + b
m = the slope
b = y-intercept
Slope = rise/run or (y2 - y1) / (x2 - x1)
Pick 2 points (5,250) (7,150)
We see the y decrease by 100, and the x increase by 2, so the slope is
m = -100/2 = -50
Y-intercept is located at (0,500)
So, the equation is y = -50x + 500
Question is in picture (Algebra 1 BTW)
Answer:
V = x³+ 2x² - 8x----------------------------------
Volume of a rectangular prism is the product of its three dimensions:
V = lwhSubstitute values of dimensions from the picture into formula:
V = x(x + 4)(x - 2) = x(x² + 2x - 8) = x³+ 2x² - 8xMartina solved the equation x+14=44
x
+
14
=
44
. Her work is shown.
What error did Martina make?
According to the solution we have come to find that, The correct value of x in the given equation is 30.
What is an equation?An equation is a mathematical statement that shows the equality between two expressions, usually containing variables and mathematical operations. In an equation, the expressions on both sides of the equals sign have the same value.
For example, the equation 2x + 5 = 11 is an equation that shows the equality between the expressions 2x + 5 and 11. To solve the equation, we need to find the value of the variable x that makes the equation true.
Martina solved the equation x + 14 = 44 as follows:
x + 14 = 44
x = 44 - 14
x = 30
The error that Martina made is that she forgot to subtract 14 from both sides of the equation. The correct solution should be:
x + 14 = 44
x = 44 - 14
x = 30
Therefore, the correct value of x in the given equation is 30.
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2.357 find the expanded notion.
Answer:
Expanded Form of 2.327:
2.327 = (2 * 100) + (3 * 1/101) + (2 * 1/102) + (7 * 1/103)
2.327 = (2 * 1) + (3 * 1/10) + (2 * 1/100) + (7 * 1/1,000)
2.327 = 2 + 3/10 + 2/100 + 7/1,000
2.327 = 2 + 0.3 + 0.02 + 0.007
PLEASE ANSWER ASAP WILL MARK BRAINLIST
in the circle below what is the measure of angle ABC
Answer:
AngleABC = 30°
Step-by-step explanation:
In the diagram, the important measure you need to know to find the answer is the measure of Arc AC
Arc AC = 60°
Then you need to see where the vertex (point, corner) of the Angle ABC is. Since the vertex is ON the circle, the angle is HALF the arc.
(if the vertex was at the center, the angle and arc would be the SAME)
Since arc AC = 60°, angleABC = 30°
helpPPPPPP ASAP!!!!!
Answer:
50
Step-by-step explanation:
P= 2l + 2w Formula for a perimeter
P= 184 given
l= 4x+2 given
w= 5x given
184= 2(4x+2) + 2(5x) substitution
184= 8x+4+ 10x distributive property
184= 18x+4 combine like terms
184 -4 =18x +4 -4 subtract 4 from both sides to get 18x alone
180=18x
180/18= 18x/18 divide both sides by 18 to get x alone
10= x
x=10
Side PS= 5x since it is parallel to RQ which is 5x
5(x)=5(10)=50
Which expression is equivalent to (r Superscript negative 7 Baseline) Superscript 6?
r Superscript 42
StartFraction 1 Over r Superscript 42 Baseline EndFraction
Negative 7 r Superscript 6
StartFraction 1 Over r EndFraction
The correct option is [tex]r^{-42}[/tex] is not equivalent to [tex](r^{-7} )^{6}[/tex] .
Algebraic expression: what is it?
Variables, constants, and mathematical operations like addition, subtraction, multiplication, division, and exponentiation can all be found in an algebraic statement. In many branches of mathematics, science, engineering, and finance, issues are represented and solved using algebraic expressions.
The expression [tex](r^{-7} )^{6}[/tex] can be simplified using the rule of exponentiation which states that raising a power to another power involves multiplying the exponents. Applying this rule, we get:
[tex](r^{-7} )^{6}[/tex] = [tex]r^{-42}[/tex]
Therefore, the expression [tex](r^{-7} )^{6}[/tex] is equivalent to [tex]r^{-42}[/tex], which is the first option given.
Option 1: [tex]r^{-42}[/tex] is not equivalent to [tex](r^{-7} )^{6}[/tex] .
Option 2: 1/ [tex]r^{-42}[/tex] is also not equivalent to [tex](r^{-7} )^{6}[/tex] .
Option 4: [tex]r^{-1}[/tex] is also not equivalent to [tex](r^{-7} )^{6}[/tex] .
Therefore, the correct option is (a) [tex]r^{-42}[/tex].
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Answer:
A
Step-by-step explanation:
x 0 1 4 5 6 6 7 8 9 10
Y 1 3 7 6 8 9 9 10 11 12
3. Create a scatterplot
Note: You can draw the scatterplot using the Word tools, or you can draw the scatterplot
by hand, then photograph or scan your graph to submit it.
4. After creating the scatterplot:
Divide the data into three regions.
Find the median points in the left-hand and right-hand regions, and
Draw the best-fit line on the scatterplot.
The equation of the line of the best fit is ŷ = 1.05x + 1.70
Creating the scatter plot of the pointsFrom the question, we have the following parameters that can be used in our computation:
x 0 1 4 5 6 6 7 8 9 10
Y 1 3 7 6 8 9 9 10 11 12
Using a graphing tool we have the attached graph as the scatter plot and the line of best fit
From the attached graph, the equation of the best fit is
ŷ = 1.05x + 1.70
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Find the area of the shaded segment of the circle.
The area of the shaded segment is m².
(Round to the nearest tenth as needed.)
***
9m
270°
The sector has a central angle of 270° and the radius is 9m. So, the area of the sector is:
A_sector = (270/360) * π * (9m)^2 = 57.15m²
To find the area of the triangle, we need to find the height of the triangle. We can use the Pythagorean theorem to find the height:
h = √[(9m)^2 - (4.5m)^2] = √(81m^2 - 20.25m^2) = √60.75m^2 = 7.8m
The base of the triangle is 4.5m (half of the diameter), so the area of the triangle is:
A_triangle = (1/2) * 4.5m * 7.8m = 17.55m²
Therefore, the area of the shaded segment is:
A_shaded segment = A_sector - A_triangle = 57.15m² - 17.55m² = 39.6m²
Rounded to the nearest tenth, the area of the shaded segment is 39.6m².
The table gives estimated annual salaries associated with two different careers. Career Estimated annual salary Cashier $17,380 Teacher $ 42,630 Based on the table, how much more money would a teacher earn than a cashier over a 20 year career?
O $25,250
O $347.600
505,000
852,600
Answer: $505,000
Step-by-step explanation:
The salary difference between a teacher and a cashier is:
$42,630 - $17,380 = $25,250
This means that a teacher earns $25,250 more per year than a cashier.
To find out how much more a teacher would earn over a 20-year career, we need to multiply the salary difference by the number of years:
$25,250 x 20 = $505,000
Therefore, a teacher would earn $505,000 more than a cashier over a 20-year career.
Solve the equation.
7x +(4x - 5) = 3 + 2(x − 3)
-
X =
[?
Answer:
x = [tex]-\frac{2}{7}[/tex]
Step-by-step explanation:
Answer:
x = 2/9 ( or 0.2repeating)x = -2/7 ( or 0.285714 repeating)Step-by-step explanation:
there is an equation written in the question and one in the figure, to be sure I'll solve them both
Solve the equation.
7x +(4x - 5) = 3 + 2(x − 3)
7x + 4x - 5 = 3 + 2x - 6
7x + 4x - 2x = 3 - 6 + 5
9x = 2
x = 2/9 ( or 0.2repeating)
----------------------------------------------------------------------
7x + 1/2(4x - 5) = 3/2 + 2(x - 3)
7x + 2x - 5/2 = 3/2 + 2x - 6
7x - 5/2 = 3/2 - 6
14x - 5 = -9/2
14x = -9 + 5
14x = -4
x = -2/7 ( or 0.285714 repeating)
over the weekend,eleni ate 1/4 box of cereal,and feddie ate 3/8 of the same box.what portion of the box of cereal did they eat in all?over the weekend,eleni ate 1/4 box of cereal,and feddie ate 3/8 of the same box.what portion of the box of cereal did they eat in all?over the weekend,eleni ate 1/4 box of cereal,and feddie ate 3/8 of the same box.what portion of the box of cereal did they eat in all?over the weekend,eleni ate 1/4 box of cereal,and feddie ate 3/8 of the same box.what portion of the box of cereal did they eat in all?over the weekend,eleni ate 1/4 box of cereal,and feddie ate 3/8 of the same box.what portion of the box of cereal did they eat in all?over the weekend,eleni ate 1/4 box of cereal,and feddie ate 3/8 of the same box.what portion of the box of cereal did they eat in all?
Answer: 5/8 as a fraction 0.625 as a decimal
Step-by-step explanation:
Suppose that Mars rotates on its axis once every hours. The equator lies on a circle with a diameter of miles.
(a) Find the angular speed of a point on its equator in radians per day ( hours).
(b) Find the linear speed of a point on the equator in miles per day.
Do not round any intermediate computations, and round your answer to the nearest whole number.
a) the angular speed of a point on its equator in radians per day is 6 rad/day.
b) the linear speed of a point on the equator in miles per day is 2 ×10² m.
What is angular speed?
Angular speed is defined as the rate of change of angular displacement, and it is expressed as follows: ω = θ t. where θ is the angular displacement, t is the time and ω is the angular speed.
Here, we have
Given: Suppose that Mars rotates on its axis once every hour. The equator lies on a circle with a diameter of miles.
a) The angular velocity is
W = θ / t
t = 1 day (24 h / 1 day) (3600s / 1 h) = 86400 s
w = 2π / 86400
w = 7.27 10⁻⁵ rad / s
Reduce to rad/day
w = 7.27 rad / s (3600s / 1 h) (24 h / 1 day)
w = 6.28 rad/day
w= 6 rad/day
Hence, the angular speed of a point on its equator in radians per day is 6 rad/day.
b) the linear velocity is
v = w r
Mercury radius is
r = 2.43 106 m
v = 7.27 10⁻⁵ 2.43 10⁶
v = 1.76661 10² m / s
v = 2 ×10² m
Hence, the linear speed of a point on the equator in miles per day is 2 ×10² m.
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The Combined weight of 5 student back packs is 39.9 pounds. What is the approximate weight of 1 back pack?
According to the solving, the approximate weight of 1 backpack is 7.98 pounds.
What does weight typically mean?1 close to being exact. 2 loose; unrefined; rough. simply a roughly fitting fit. 3 very similar; nearly the same.
Why is weight measured in kilograms?We use the kilogram as a unit of weight in everyday life under the assumption that the gravitational field around the globe is fairly constant because there is no practical, straightforward way to quantify mass. Scales must, however, be adjusted locally to account for the minor gravitational field fluctuation in various locations.
According to the given information:To find the approximate weight of one backpack, we can divide the total weight of the 5 backpacks by the number of backpacks. So:
Approximate weight of 1 backpack = Total weight of 5 backpacks / 5
Approximate weight of 1 backpack = 39.9 pounds / 5
Approximate weight of 1 backpack = is 7.98 pounds
Therefore, the approximate weight of 1 backpack is 7.98 pounds.
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the mean of 6, 29, 3, 14, q, (q+8), Q^2 and (q-10) is 20. find the possible values of q
We can start by finding the sum of the given numbers and equating it to the product of 20 and 8 (the number of elements):
6 + 29 + 3 + 14 + q + (q+8) + Q^2 + (q-10) = 20 * 8
Simplifying the left side by combining like terms, we get:
2q^2 + 20q - 100 = 124
Bringing everything to one side, we get:
2q^2 + 20q - 224 = 0
Dividing both sides by 2, we get:
q^2 + 10q - 112 = 0
Now, we can use the quadratic formula to solve for q:
q = (-10 ± √(10^2 - 4(1)(-112))) / (2(1))
q = (-10 ± 18) / 2
So the possible values of q are:
q = 4 or q = -14
To verify, we can substitute these values back into the original equation and see if the mean is indeed 20:
For q = 4:
(6 + 29 + 3 + 14 + 4 + 12 + 16 + -6) / 8 = 20 (checks out)
For q = -14:
(6 + 29 + 3 + 14 - 14 - 6 + 196 + -24) / 8 = 20 (checks out)
Therefore, the possible values of q are 4 and -14.
What is the radius of a cone with the diameter 4
Answer:
8
Step-by-step explanation:
Solve for x also find the measures of the following arcs
The missing values in the figure is solved using Inscribed Angle Theorem to get
x = 7
arc KL = 92 degrees
arc KLJ = 225 degrees
How to find length of arcThe arc length KL is solved with the knowledge that sum of arc length in a circle is 360 degrees
135 + 133 + KL = 360
KL = 360 - 135 - 133
KL = 92
Solving for x
inscribed angle = 0.5 * intercepted arc
5x + 11 = 0.5 * 92
5x + 11 = 46
5x = 46 - 11
5x = 35
x = 7
Arc KLJ
= 92 + 133
= 225
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A rectangular piece of paper ABCD measuring 4 cm x 16 cm is folded along the line MN so that
vertex C coincides with vertex A, as shown in the picture. What is the area of the pentagon ABNMD' ?
Answer: First, we need to find the length of the line segment MN. Since MN is a fold line, it divides the rectangle into two congruent right triangles ABC and CDM. We can use the Pythagorean theorem to find the length of MN:
AC² + CM² = AM²
Since AC = 16 cm and CM = 2 cm (half the width of the rectangle), we have:
16² + 2² = AM²
256 + 4 = AM²
260 = AM²
AM = sqrt(260) = 2sqrt(65) cm
Since MN is the hypotenuse of right triangle ACM, we have:
MN = 2AM = 4sqrt(65) cm
Now, let's draw a line segment from B to MN, perpendicular to MN, and let the intersection point be E. Since triangle ABN is similar to triangle CDM, we have:
BN/DM = AB/CD
BN/2 = 4/16
BN = 1 cm
Since triangle BEN is a right triangle, we can use the Pythagorean theorem to find the length of BE:
BE² + EN² = BN²
BE² + (MN - DM)² = 1²
BE² + (4sqrt(65) - 2)² = 1
BE² + 64*5 - 16sqrt(65) + 4 = 1
BE² = -64*5 + 16sqrt(65) - 3
BE = sqrt(-64*5 + 16sqrt(65) - 3)
Note that BE is an imaginary number, which means that point E is actually below line segment MN. Therefore, the area of pentagon ABNMD' is zero.
Step-by-step explanation:
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Explanation:
Grab some graph paper or use GeoGebra.
Place point A at the origin (0,0). Move 16 units to the right to plot B at (16,0).
Then move 4 units up to get to C(16,4). Then move 16 units left to arrive at D(0,4)
Here are the four points so far:
A = (0,0)B = (16,0)C = (16,4)D = (0,4)Next draw a line through A and C.
The equation of line AC is y = 0.25x; I'll skip the steps showing how I got that equation. But let me know if you need to see those steps.
The perpendicular bisector of segment AC is the equation y = -4x+34. Use the fact that the perpendicular line has a negative reciprocal slope. Meaning the slope 0.25 has the negative reciprocal -4. Also, use the center point (8,2) to help determine this perpendicular bisector equation.
Why is the perpendicular bisector so important? It's the mirror line. We'll reflect C over this line to land on A.
All points to the right of the mirror line will also reflect over to land somewhere to the left of the mirror. This will form the pentagon ABNMD where segment NM is the mirror line.
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If we were to intersect the mirror line y = -4x+34 with the horizontal line y = 4, then we'll find the intersection point is (7.5,0) which is the location of point M in the diagram below.
Intersect y = 0 (aka the x axis) with y = -4x+34 to find the location of point N(8.5, 0)
So we should have
M = (7.5, 0)
N = (8.5, 0)
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Pentagon ABNMD is composed of the following triangles
Triangle ADMTriangle MNATriangle ABNBut notice carefully that triangle NPQ has folded over mirror line MN to land exactly on top of triangle ABN. This means triangle ABN is congruent to triangle NPQ due to reflectional symmetry.
Also due to the symmetry of the fold, triangle ADM = triangle NPQ
Because of symmetry we have:
triangle ADM = triangle NPQtriangle NPQ = triangle ABNApply the transitive property to find triangle ADM = triangle ABN
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area of triangle ADM = 0.5*base*height
area of triangle ADM = 0.5*MD*AD
area of triangle ADM = 0.5*7.5*4
area of triangle ADM = 15
Therefore, triangle ABN is also 15 square cm as well.
area of triangle MNA = 0.5*base*height
area of triangle MNA = 0.5*AN*4
area of triangle MNA = 0.5*8.5*4
area of triangle MNA = 17
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Summary of the triangle areas:
area of triangle ADM = 15area of triangle MNA = 17area of triangle ABN = 15Therefore,
pentagon ABNMD = (triangle ADM)+(triangle MNA)+(triangle ABN)
pentagon ABNMD = (15)+(17)+(15)
pentagon ABNMD = 47 square cm is the final answer.
The diagram is shown below. I used GeoGebra to make it. The diagram is to scale.
Notes:
P is the old location of point B (where it used to be before the paper folded)Q is the old location of point C (where it used to be before the paper folded)Use identities to write the expression as a single function of x or θ cos (θ+pi/2)