The height of the thrown bottle becomes zero at the t = 1/2 that is half a second and at t = 3 seconds
How do you form an equation?An equation is formed by expressing the relationship between two or more mathematical expressions. It can be formed by using mathematical symbols, operations, and variables. Typically, equations are used to solve problems or find unknown quantities by setting up an equation that represents the problem and then manipulating the equation using algebraic operations until the solution is obtained. For example, to solve the equation 2x + 3 = 7, you would use algebraic operations to isolate the variable x on one side of the equation, giving you the solution x = 2.
To find when the height of the thrown bottle is equal to 0, we need to solve the given equation:
h = -16t² + 56t - 24 = 0
Takin -8 common out we get,
-8(2t² - 7t + 3) = 0
Now we can use the zero product property, which states that if the product of two factors is equal to zero, then at least one of the factors must be zero:
-8 = 0 or 2t² - 7t + 3 = 0
But -8 ≠ 0, so we only possible one is,
2t² - 7t + 3 = 0
(2t - 1)(t - 3) = 0
Using the zero product property, we get,
2t - 1 = 0 or t - 3 = 0
Solving for t we get,
t = 1/2 or t = 3
Therefore, the height of the thrown bottle is equal to 0 at t = 1/2 seconds and t = 3 seconds.
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The height of the thrown bottle becomes zero at the t = 1/2 that is half a second and at t = 3 seconds
How do you form an equation?An equation is formed by expressing the relationship between two or more mathematical expressions. It can be formed by using mathematical symbols, operations, and variables. Typically, equations are used to solve problems or find unknown quantities by setting up an equation that represents the problem and then manipulating the equation using algebraic operations until the solution is obtained. For example, to solve the equation 2x + 3 = 7, you would use algebraic operations to isolate the variable x on one side of the equation, giving you the solution x = 2.
To find when the height of the thrown bottle is equal to 0, we need to solve the given equation:
h = -16t² + 56t - 24 = 0
Takin -8 common out we get,
-8(2t² - 7t + 3) = 0
Now we can use the zero product property, which states that if the product of two factors is equal to zero, then at least one of the factors must be zero:
-8 = 0 or 2t² - 7t + 3 = 0
But -8 ≠ 0, so we only possible one is,
2t² - 7t + 3 = 0
(2t - 1)(t - 3) = 0
Using the zero product property, we get,
2t - 1 = 0 or t - 3 = 0
Solving for t we get,
t = 1/2 or t = 3
Therefore, the height of the thrown bottle is equal to 0 at t = 1/2 seconds and t = 3 seconds.
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what is the circumference i need a refresh of how to do it i kinda forgot sum1 give answer and explanation
Answer:
102
Step-by-step explanation:
34 is d so 34×3=102
pls correct me if im wrong
(4a-ts)(2a+2ts) what is the solution
Answer:
8a^2 + 6ats - 2t^2s^2
Step-by-step explanation:
(4a-ts)(2a+2ts)
4a(2a+2ts) -ts(2a+2ts)
8a^2 + 8ats - 2ts - 2t^2s^2
= 8a^2 + 6ats - 2t^2s^2
For a 20-year endowment insurance on (65) with face amount 1,000:
benefits are payable at the moment of death, or at the end of 20 years if the
insured survives.
− Mortality follows the Illustrative Life Table with = 0.06.
− Deaths are uniformly distributed between integral ages.
− Gross premiums are payable at the beginning of each year for 20 years, and are
determined by the equivalence principle.
− Commissions and taxes are 35% of the gross premium in the first year, 5% in the
other years, and are paid at the beginning of each year.
− Other expenses are 1 in all years, paid at the beginning of the year.
Calculate the gross premium for the endowment insurance
The gross premium for the endowment insurance is 73.
How to solveTo calculate the gross premium for the endowment insurance, we can use the equivalence principle, which states that the present value of the premiums must equal the present value of the benefits.
We can break down the problem into two parts: the present value of the death benefit and the present value of the endowment benefit.
First, let's calculate the present value of the death benefit. The death benefit is payable at the moment of death, and the probability of dying at age x is given by qx, where qx is the probability of surviving to age x.
Since deaths are uniformly distributed between integral ages, we can assume that deaths occur at the midpoint of the age interval. Therefore, the probability of dying at age x can be approximated by
.
Let's denote the present value of the death benefit as DB. Using the equivalence principle, we can write:
DB = A*(1+0.35) - E0 - C1 - 0.05A(1+0.05)
- 0.05A(1+0.05)
- ... - 0.05A(1+0.05)
where A is the annual premium, E0 is the expense paid at the beginning of the first year, and C1 is the commission and tax paid at the beginning of the first year.
The factor (1+0.35) accounts for the commission and tax on the first year premium.
Using the formula for the present value of a perpetuity, we can simplify the above equation to:
DB =
= 116.22
where q65/2 is the probability of dying at age 65. We can obtain q65/2 from the Illustrative Life Table, which gives
q65/2 = 0.112488.
Therefore, the gross premium for the endowment insurance is 73.
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I don’t understand how to do this ?
The value of the derivative of the composite function h(x) = f[g(x)] is equal to h'(3) = 2.
How to find the exact value of the derivative of a composite function
In this problem we find a function in the form of a composite function of the form h(x) = f[g(x)], whose derivative is defined by the chain rule:
h'(x) = f'(g) · g'(x)
If we know that g'(3) = 2 and f'(2) = 1, then the exact value of the derivative of the composite function is:
h'(3) = 2 · 1
h'(3) = 2
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Compound Inequalities
The manufacturers of Cheerios only ship boxes that weigh more than 13.98 and less than 14.02 oz.
a. Represent the weights of boxes that will ship in three forms.
b. Represent the weights of boxes that will not ship in three forms.
*In Symbolic/Inequality and number line*
An compound inequality that describes the acceptable weights for a 20 oz box is 19.25 ≤ x ≤ 20.75.
We have,
The absolute value of a number is defined as its magnitude irrespective of the sign of the number. To determine the absolute value of a real number, we consider only the number and remove the sign. It can only be a non-negative value.
We are Given that, cereal manufacturer has a tolerance of 0.75 oz for a box of cereal that is supposed to weigh 20 oz.
Let the weight of cereal box be; w
Now, the inequality is
20-0.75 ≤ x ≤ 20+0.75
Add 20 on both sides of an inequality, we get
19.25 ≤ x ≤ 20.75
Here, 19.25 ≤ x ≤ 20.75
Therefore, an absolute value inequality to find the range of acceptable weights is 19.25 ≤ x ≤ 20.75.
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There have been many cases where long-standing issues that divided societies have been resolved, such as the reunification of Germany and the end
of apartheid in South Africa. Both of these benefitted society as whole, but one of the short-term results was an increase in crime. Why is this the case?
O A. Both experienced racially motivated crime.
OB.
O C.
O D.
Both involved a sudden reorganization of society.
Both came after a massive reform in public education.
Both are the result of implementing social service programs.
The increase in crime was caused by D. Both are the result of implementing social service programs.
What led to the increase in crime ?Significant modifications made to a society's social and political structures can cause periods of instability and uncertainty, leading to a rise in crime rates. The daunting tasks of restructuring society that came with the abolishment of apartheid in South Africa and Germany's reunification represent examples of such changes.
When power structures, economic systems, and social norms abruptly shift, gaps in authority may arise, thereby presenting opportunities for criminal activity to escalate. Criminal elements can capitalize on these unexpected power vacuums.
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At Downtown Pizza, 4 of the last 6 pizzas sold had pepperoni. What is the experimental probability that the next pizza sold will have pepperoni?
The answer of the given question based on the experimental probability is the next pizza sold at Downtown Pizza will have pepperoni is 2/3 or approximately 0.67.
What is Experimental probability?Experimental probability is ratio of number of times event occurs to the total number of trials or observations conducted in experiment or real-life situation. It is based on actual observations or data and is also known as empirical probability.
Experimental probability is used when it is not possible or feasible to determine the theoretical probability of an event, or when the theoretical probability does not match the actual outcomes observed in a given situation. In such cases, experimental probability provides an estimate of the likelihood of an event occurring based on real-life data.
The experimental probability that the next pizza sold will have pepperoni is the ratio of the number of pepperoni pizzas sold to the total number of pizzas sold in the recent past.
According to the problem, 4 of the last 6 pizzas sold had pepperoni. So, the experimental probability of the next pizza sold having pepperoni is:
experimental probability = number of pepperoni pizzas sold / total number of pizzas sold
experimental probability = 4 / 6
experimental probability = 2 / 3
Therefore, the experimental probability that the next pizza sold at Downtown Pizza will have pepperoni is 2/3 or approximately 0.67.
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View photo below. This is timed.
Answer: ∠EFB, ∠DFE, and ∠CFB
Step-by-step explanation:
Starting Angle: ∠CFD
Since its in an X formation: ∠EFB, ∠DFE, and ∠CFB would all work.
Use the graph to answer the question.
graph of polygon ABCD with vertices at 0 comma 0, 5 comma 2, 5 comma negative 5, 0 comma negative 3
Determine the coordinates of polygon A′B′C′D′ if polygon ABCD is rotated 90° counterclockwise.
A′(0, 0), B′(−2, 5), C′(5, 5), D′(3, 0)
A′(0, 0), B′(−2, −5), C′(−5, 5), D′(−3, 0)
A′(0, 0), B′(−5, −2), C′(5, −5), D′(3, 0)
A′(0, 0), B′(−5, −2), C′(−5, −5), D′(0, 3)
the Correct option of coordinates of polygon A′B′C′D′ if polygon ABCD is rotated 90° counterclockwise is A
What is coordinate?A coordinate is a set of numbers that represents the position of a point in space. It is often used to describe the location of a point on a graph or a map
According to given informationTo determine the coordinates of polygon A′B′C′D′, we need to rotate each vertex of polygon ABCD 90° counterclockwise.
We can do this by using the following formulas for a 90° counterclockwise rotation of a point (x, y):
x' = -y
y' = x
Using these formulas, we can find the coordinates of each vertex of polygon A′B′C′D′ as follows:
A′(0, 0): Since (0, 0) is the origin, a 90° counterclockwise rotation will still result in (0, 0).
B′(-2, 5): To rotate the point (5, 2) 90° counterclockwise, we have x' = -y = -2 and y' = x = 5. So, B′ is (-2, 5).
C′(5, 5): To rotate the point (5, -5) 90° counterclockwise, we have x' = -y = 5 and y' = x = 5. So, C′ is (5, 5).
D′(3, 0): To rotate the point (0, -3) 90° counterclockwise, we have x' = -y = 0 and y' = x = 3. So, D′ is (3, 0).
Therefore, the coordinates of polygon A′B′C′D′ are A′(0, 0), B′(-2, 5), C′(5, 5), and D′(3, 0).
So, the answer is A) A′(0, 0), B′(−2, 5), C′(5, 5), D′(3, 0).
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The heights of 18 year old men are approximately normally distributed with mean 68 inches in standard deviation 3 inches what is the probability that an 18-year-old man do that random is greater than 74 inches tall
The probability that an 18-year-old man picked at random is taller than 74 inches is approximately 0.0228 or 2.28%.
We have,
We can use the standard normal distribution to solve this problem by standardizing the height value using the formula:
z = (x - μ) / σ
where:
x = the height value (in inches)
μ = the mean height (in inches)
σ = the standard deviation (in inches)
Substituting the given values, we get:
z = (74 - 68) / 3
z = 2.0
Using a standard normal table or calculator, we can find the probability that a random standard normal variable is greater than 2.0 to be approximately 0.0228.
Therefore,
The probability that an 18-year-old man picked at random is taller than 74 inches is approximately 0.0228 or 2.28%.
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If the image is rotated about the x-axis, which of the following images best represents the result?
A. Y
B. Z
C. X
D. W
The images best represents the result is image W.
We have, the image is rotated about the x-axis.
Now, rotating around x axis can give the other half of the image.
Also, rotating about x axis gives the mirror image of the figure.
So, the rotation leads to a circle.
and, In three dimensional we can say that Sphere.
Thus, the required image is W.
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A genetic experiment with peas resulted in one sample of offspring that consisted of 427 green peas and 154 yelow peas. a. Construct a 90% confidence interval to estimate of the percentage of yellow peas. b. Based on the confidence interval, do the results of the experiment appear to contradict the expectation that 25%, of the offspring peas would be yellow? a. Construct a 90% confidence interval. Express the percentages in decimal form.
The initial temperature of a liquid in a beaker is 19°C. The beaker is heated and its temperature increased by 71°C after half an hour. Then, the beaker is left and the temperature decreased by 2°C every minute for half an hour. What is the final temperature of the liquid after one hour?
Answer:
Step-by-step explanation:
temperature is heated half an hour
19°c+71°c=90°c
{since its increased by 71°c}
90°c-[2°c*30]=30°c
so the final temperature is 30°cWhat’s the equation of the line shown above ? A y=2x+3 B y=2x+1/3 C y=3x+2 D y=1/3x+2
It should be noted that to find the equation of a line, two pieces of information are necessary: the slope and coordinates of a point on the line.
How to find the equationAlso, to acquire this, one can leverage either the point-slope form or the slope-intercept form to compose it.
If both the slope (m) and coordinates of the point (x1, y1) on the line are known, then one can express the equation in its point-slope style as follows:
y - y1 = m(x - x1)
Or, if only the slope (m) and the y-intercept (b), which is the value of y where the line crosses the y-axis, are accessible, then the equation can be expressed via the slope-intercept form with: y = mx + b
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) Find the average gold medal winnings of those NOCs who have been
awarded fewer than ten bronze medals.
Count the number of those NOCs that have been awarded ten or more
silver medals.
This question requires the construction and interpretation of a pie chart based
on the data throughout this task.
a) Create a suitable and fully labelled pie chart to show the total medal
winnings of the Top 10 NOCs at the Beijing 2022 Winter Olympic games.
Interpret the pie chart produced in Question 8a). Discuss two probable
justifications for the behaviour of the data.
a. To create a pie chart to show the total medal winnings of the Top 10 NOCs at the Beijing 2022 Winter Olympic games:
Obtain the data for the medal winnings of the Top 10 NOCs
Find the average gold medal winnings of those NOCs who have beenawarded fewer than ten bronze medals.?Calculate the total medal winnings for each NOC (i.e., the sum of gold, silver, and bronze medals)
Create a pie chart in Excel or any other software by selecting the data and choosing the pie chart option
Add a title and labels to the chart to show the NOCs and their corresponding medal winnings
b. The interpretation of the pie chart produced in Question 8a) depends on the shape and distribution of the chart. Two probable justifications for the behavior of the data are:
Dominance of certain NOCs: The pie chart may show that certain NOCs have a significantly higher share of medal winnings compared to others. This may be due to the dominance of these NOCs in specific sports or events, or due to their superior training and preparation.
Balanced distribution of medals: The pie chart may show a relatively balanced distribution of medal winnings among the Top 10 NOCs. This may indicate a high level of competitiveness and equal opportunities for success among the NOCs.
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A ribbon is 29 centimeters long. How many millimeters long is the ribbon?
2.9 millimeters
2,900 millimeters
290 millimeters
58 millimeters
Answer: The 290 millimeters long is the ribbon.
Step-by-step explanation:
To convert centimeters to millimeters
1 cm =10 millimeters
therefore,
29cm= 290 millimeters
Therefore, the ribbon is 290 millimeters long.
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sketch the graphs y=|x|, y=|x|+2, y=|x|-1
The graph of y=|x| is a v-shaped graph that starts at the origin and continues infinitely in both the positive and negative x-directions. The graph of y=|x|+2 is the same v-shaped graph, but it has been shifted up two units so that it no longer intersects the x-axis. Finally, the graph of y=|x|-1 is a v-shaped graph that has been shifted down one unit so that it intersects the x-axis at (1,0) and (-1,0). All three graphs are symmetric with respect to the y-axis.
For each of the following, place a decimal point in the number to make the sentence reasonable. a. The basketball player is 1950 cm tall. b. A new piece of chalk is about 8100 cm long. c. The speed limit in town is 400 km/hr.
a. The basketball player is 19.50 cm tall. (Assuming you meant to write
195 cm instead of 1950 cm)
b. A new piece of chalk is about 81.00 cm long.
c. The speed limit in town is 40.0 km/hr. (Assuming you meant to write 40 km/hr instead of 400 km/hr).
What is the equivalent expression?Equivalent expressions are expressions that perform the same function despite their appearance. If two algebraic expressions are equivalent, they have the same value when we use the same variable value.
By placing a decimal point in the appropriate location, we can make the numbers more reasonable.
a. The basketball player cannot be 1950 cm tall, as that is almost 20 meters or over 65 feet.
The average height of a basketball player in the NBA is around 6'7'' (or about 2 meters).
Therefore, we can assume that the correct height of the basketball player is 195 cm, or about 6'5''.
b. A piece of chalk that is 8100 cm long is over 80 meters long, which is clearly too long for a piece of chalk.
Therefore, we can assume that the correct length of the chalk is 81.00 cm or about 32 inches.
c. A speed limit of 400 km/hr is unreasonably high, as it is equivalent to over 248 miles per hour.
The highest speed limits in the world are typically around 80-85 mph (130-140 km/hr), so 400 km/hr is clearly incorrect.
Therefore, we can assume that the correct speed limit is 40.0 km/hr, which is a more reasonable speed limit for a town or residential area.
Hence,
a. The basketball player is 19.50 cm tall. (Assuming you meant to write
195 cm instead of 1950 cm)
b. A new piece of chalk is about 81.00 cm long.
c. The speed limit in town is 40.0 km/hr. (Assuming you meant to write 40 km/hr instead of 400 km/hr).
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sine law vector method
Answer: Well that wouldn't work quite as well.
Step-by-step explanation:
Find the weighted mean. Round your answer to the nearest tenth.
The weighted mean of the deliveries each week can be found to be 5. 38 .
How to find the weighted mean ?The weighted mean is a type of mean that takes into account, the weight of the various items in question.
First, find the total frequency :
= 1 + 5 + 4 + 3
= 13
The weighted mean would then be:
= ∑ ( weight of delivery x number of deliveries )
= ( 1 / 13 x 2 ) + ( 5 / 13 x 4 ) + ( 4 / 13 x 6 ) + ( 3 / 13 x 8 )
= 0. 15 + 1. 54 + 1. 84 + 1. 85
= 5. 38
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Choose which sampling technique 1s used. (R) Random (STR) Stratined (CLS) Cluster
(CON) Convenience (SYS) Systematic
1. I ask everyone in my 5th period class who has more than one computer
at home in a study about all of my students for the year.
2. I collect data from every 15th student on my list of the entire school
population.
3. After using a random number table to generate two-digit numbers, I
decide on 10 people to choose from the population.
Answer:
1. CON (convenience sampling) - This sampling technique involves selecting participants who are readily available and willing to participate. In this case, the researcher is only selecting from their 5th period class, which is not a representative sample of the entire student population.
2. SYS (systematic sampling) - This sampling technique involves selecting every nth element from a population. In this case, the researcher is selecting every 15th student on their list of the entire school population.
3. R (random sampling) - This sampling technique involves selecting participants at random from the population. In this case, the researcher is using a random number table to generate two-digit numbers and selecting 10 people from the population.
Hope this helps!
Answer:
1. CON (convenience sampling) - This sampling technique involves selecting participants who are readily available and willing to participate. In this case, the researcher is only selecting from their 5th period class, which is not a representative sample of the entire student population.
2. SYS (systematic sampling) - This sampling technique involves selecting every nth element from a population. In this case, the researcher is selecting every 15th student on their list of the entire school population.
3. R (random sampling) - This sampling technique involves selecting participants at random from the population. In this case, the researcher is using a random number table to generate two-digit numbers and selecting 10 people from the population.
Hope this helps!
A room is 18 feet long and 12 feet wide. At $13.25 per square yard, what will be the cost for wall-to-wall
carpeting?
The cost for wall-to-wall carpeting is equal to $ 318.
How to compute the cost for wall-to-wall carpeting
In this problem we need to compute the cost for wall-to-wall carpeting, that is, the cost of covering the floor of the entire room. This can be done by using the following formula:
C = (1 / 9) · c · w · l
Where:
C - Cost, in monetary units. c - Unit cost, in monetary units per yard.w - Width, in feetl - Length, in feetIf we know that c = 13.25, w = 12 and l = 18, then the cost of the wall-to-wall carpeting is:
C = (1 / 9) · 13.12 · 12 · 18
C = 318
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Find The volume of this 7th Grade Mathematical 3D Triangular Prism!!!!!!!!
The volume of the triangular base prism is 120 inches³.
How to find the volume of a triangular prism?The prism above is a triangular base prism. The volume of the triangular base prism can be calculated as follows:
volume of the prism = base area × height
Therefore,
base area = 1 / 2 bh
base area = 1 / 2 × 8 × 10
base area = 80 / 2
base area = 40 inches²
Therefore,
volume of the prism = 40 × 3
volume of the prism = 120 inches³
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In a game of cards you win $1 if you draw a heart (except the ace), $5 if you draw an ace (including the ace of hearts), $10 if you draw the king of spades and $0 for any other card. Let W be the amount you win playing this game, what the probability distribution of W?
There are four possible outcomes in this game: winning $1, winning $5, winning $10, or winning $0.
The probability of winning $1is the probability of drawing one of the 11 hearts that are not the ace of hearts. There are 52 cards in the deck, so the probability of drawing any one of them is 1/52. However, there are 11 hearts that win $1, so the probability of winning $1 is:
P(W = $1 ) = 11/52
The probability of winning $5 is the probability of drawing the ace of hearts, which is 1/52.
P(W = $5) = 1/52
The probability of winning $10 is the probability of drawing the king of spades, which is also 1/52.
P(W = $10) = 1/52
The probability of winning $0 is the probability of drawing any other card, which is 39/52.
P(W = $0) = 39/52
Write an equation for the transformed logarithm shown below, that passes through (1,0) and (-3,3)
The equation of the transformed logarithmic function that passes through the points (1,0) and (-3,3) is f(x) = 4.366*log10(x+4)
To find the equation of the transformed logarithmic function that passes through the points (1,0) and (-3,3), we need to determine the values of a, b, h, and k.
First, we know that the function passes through (1,0), so we can substitute x=1 and y=0 in the general equation of the transformed logarithmic function to get:
0 = a x log b(1-h) + k
Since log b(1-h) = 0 for any base b and any value of h, we can simplify the equation to:
0 = k
Therefore, k=0 and the equation becomes:
f(x) = a x log b(x-h)
Now, we have two equations with two unknowns (a and h), which we can solve by substitution or elimination. For simplicity, we can substitute b=10 (base 10 logarithm) since it is a common base and the values of a and h do not depend on the base of the logarithm. We get:
10⁶/ᵃ = 3-h
Since h is the horizontal shift of the graph, we can choose any value of h that makes the expression inside the logarithm positive. For example, we can choose h=-4, which makes x-h=-3-(-4)=1, the same as the x-coordinate of the point (1,0).
Taking the logarithm of both sides with base 10, we get:
log(4) = (6/a)*log(10)
Solving for a, we get:
a = 6/(log(4)*log(10)) ≈ 4.366
Now, we can substitute the values of a and h in the equation of the transformed logarithmic function to get:
f(x) = 4.366*log10(x+4)
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Help with this math please.
Find the sum of 10x² + 7x + 6 and 6x + 5.
Answer:
Submit Answer
M
DELL
10x² + 13x + 11 is the sum of linear equation .
What is a linear equation in mathematics?
The algebraic equation y=mx+b is known as a linear equation. All of its terms are constant, first-order linear ones. where m denotes the slope and b the y-intercept.
The aforementioned is occasionally referred to as a "linear equation in two variables" where y and x are variables. One or more variables may be present in a linear equation. The term "bivariate linear equation" refers to a linear equation with two variables.
two equations are 10x² + 7x + 6 and 6x + 5.
now add both equations
= (10x² + 7x + 6 )+ (6x + 5)
= 10x² + 7x + 6 + 6x + 5
= 10x² + 13x + 11
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In the diagram shown, ABC is equilateral. AD is an altitude.
10). ABD = _____.
11). ADB = _____.
12). BAD = _____.
13). CAD = _____.
Suppose AB = 6. Find:
14). BD
15). AD
16). Area of ABC
All the correct values are,
⇒ ∠ABD = 60°
⇒ ∠ADB = 90°
⇒ ∠BAD = 30°
⇒ ∠CAD = 30°
⇒ BD = 3
⇒ AD = √27
⇒ The value of area of triangle ABC is, 3√27
Given that;
In the diagram shown, ABC is equilateral. AD is an altitude.
Hence, We get;
All the angles of a equilateral triangle are,
⇒ 60°
Thus, We get;
⇒ ∠ABD = 60°
Since, AD is an altitude.
⇒ ∠ADB = 90°
And,
⇒ ∠BAD = 60 / 2 = 30°
⇒ ∠CAD = 60/2 = 30°
By Pythagoras theorem;
AB² = AD² + BD²
6² = AD² + 3²
36 = AD² + 9
AD² = 27
AD = √27
Thus, The value of area of triangle ABC is,
⇒ 1/2 × AD × BC
⇒ 1/2 × √27 × 6
⇒ 3√27
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Triangle D has been dilated to create triangle D′. Use the image to answer the question.
image of a triangle labeled D with side lengths of 3.8, 4.8, 4.2 and a second triangle labeled D prime with side lengths of x, 2.4, 2.1
Determine the scale factor used.
one half
2
one fourth
3
Answer:
To determine the scale factor used, we need to find the ratio of corresponding side lengths in both triangles.
Let's compare the length of the first side in both triangles:
x/3.8 = 2/4.8
Cross-multiplying, we get:
4.8x = 7.6
Dividing both sides by 4.8, we get:
x = 1.58
Now, let's compare the length of the second side in both triangles:
2.4/4.2 = 0.57
Finally, let's compare the length of the third side in both triangles:
2.1/4.8 = 0.44
Since the scale factor is the same for all corresponding side lengths, we can take the average of the ratios:
(scale factor) = (1.58/3.8 + 0.57 + 0.44)/3
(scale factor) = 1.01
Therefore, the scale factor used is approximately 1.
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As we have shown that both AE is congruent to CE and BE is congruent to DE.
Now, let's focus on the diagonals of the parallelogram, AC and BD. We are told that they intersect at point E.
To prove that AE is congruent to CE, we can use the fact that opposite sides of a parallelogram are congruent. Since AC is a diagonal of the parallelogram, it divides it into two congruent triangles, AEC and CED. This is because AE is parallel to CD (as they are opposite sides of the parallelogram), and EC is common to both triangles.
Therefore, by the Side-Angle-Side (SAS) congruence theorem, we can conclude that triangle AEC is congruent to triangle CED. In particular, this means that AE is congruent to CE.
Similarly, to prove that BE is congruent to DE, we can use the same reasoning. Since BD is a diagonal of the parallelogram, it divides it into two congruent triangles, AEB and DEC. This is because BE is parallel to AC (as they are opposite sides of the parallelogram), and ED is common to both triangles.
Therefore, by the SAS congruence theorem, we can conclude that triangle AEB is congruent to triangle DEC. In particular, this means that BE is congruent to DE.
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