We have proven that the sum of the squares of the diagonals of a parallelogram is equal to the sum of squares of its sides.
Parallelograms are quadrilaterals with opposite sides parallel to each other. They have many interesting properties, one of which is the relationship between the sum of the squares of the diagonals and the sum of squares of their sides. In this explanation, we will prove this relationship mathematically.
Let us consider a parallelogram with diagonals AC and BD, and sides AB, BC, CD, and DA.
To begin, let us draw line segments from the midpoints of each side to the midpoint of the opposite side. This divides the parallelogram into four congruent triangles, as shown below:
We can now use the Pythagorean theorem to find the length of each line segment. Let us denote the midpoint of AB as M, the midpoint of BC as N, and the midpoint of CD as P. Then we have:
AM² = AB²/4 + BM²
BN² = BC²/4 + CN²
CP² = CD²/4 + DP²
DM² = DA²/4 + AM²
Note that the line segments AM, BN, CP, and DM are half the length of the diagonals AC and BD. We can now add all these equations together:
=> AM² + BN² + CP² + DM² = (AB² + BC² + CD² + DA²)/4 + (AC² + BD²)/4
Rearranging this equation gives us:
=> AC² + BD² = 2(AM² + BN² + CP² + DM²) - (AB² + BC² + CD² + DA²)
Now, recall that the four triangles we divided the parallelogram into are congruent. Therefore, AM = BN = CP = DM, and AB = CD, BC = DA. Substituting these equalities into the above equation, we get:
=> AC² + BD² = 4(AM² + BN²) - 4(AB² + BC²)
We can further simplify this expression using the fact that the diagonals of a parallelogram bisect each other. Therefore, AM = CP and BN = DM. Substituting these equalities gives us:
AC² + BD² = 4(AM² + BN²) - 4(AB² + BC²)
= 4(AM² + BN² - AB² - BC²)
= 4(AN² + BM²)
Recall that AN and BM are half the length of the sides of the parallelogram. Therefore, we can write:
=> AC² + BD² = 4(AN² + BM²) = 4(AB² + BC² + CD² + DA²)
This completes the proof. We have shown that the sum of the squares of the diagonals of a parallelogram is equal to the sum of squares of its sides.
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: A major corporation is building a 4325 acre complex of homes, offices, stores, schools, and churches in the rural community of Glen Cove. As a result of this development, the planners have estimated that Glen Clove's population (in thousands) t yr from now will be given by the following function.P(t) = 35t^2 + 125t + 200/t^2 + 4t + 40(a) What is the current population of Glen Cove? (b) What will be the population in the long run?
The current population of Glen Cove is 5,000 people and in the long-run, the population of Glen Cove will approach 35,000 people.
Corporation:A corporation may be a corporate entity whose shareholders elect a board of directors to supervise its operations. the dissimilarity between the business and an organization is that an organization may be a sort of business that's suitable for smaller businesses and organizations, whereas a corporation may be a kind of business that's appropriate for larger enterprises and entities.
The population in Glen Clove's population that is t years from now will be
P(t) = [tex]\frac{35t^2 + 125t + 200}{t^2 + 4t + 40}[/tex]
The current population can be calculated by substituting t = 0 in P(t).
So,
P(0) = [tex]\frac{35(0)^2 + 125(0) + 200}{(0)^2 + 4(0) + 40}[/tex]
P(0) = 5
Thus, the current population of Glen Clove is 5 thousand.
(b)For the long run, t tends to infinity. So, evaluate the infinite limit for the given function as follows:
[tex]\lim_{t \to \infty} P(t)= \lim_{t \to \infty}P(t)\frac{35t^2 + 125t + 200}{t^2 + 4t + 40}[/tex]
=[tex]\lim_{t \to \infty}\frac{t^2(35+\frac{125}{t} +\frac{200}{t} )}{t^2(1+\frac{4}{t} +\frac{40}{t^2} )}[/tex]
= [tex]\lim_{t \to \infty}\frac{(35+\frac{125}{t} +\frac{200}{t} )}{1+(\frac{4}{t} +\frac{40}{t^2} )}[/tex]
Now, solve further by putting the limits as follows:
As t approaches infinity, the terms in the numerator approach 0, while the terms in the denominator approach 1, so the limit of P(t) as t approaches infinity is:
lim t -> infinity P(t) = 35 / 1 = 35
So, in the long-run, the population of Glen Cove will approach 35,000 people.
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Determine which variable is the likely explanatory variable and which is the likely response variable.-The explanatory variable is the miles per gallon and the response variable is the weight.-The explanatory variable is the weight and the response variable is the miles per gallon.
For the first statement, the likely explanatory variable is the miles per gallon and the likely response variable is the weight.
For the second statement, the likely explanatory variable is the weight and the likely response variable is the miles per gallon.
In statistical analysis, an explanatory variable is a variable that is presumed to have an effect on another variable. It is also known as an independent variable. On the other hand, a response variable is the variable that is presumed to be affected by the explanatory variable. It is also known as a dependent variable.
In the first statement, it is stated that the explanatory variable is the miles per gallon, which means that we are looking at how the fuel efficiency of a vehicle affects its weight. Therefore, the weight is the likely response variable, as it is presumed to be affected by the miles per gallon.
In the second statement, it is stated that the explanatory variable is the weight, which means that we are looking at how the weight of a vehicle affects its fuel efficiency. Therefore, the miles per gallon is the likely response variable, as it is presumed to be affected by the weight.
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Please Help!
Find The Point-Slope And Slope Intercept of the following image.
The equation that represents the relationship between the number of miles to tread thickness will be y = - 0.12x + 9.
What is the equation of a line passing through two points?Let the equation of the line pass through (x₁, y₁) and (x₂, y₂). Then the equation of the line is given as,
[tex]\rm (y - y_2) = \left (\dfrac{y_2 - y_1}{x_2 - x_1} \right ) (x - x_2)[/tex]
Let 'x' be the number of miles and 'y' be the tread thickness. Then the two points are (15, 7.2) and (35, 4.8). Then the equation is given as,
(y - 7.2) = [(7.2 - 4.8) / (15 - 35)] (x - 15)
y - 7.2 = -0.12(x - 15)
y - 7.2 = - 0.12x + 1.8
y = - 0.12x + 9
The equation that represents the relationship between the number of miles to tread thickness will be y = - 0.12x + 9.
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The existing entries in the following table show the maximum amount of goals scored and the amount of shots saved that Mia Hamm and Alyssa Naeher could make as soccer players during a year.
Goals Scored
Amount of shots saved as a goalie
Mia Hamm
50
30
Alyssa Naeher
20
120
Please answer the following questions below:
a. Does either Mia or Hamm have an absolute advantage in goals scored and amounts?
b. Calculate the opportunity cost of goals scoring 1 goal for Mia and Alyssa.
c. Calculate the opportunity cost of saving 1 shot as a goalie for Mia and Alyssa.
d. Who has the comparative advantage in scoring goals? Who has the comparative advantage in saving shots as a goalie? Why?
Please show all your work and calculations.
a. Yes, Mia Hamm has an absolute advantage in both scoring goals and saving shots as a goalie,
b. The opportunity cost of scoring 1 goal for Mia is 0.6 shots
c. The opportunity cost of saving 1 shot as a goalie for Mia is 1.67 goals . The opportunity cost of saving 1 shot as a goalie for Alyssa is 0.17 goals scored.
d. Mia has the comparative advantage in scoring goals, since her opportunity cost of scoring 1 goal is lower than Alyssa's. Alyssa has the comparative advantage in saving shots as a goalie, since her opportunity cost of saving 1 shot is lower than Mia's
How do we calculate?b. The opportunity cost of scoring 1 goal for Mia is 0.6 shots saved as a goalie which is calculated as
30 shots/50 goals = 0.6).
The opportunity cost of scoring 1 goal for Alyssa is 6 shots saved as a goalie which is calculate as
opportunity cost 120 shots/20 goals = 6.
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A Ferris wheel is 24 meters in diameter and is boarded from a platform that is 1 meter above the ground. The six o'clock position on the Ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 16 minutes. The function h (t) gives a person's height in meters above the ground t minutes after the wheel begins to turn.
a. Find the amplitude, midline, and period of h (t).
b. Assume that a person has just boarded the Ferris wheel from the platform and that the Ferris wheel starts spinning at time t=0. Find a furmula for the height function h(t).
c. If the Ferris whell continues to turn, how high off the ground is a person after 52 minutes?
a. The amplitude is 12m, the midline is 13m, and the period of h (t) is 16 minutes.
How to solve these?a. The amplitude of the height function h(t) is 12 meters (24 meters diameter / 2).
The midline of the height function is 12 meters (24 meters diameter / 2) + 1 meter (height of the platform).
The period of the height function is the time it takes for the Ferris wheel to complete one full cycle, which is 16 minutes.
b. The height function h(t) can be modeled as a sinusoidal function, where h(t) = 12 cos (2πt/16) + 13.
The cosine function models the cyclical change in height as the Ferris wheel turns.
The 2π in the argument of the cosine function represents the full revolution of the Ferris wheel, and the 16 in the argument of the cosine function represents the time it takes for the Ferris wheel to complete one revolution.
The 13 at the end of the equation is the midline of the height function, which represents the average height of the person above the ground.
c. To find the height of a person after 52 minutes, we substitute t = 52 into the height function h(t) = 12 cos (2πt/16) + 13:
h(52) = 12 cos (2π x 52/16) + 13
h(52) = 12 cos (13π) + 13
h(52) = 12(-1) + 13
h(52) = 1 meter
So, a person would be 1 meter above the ground after 52 minutes.
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S A black bear was tracked across Central Florida logging approximately 120 miles in 30 days.
For 24 miles of M34’s journey, he was tracked walking along the Kissimmee River.
How many days was he traveling along the Kissimmee River?
Answer: 6 days
120 miles divided into 30 days is 4 miles per day.
24 miles divided by 4 miles is 6 , which means the bear spent 6 days walking along Kissimmee River.
I hope this helped & Good Luck <3 !!
Need help!! with this question please
The statement true for A, C and E is the second statement r → p ∨ q.
What is Truth Table?.Truth table is a table which describes the truth value of a complex statement regarding to the truth value of single statements.
Given is a truth table.
Consider r → p ∨ q, for the statements A, C and E.
For A, p is true and q is true, then p ∨ q is true. So r is true.
For C, p is true and q is false, then p ∨ q is true. So r is true.
For E, p is false and q is true, then p ∨ q is true. So r is true.
Hence the correct statement is r → p ∨ q.
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The region bounded by the given curves is rotated about the specified axis. Find the volume V of the resulting solid by any method.
x = (y − 6)^2, x = 25; about y = 1
The volume V of the resulting solid is [tex]$\frac{5000 \pi}{3}[/tex]
As per the data given:
The region bounded by the given curves is rotated about the specified axis.
Here we have to determine the volume V of the resulting solid by any method.
The given equations are x = [tex](y -6)^2[/tex], x = 25; about y = 1.
To find the volume of resulting solid we are using cylinder method
To derive the above solution, we need to use the old fashioned quadratic method.
Substitute the values of x then we get
25 = [tex](y-6)^2[/tex]
[tex]\rightarrow 5^2=(y-6)^2[/tex]
y - 6 = 5
y = 6 + 5
y = 11
So, 1 ≤ y ≤ 11
Graph attached at end of the solution
Using cylinder method and integrate along y -axis from y = 1 to y = 11
Volume [tex]}=\int_a^b 2 \pi r h d y[/tex]
Volume [tex]& =2 \pi \int_1^{11}(y-1)\left(25-(y-6)^2\right) d y[/tex]
Volume [tex]=2 \pi \int(y-1)\left(25-y^2-36+12 y\right) d y \\[/tex]
Volume [tex]& =2 \pi \int_1^{11}\left(-y^3+13 y^2-23 y+11\right) d y[/tex]
Volume [tex]=2 \pi\left[-\frac{y}{4}+\frac{13 y}{3}-\frac{23 y}{2}+11 y\right]_1^{11} \\[/tex]
Volume [tex]& =2 \pi \cdot\left(\frac{10043}{12}-\frac{43}{12}\right)=2 \pi\left(\frac{10000}{12}\right)=\frac{5000\pi }{3}[/tex]
Thus the volume of resulting solid is:
[tex]$}=\frac{5000 \pi}{3}[/tex]
Therefore the volume is [tex]$\frac{5000 \pi}{3}[/tex]
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50 POINTS PLEASE HELP SOLVE
Given that an investment into a college fund is represented by A=12,000e^(0.023*12) , match each value to the correct description of the value in the context of the question. 12,000 The amount of money in the account at the end of the time given. e
The initial amount of money invested into the college fund 0.023
The interest rate.
The interest is compounded continuously.
The amount of years the money is invested.
As per compound interest, the account balance at the end of 12 years is approximately $17,217.
Compound interest is an important concept in finance and investing, and it plays a critical role in understanding how your money can grow over time. When you invest money, your initial investment earns interest, and over time, that interest can also earn interest, resulting in exponential growth.
In the context of this question, the investment into a college fund is represented by the equation
=> [tex]A = 12,000e^{0.023*12}[/tex]
where A is the amount of money in the account at the end of the time given. Let's break down the different components of this equation and see how they relate to compound interest.
First, we have the initial investment, which is not explicitly given in the equation, but we can infer that it is 12,000. This is the amount of money that is initially invested into the college fund.
Next, we have the interest rate, which is 0.023.
Finally, we have the amount of time that the money is invested, which is 12 years in this case. The longer the money is invested, the more time there is for compound interest to work its magic and grow the account balance.
Putting all of these components together, we can use the equation
=> [tex]A = 12,000e^{0.023*12}[/tex]
to calculate the amount of money in the college fund at the end of 12 years.
By plugging in the values for the initial investment, interest rate, and time, we can see that the account balance at the end of 12 years is approximately $17,217.
This represents the power of compound interest and how it can turn a modest initial investment into a substantial sum over time.
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Can I get help please
Answer:
B. Vertex
Step-by-step explanation:
The midpoint between the two x-intercepts of a parabola will provide the x-coordinate of the vertex. Line of symmetry is drawn along this particular x-coordinate. This x-coordinate is substituted into the equation of the graph to find its corresponding y-coordinate.
We determined that the unconditional probability distribution for Y, the number of defects per yard in a certain fabric, is P(y) = - (1/2)^y+1 y = 0, 1, 2, .... Find the expected number of defects per yard.
The expected number of defects per yard is 1.
The unconditional probability distribution for Y, the number of defects per yard in a certain fabric, is P(y) = - (1/2)^y+1 y = 0, 1, 2, ....
To calculate the expected number of defects per yard, we use the formula E(Y) = ΣyP(y), where y is the number of defects per yard and P(y) is the probability of that number of defects per yard.
We can then substitute the values for P(y) into the formula and sum for all values of y from 0 to ∞. This yields E(Y) = Σy(-(1/2)^y+1) = 1. Thus, the expected number of defects per yard is 1.
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Need help PLSSS THANK YOU
Using the figure the true trigonometry statements are
A. (sin A)² + (cos A)² = 1
C. 1/(cos A)² - (tan A)² = 1
D. (sin A)² + (cos A)² + (tan A)² = 1/(cos A)²
How to show the statements are trueA. (sin A)² + (cos A)² = 1 this is a fundamental trigonometry identity
say angle A = 30
(sin 30)² + (cos 30)² = 1
C. 1/(cos A)² - (tan A)² = 1
1/(cos A)²= 1 + (tan A)²
1/(cos A)²= 1 + (sin A)² / (cos A)²
1/(cos A)²= ((cos A)² + (sin A)²) / (cos A)²
and (cos A)² + (sin A)² = 1, hence
1/(cos A)²= 1/(cos A)² this is true
D. (sin A)² + (cos A)² + (tan A)² = 1/(cos A)²
recall (sin A)² + (cos A)² = 1
1 + (tan A)² = 1/(cos A)² same as line 2 in C hence this is true
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Finding the Volume of a Solid In Exercises 23 and 24, use the disk method or the shell method to find the volumes of the solids generated by revolving the region bounded by the graphs of the equations about the given lines. 23. y = x, y = 0, x= 3 (a) the x-axis (b) the y-axis (c) the line x = 3 (d) the line x = 6 24. y = x, y = 2, X (a) the x-axis (b) the line y = 2 (c) the y-axis (d) the line x = -1
(a) Revolving about the x-axis, the volume of the solid is 3Pi
(b) Revolving about the y-axis, the volume of the solid is 18 Pi
(c) Revolving about the line x=3, the volume of the solid is 0
(d) Revolving about the line x=6, the volume of the solid will be 32/3 Pi
The region bounded by y=x, y=0, and x=3 is a triangle in the first quadrant.
To find the volume of the solid generated by revolving this region about an axis, we can use the disk or washer method.
(a) Revolving about the x-axis:
Each cross-section of the solid perpendicular to the x-axis is a disk with radius x and thickness dx.
The volume of each disk is [tex]\pi x^2 dx[/tex].
The limits of integration are 0 and 3, the x-coordinates of the intersection points of the two curves.
Hence, the volume of the solid is
[tex]V &= \int_0^3 \pi x^2 dx\\\\\&= \left[\frac{\pi}{3}x^3\right]_0^3\\\\\&= \frac{9\pi}{3}\\\\\&= 3\pi.[/tex]
(b) Revolving about the y-axis:
Each cross-section of the solid perpendicular to the y-axis is a washer with outer radius 3 and inner radius x, and thickness dx.
The volume of each washer is [tex]\pi(3^2-x^2)dx.[/tex]
The limits of integration are 0 and 3.
Hence, the volume of the solid is
[tex]V &= \int_0^3 \pi(3^2-x^2)dx\\\\\&= \left[\pi(9x-\frac{x^3}{3})\right]_0^3\\\\\&= \pi(27-\frac{27}{3})\\\\\&= 18\pi.[/tex]
(c) Revolving about the line x=3:
Each cross-section of the solid perpendicular to the line x=3 is a washer with outer radius 3-x and inner radius 3-x-|x|, and thickness dx.
The volume of each washer is[tex]\pi((3-x)^2-(3-x-|x|)^2)dx[/tex].
The limits of integration are -3 and 3.
Hence, the volume of the solid is
[tex]V &= \int_{-3}^3 \pi((3-x)^2-(3-x-|x|)^2)dx\\\\\&= \left[8\pi\int_0^3 (3-x-|x|)dx\right]\\\\\&= 8\pi\int_0^3 (3-2x)dx\\\\\&= 8\pi\left[3x-x^2\right]_0^3\\\\\&= 8\pi(9-9)\\\\\&= 0.\\\\[/tex]
(d) Revolving about the line x=6:
Each cross-section of the solid perpendicular to the line x=6 is a washer with outer radius |3-x| and inner radius |x-6|, and thickness dx.
The volume of each washer is[tex]\pi(|3-x|^2-|x-6|^2)dx.[/tex]
The limits of integration are 0 and 3.
Hence, the volume of the solid is
[tex]V &= \int_0^3 \pi(|3-x|^2-|x-6|^2)dx\\\\\&= \left[\frac{32\pi}{3}\right]\\\\\&= \frac{32}{3}\pi.[/tex]
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A single-factor ANOVA study consists ofr=6treatments with sample sizesni≡10. a. Assuming that pairwise comparisons of the treatment means are to be made with a 90 percent family confidence coefficient, find theT,S, andBmultiples for the following numbers of pairwise comparisons in the family:g=2,5,15. What generalization is suggested by your results? b. Assuming that contrasts of the treatment means are to be estimated with a 90 percent family confidence coefficient, find theSandBmultiples for the following numbers of contrasts in the family:g=2,5,15. What generalization is suggested by your results?
The generalization suggested by these results is that as the number of contrasts increases, the multiples decrease.
a. For each of the g numbers of pairwise comparisons in the family (2, 5, and 15), the T, S, and B multiples can be calculated using the following formulas:
[tex]T = t α/2 (g-1)S = t α/2 (g-1) √2/(n-1)B = t α/2 (g-1) √(2/n)[/tex]
Where tα/2 is the t-value associated with the given family confidence coefficient (in this case, 90%) and g is the number of pairwise comparisons.
For g=2, the T, S, and B multiples are 3.078, 2.179, and 2.500 respectively. For g=5, the multiples are 2.571, 1.711, and 2.001 respectively. For g=15, the multiples are 2.228, 1.429, and 1.641 respectively.
The generalization suggested by these results is that as the number of pairwise comparisons increases, the multiples decrease.
b. For each of the g numbers of contrasts in the family (2, 5, and 15), the S and B multiples can be calculated using the following formulas:
[tex]S = t α/2 (g-1) √2/nB = t α/2 (g-1) √(2/n)[/tex]
Where tα/2 is the t-value associated with the given family confidence coefficient (in this case, 90%) and g is the number of contrasts.
For g=2, the S and B multiples are 2.179 and 2.500 respectively. For g=5, the multiples are 1.711 and 2.001 respectively. For g=15, the multiples are 1.429 and 1.641 respectively.
The generalization suggested by these results is that as the number of pairwise comparisons or contrasts increases, the multiples decrease.
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find x in the png below
Answer:
The x-intercept is 27/8, or 3.375.
Step-by-step explanation:
0 = -8/3x + 9
-8/3x = -9
-9 * -3/8 = 27/8
x = 27/8
This should be easy: Marcus bought 5 crates of apples. Each crate held 2.9 pounds of apples. How many pounds of apples did Marcus buy in all?
Answer:
14.5
Step-by-step explanation:
2.9(5)=14.5
Answer:
14.5p
here,
lets start off by looking at the full scale problem. We already know that each crate can hold 2.9 apples at a time. We also know that we have 5 crates in total.
hence,
we would multiply [tex]5(2.9)=14.5[/tex] or [tex]5x2.9=14.5[/tex]
factor the expression using GCF 16 X plus 8Y
Factoring the expression 16x+ 8y using GCF would be, 8(2x + 1).
What are GCF and distributive property?The GCF of two or more than two numbers is the highest number that divides the given two numbers completely.
We also know that distributive property states a(b + c) = ab + ac.
Given, An expression 16x+ 8y.
Now, The HCF of 16 and 8 is 8, Because 8 is the highest number that divides 16 and 8 completely.
Therefore, The factored form of the expression 16x+ 8y would be,
8(2x + 1).
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simplify √8/18
answers:
a. -1/3
b. 2/3
c. 3/2
d. 2
Answer:
b. 2/3
Step-by-step explanation:
8 = 4 x 2
√8 = √4 x √2
√4 = 2
√8 = 2 x √2 = 2√2
18 = 9 x 2
√18 = √9 x √2
√9 = 3
√18 = 3 x √2 =3√2
√(8/18) = 2√2 / 3√2 = 2/3
You will need a pencil and paper.
Shalonda is deciding what to wear. She has three pairs of jeans and wants to wear either a blue shirt or a red shirt. How many different combinations are possible?
5 combinations
6 combinations
7 combinations
8 combinations
Answer:
6 combinations
Step-by-step explanation:
3 pairs of Jeans & 2 shirts
3 x 2 = 6
6 combinations
Vera rents bicycles to tourists. She recorded the height (in \text{cm}cmstart text, c, m, end text) of each customer and the frame size (in \text{cm}cmstart text, c, m, end text) of the bicycle that customer rented.
After plotting her results, Vera noticed that the relationship between the two variables was fairly linear, so she used the data to calculate the following least squares regression equation for predicting bicycle frame size from the height of the customer:
\hat{y} = \dfrac{1}{3}+\dfrac{1}{3}x
y
^
=
3
1
+
3
1
xy, with, hat, on top, equals, start fraction, 1, divided by, 3, end fraction, plus, start fraction, 1, divided by, 3, end fraction, x
What is the residual of a customer with a height of 155\,\text{cm}155cm155, start text, c, m, end text who rents a bike with a 51\,\text{cm}51cm51, start text, c, m, end text frame?
The residual of a customer with a height of 155 cm who rents a bike with a 51 cm frame is -1.
What is Residual?The discrepancy between a response variable's observed value and its expected value as predicted by the regression line.
We have the Equation: y'=1/3x + 1/3.
The regression equation is given as:
y'=(1/3)x + (1/3)
Since the height is given as 155cm, x=155 cm
The predicted frame size,
y'=(1/3)x + (1/3)
y'=(1/3) × 155+ (1/3)
= 51 2/3 + 1/3
= 52
So, Residual = Observed y- predicted y
= 51-52
= -1
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Answer:
Step-by-step explanation:
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Find x from two right triangles
The length of x is 7.
Describe Triangles?In geometry, a triangle is a three-sided polygon made up of three line segments that join at three points called vertices. Triangles are one of the most basic and important shapes in geometry, and they are used in many areas of mathematics, science, and engineering.
Using the property of a median, we know that BD divides AC into two equal parts. Let's call the length of CD as y.
So, we have:
AC = AD + CD
x = 3 + y
Using the Pythagorean Theorem, we can find the length of BD:
BD² = AB² + AD² - 2(AB)(AD)(cos(B))
BD² = 4² + 3² - 2(4)(3)(cos(B))
BD² = 25 - 24cos(B)
BD = √(25 - 24cos(B))
Since BD is a median, we know that it divides BC into two equal parts. Let's call the length of BD as z.
So, we have:
BC = 2z
8 = 2z
z = 4
Now, using the property of medians again, we know that:
4² + y² = z²
16 + y² = 16
y² = 0
y = 0
Therefore, CD is also 4, and:
x = 3 + y
x = 3 + 4
x = 7
So, the length of AC is 7.
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What is the true solution to the logarithmic equation?
log₂[log₂ (√4x)] = 1
O x=-4
O x=0
O x=2
Ox=4
Step-by-step explanation:
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The only solution to the given logarithmic equation is x = 4.
What is Logarithmic?An example of a logarithmic function is one that expresses the connection between two values as the logarithm of one quantity to a certain base.
A logarithmic function's basic form is:
f(x)=log base b (x)
where x is the function's input, b is the logarithm's base, and f(x) is the function's value.
We can solve the logarithmic equation step by step as follows:
log₂[log₂ (√4x)] = 1
First, we simplify the expression inside the logarithm:
log₂[log₂ (√4x)] = 1
log₂[log₂ (2√x)] = 1
Next, we use the definition of logarithms to rewrite the equation in exponential form:
log₂ (2√x) = 2¹ = 2
2√x = 2² = 4
√x = 2
x = 4
Thus, the solution is x=4.
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Wendy will have new carpet installed on her rectangular bedroom floor. The floor is 15 feet in length and 12 feet in width. The carpet costs $27.50 per square yard, and installation costs $4.15 per square yard. Which statement about the total cost of the carpet and installation is true?
a. The total cost is $379.80 since $31.65×12 = $379.80.
b. The total cost is $633.00 since $31.65×20 = $633.00.
c. The total cost is $474.75 since $31.65×15 = $474.75.
d. The total cost is $4,950.00 since $27.50×180 = $4,950.00
The statement that is true about the total cost of the carpet and installation is: The total cost of the carpet and installation is $633.
What is Algebraic expression ?
Algebraic expression can be defined as combination of variables and constants.
To determine the total cost of the carpet and installation, we first need to calculate the area of Wendy's bedroom floor, which is:
Area = Length x Width
Area = 15 ft x 12 ft
Area = 180 square feet
We can then convert the area to square yards by dividing by 9:
Area in square yards = 180 sq ft / 9
Area in square yards = 20 sq yd
The cost of the carpet is $27.50 per square yard, so the cost of the carpet alone is:
Carpet cost = 20 sq yd x $27.50/sq yd
Carpet cost = $550
The installation cost is $4.15 per square yard, so the cost of the installation alone is:
Installation cost = 20 sq yd x $4.15/sq yd
Installation cost = $83
To find the total cost, we need to add the cost of the carpet and the installation
Total cost = Carpet cost + Installation cost
Total cost = $550 + $83
Total cost = $633
Therefore, the statement that is true about the total cost of the carpet and installation is: The total cost of the carpet and installation is $633.
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The manager of a furniture factory finds that it costs $2,200 to manufacture 100 chairs in one day and $4,800 to produce 300 chairs in one day.
a) Express the cost as a function of the number of chairs produced, assuming that it is linear.
b) Sketch the graph of the function obtained in part (a).
c) What is the slope of the graph and what does it represent?
d) What is the y-intercept of the graph and what does it represent?
a) C(x) = 0.02x + 2200, b) A straight line with positive slope, c) Slope is 0.02, represents cost per chair, d) Y-intercept is 2200, represents fixed cost.
a) The cost C(x) can be expressed as a linear function of the number of chairs x produced. We can solve for the slope and y-intercept using the two points (100, 2200) and (300, 4800). First, calculate the slope of the line using the formula (y2 - y1)/(x2 - x1). This gives us (4800 - 2200)/(300 - 100) = 0.02. Then, we can use the slope and one of the points to solve for the y-intercept using the equation y = mx + b, where m is the slope and b is the y-intercept. Substituting the values from the first point (100, 2200) into the equation gives us 2200 = 0.02(100) + b, so the y-intercept is 2200. Therefore, the cost function can be expressed as C(x) = 0.02x + 2200.
b) The graph of this function is a straight line with a positive slope, as shown in the diagram.
c) The slope of the graph is 0.02, which represents the cost per chair.
d) The y-intercept is 2200, which represents the fixed cost of manufacturing the chairs.
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What is the equation for the line of reflection?
ty
2
27
Mark this and return
2
D
B
C
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C₁
B
D'
00
Save and Exit
A'
10
X
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Submit
The solution is, the line of reflection is y = 1.
What is line of reflection?When reflecting a figure in a line or in a point, the image is congruent to the preimage. A reflection maps every point of a figure to an image across a fixed line. The fixed line is called the line of reflection.
here, we have,
The given figure shows two triangles reflected horizontally.}
We can find the line reflection because it would be an horizontal line half way. Remember that horizontal lines have the form y = k, where k = 1 in this case,
Hence, the line of reflection is y = 1.
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which expression represents “10 less than the product of a and b”
1. (a+ b) 10
2. ab-10
3. 10- ab
4 a (b-10)
Answer:
2
Step-by-step explanation:
ab - 10. this means 10 is less than the product of a and b
How do I answer this?
What is k = ?
What is X = ?
How do I slove?
The value of the scale factor of dilation (k) is 5/3 and the value of x is 21
How to determine the value of kThe variable k is the scale factor of dilation of the triangles
This is calculated as
k = Image triangle/Pre-image triangle
Substitute the known values in the above equation, so, we have the following representation
k = 15/9
So, we have
k = 5/3
How to determine the value of xThis is calculated as
x * 5/3 = 35
So, we have
x = 3 * 35/5
Evaluate
x = 21
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John buys 2 shirts for $7.89, he also buys a pair of shoes for $15.If he has a sales tax of 8.9%. What would be his final cost?
Answer:
$33.52
Step-by-step explanation:
7.89+7.89+15= 30.78
30.78x 0.089=2.73942 rounded off to the nearest hundredth which is 2.74
30.78+2.74= $33.52
what is 2/3 times 2/7?
Answer:
2/3 x 2/3 as a fraction is 4/9.
Step-by-step explanation:
We are performing simple fractional multiplication to get the answer, i.e., 2 x 2 = 4. The numerator of the fraction is multiplied with the numerator of the other, i.e., 3 x 3 = 9.
Answer:
4/21
Step-by-step explanation:
Just line them up side by side, then multiply 2×2 and 3×7. Hope this helps