Determine the point on the graph of y = In 2x at which the tangent line is perpendicular to
x+4y=1.
please show all workings:)
Answers
Answer:
(1/4, ln(1/2))
Step-by-step explanation:
The slope of the given line is -1/4, so the perpendicular line will have a slope of -1/(-1/4) = 4.
The slope of the given function is its derivative:
y' = 2/(2x) = 1/x
That will have a value of 4 when x = 1/4.
The point on the graph where the slope is 4 is (x, y) = (1/4, ln(1/2)).
Answer:
[tex](\frac{1}{4},-\ln(2))[/tex]
Step-by-step explanation:
Let's first differentiate [tex]y=\ln(2x)[/tex].
This gives us [tex]y'=\frac{(2x)'}{2x}=\frac{2}{2x}=\frac{1}{x}[/tex]. This gives us the slope of any tangent line to any point on the curve of [tex]y=\ln(2x)[/tex].
Let [tex](a,b)[/tex] be a point on [tex]y=\ln(2x)[/tex] such that the tangent line at that point is perpendicular to [tex]x+4y=1[/tex].
Let's find the slope of this perpendicular line so we can determine the slope of the tangent line. Keep in mind, that perpendicular lines (if not horizontal to vertical lines or vice versa) have opposite reciprocal slopes.
Let's begin.
[tex]x+4y=1[/tex]
Subtract [tex]x[/tex] on both sides:
[tex]4y=-x+1[/tex]
Divide both sides by 4:
[tex]y=\frac{-x}{4}+\frac{1}{4}[/tex]
The slope is -1/4.
This means the line perpendicular to it, the slope of the line we wish to find, is 4.
So we want the following to be true:
[tex]\frac{1}{x} \text{ at } x=a[/tex] to be [tex]4[/tex].
So we are going to solve the following equation:
[tex]\frac{1}{a}=4[/tex]
Multiply both sides by [tex]a[/tex]:
[tex]1=4a[/tex]
Divide both sides by 4:
[tex]\frac{1}{4}=a[/tex]
So now let's find the corresponding [tex]y[/tex]-coordinate that I called [tex]b[/tex] earlier for our particular point that we wished to find.
[tex]y=\ln(2x)[/tex] for [tex]x=a=\frac{1}{4}[/tex]:
[tex]y=\ln(2\cdot \frac{1}{4})[/tex] (this is our [tex]b[/tex])
[tex]y=\ln(\frac{1}{2})[/tex]
[tex]y=\ln(1)-\ln(2)[/tex]
[tex]y=0-\ln(2)[/tex]
[tex]y=-\ln(2)[/tex]
So the point that we wished to find is [tex](\frac{1}{4},-\ln(2))[/tex].
---------------------Verify--------------------------------
What is the line perpendicular to the tangent line to the curve [tex]y=\ln(2x)[/tex] at [tex](\frac{1}{4},-\ln(2))[/tex]?
Let's find the slope formula for our tangent lines to this curve:
[tex]y'=\frac{1}{x}[/tex]
[tex]y'=\frac{1}{x}[/tex] evaluated at [tex]x=\frac{1}{4}[/tex]:
[tex]y'=\frac{1}{\frac{1}{4}}=4[/tex]
This says the slope of this tangent line is 4.
A line perpendicular this will have slope -1/4.
So we know our line will be of the form:
[tex]y=\frac{-1}{4}x+c[/tex]
Multiply both sides by 4:
[tex]4y=-1x+4c[/tex]
Add [tex]1x[/tex] on both sides:
[tex]4y+1x=4c[/tex]
Reorder using commutative property:
[tex]1x+4y=4c[/tex]
Use multiplicative identity property:
[tex]x+4y=4c[/tex]
As we see the line is in this form. We didn't need to know about the [tex]y[/tex]-intercept,[tex]c[/tex], of this equation.
Related Questions
Can someone please help me on this question.
Answers
Answer:
Step-by-step explanation:
[tex]3x^{2} + 5x-12=0[/tex]
[tex]-12=-5x-3x^{2}[/tex]
[tex](-12=-5x-3x^{2} ) * -1[/tex]
[tex]\sqrt{12=5x+3x^{2}[/tex]
The vertex form of the equation of a parabola is y= 3(x - 4)^2 -22. What is the standard form of the equation
Answers
Answer:
option A
Step-by-step explanation:
We can find the standard form by expanding the equation:
y = 3(x-4)^2 - 22
y = 3(x^2 - 8x + 16) - 22
y = 3x^2 - 24x + 48 - 22
y= 3x^2 - 24x + 26
So the correct option is option A
Answer:
option A) y= 3x²-24x+26 ~apex
Step-by-step explanation:
a toy company's total payment salaries for the first two months of 2005 is $21, 894. write and solve an equation to find the salaries for the second month if the first month's salaries are $10,205.
Answers
Solution: Second month = m
The equation is
m + 10205 = 21894
m = 11689
So the second month will be $11,894. :)
To solve for the second month's salaries, subtract the first month salary from the total for two months. The solution to the equation $21,894 - $10,205 gives this salary: $11,689.
Explanation:The subject of this problem is a toy company's salary expenses over a course of two months. Let's denote the salary expenses for the second month as X. We know from the problem that the total salary expenses for the first two months (January and February) are $21,894. Furthermore, we know that the salary expenses for January are $10,205.
We can write an equation to express these facts as follows: X (which is the salary expenses for February) is equal to the total salary for the two months minus the salary for January, or in mathematical notation, X = $21,894 - $10,205.
We solve this math equation by subtracting $10,205 from $21,894, giving us X = $11,689. Therefore, the second month's salaries total $11,689.
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What is the degree of x 6 + 4x – 3?
Answers
Answer:
The degree of a polynomial is the highest degree of its monomials with non-zero coefficients
Step-by-step explanation:
not sure
The maximum power of the polynomial x⁶ + 4x - 3 that is degree will be 6.
What is a polynomial?A polynomial expression is an algebraic expression with variables and coefficients. Unknown variables are what they're termed. We can use addition, subtraction, and other mathematical operations. However, a variable is not divisible.
A polynomial's degree is the greatest exponent of its variable term. The variable in this example is x, and the maximum exponent of x is 6, which is the degree of the polynomial. As a result, the polynomial x⁶ + 4x - 3 has a degree of 6.
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A fish tank is a cube. Its side length is 4 1/2 feet. The volume of water needed to completely fill the fish tank is ___ cubic feet.
Answers
The equation for the volume of a cube is V=s^3, where V is volume and s is side length. Plug in and solve:
V=4.5^3
V=4.5*4.5*4.5
V=91.125ft^3
Hope this helps!!
Please help!!!!!!!!!!
Answers
Answer:
Step-by-step explanation:
65 - Acute
72 - Acute
96 - Obtuse
Answer:
(a)
Step-by-step explanation:
Given 3 sides of a triangle a, b and c where c is the longest side, then
• If a² + b² = c² ⇒ triangle is right
• If a² + b² > c² ⇒ triangle is acute
• If a² + b² < c² ⇒ triangle is obtuse
Here a = 65, b = 72 and c = 96
65² + 72² ? 96²
4225 + 5184 ? 9216
9409 > 9216 ⇒ triangle is acute
Anna's scores for a video game are 670, 575, 400, 575, 1250, 720, and 885. Which statement about the data is correct?
Answers
Answer:
Step-by-step explanation:
what are the statements about the data?
Answer
C) The mean is greater than the median.
Step-by-step explanation:
To find the median, order the scores from LEAST to GREATEST.
400, 575, 575, 670, 720, 885, 1250.
The mean is the middle number, which is 670.
To find the mean, average the scores:
400 + 575 + 575 + 670 + 720 + 885 + 1250
7
= 725
725 > 670; therefore, the mean is greater than the median.
Find the area of the shaded sector (It is 45 degrees with a radius of 7) Help!!!
Answers
Answer:
Step-by-step explanation:
The formula for the area of a sector is
[tex]A_{s} =\frac{\theta }{360}*\pi r^2[/tex]
where theta is the angle given as 45 and r is the radius (7). Plugging in we get
[tex]A_{s}=\frac{45}{360}*\pi (7)^2[/tex]
Simplify a bit to get
[tex]A_{s}=\frac{1}{8}*49\pi[/tex]
which gives you, in terms of pi:
[tex]A_{s}=\frac{49\pi }{8}[/tex]
or if you need to use the decimal form, rounded to the hundredths position:
A = 19.24
Please help ASAP!
Answer, yes or no to state whether each data set is likely to be normally distributed.
1). the number of coupons used at a supermarket
2). the weights of the pumpkins that are delivered to a supermarket
3). the number of raisins in each 8-oz box of raisins at a supermarket
4). the amount of time customers spend waiting in the checkout line at a supermarket
Answers
The selection of "Yes" or "No" to state whether each data set is likely to be normally distributed is as follows: A) No. B) Yes. C) Yes. D) Yes
What is a normal distribution of data?A normal distribution of data occurs when the majority of data points are relatively similar and the data set has a small range of values.
1. The number of coupons used at a supermarket is no.
2. The weights of the pumpkins that are delivered to a supermarket is yes.
3. The number of raisins in each 8-oz box of raisins at a supermarket is yes.
4. The amount of time customers spend waiting in the checkout line at a supermarket is yes.
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The selection of "Yes" or "No" to state whether each data set is likely to be normally distributed is as follows: A) No. B) Yes. C) Yes. D) No
1. The number of coupons used at a supermarket is likely to follow a discrete distribution rather than a normal distribution. Customers may use 0, 1, 2, or more coupons, but the number of coupons used is not continuous and typically has a lower bound of 0. The distribution may be skewed to the right with a large number of transactions involving no coupons and a decreasing frequency as the number of coupons increases.
2. The weights of pumpkins are likely to be normally distributed because they are a result of many different factors, such as genetics, soil quality, water intake, etc., which tend to produce a bell-shaped curve when combined. This is an example of a continuous measurement that can be modeled well with a normal distribution.
3. The number of raisins in each 8-oz box is likely to be normally distributed due to the central limit theorem. Although the distribution of raisins per box might be uniform or have some other distribution, when the sample size (number of raisins per box) is large, the distribution of the sample mean (total number of raisins in many boxes) will approach a normal distribution. Since an 8-oz box contains a large number of raisins, the count in each box should be approximately normal.
4. The amount of time customers spend waiting in the checkout line is likely not to be normally distributed. This is because the waiting time is bounded below by zero and may have an upper limit depending on the store's operating hours or customer patience. The distribution of waiting times is often skewed to the right, with many customers experiencing short waits and a few experiencing longer waits. This results in a distribution with a long tail on the right side, which is not characteristic of a normal distribution.
(05.06)
Which of the following points lie in the solution set to the following system of inequalities?
y ≤ x − 5
y ≥ −x − 4
A) (−5, 2)
B) (5, −2)
C) (−5, −2)
D) (5, 2)
Answers
the answer for this question is (B) 5,-2
Given the similarity statement ΔDEF ∼ ΔXYZ, which angle corresponds with ∠Z?
A. ∠F
B. ∠Y
C. ∠E
D. ∠D
Answers
Answer:
F
Step-by-step explanation:
In congruent triangles, or similar triangles, the letters in the same place will also be congruent
Answer:
the answer is A
Step-by-step explanation:
PLEASE HELP ASAP!!! CORRECT ANSWER ONLY PLEASE!!!
An organization of berry farmers releases a study reporting that people who eat blueberries every day have a lower cholesterol level.
Which statement describes the best conclusion to draw from the study?
Answers
Answer:
C
Step-by-step explanation:
I would say that the study reveals that there may be a correlation between eating blueberries and a lower cholesterol level.
The best conclusion from the study is that the consumption of blueberries might be associated with lower cholesterol levels. However, it's important to note that correlation does not mean causation and other factors could be at play.
Explanation:The statement that comes closest to an appropriate conclusion from the study is that the possible consumption of blueberries might be associated with lower cholesterol levels. However, it is important to remember that correlation does not necessarily imply causation. While the organization of farmers showed a connection between eating blueberries and lower cholesterol, this does not necessarily mean eating blueberries directly causes lower cholesterol. Other factors could be involved. For instance, those who eat blueberries daily might generally lead healthier lifestyles, contributing to lower cholesterol levels.
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A rectangular prism has a volume of 2320.5 in.? a width of 12.75 inches and a height of 14 inches what is the length?
Answers
Answer:
13
Step-by-step explanation:
Volume = length x width x height
2320.5 = l x 12.75 x 14
2320.5 = l x 178.5
2320.5/178.5 = l
l = 13
Find the period of the function. y=3 sin x/8
Answers
Answer:
The period of given function is [tex]Period = 16\pi [/tex]
So, Option B is correct.
Step-by-step explanation:
In this question we need to find the period of the function y= 3 sin x/8
The formula used to find period of function is: [tex]\frac{2\pi }{b}[/tex]
We need to know the value of b.
To find the value of b we compare the standard equation with the equation of function given.
Standard Equation: y = a sin(bx - c) +d
Given Equation: y= 3 sin(x/8)
Comparing we get:
a= 3
b= 1/8
c= 0
d=0
So, we get the value of b i.e 1/8. Putting it in the formula to find period of given function.
[tex]Period = \frac{2\pi }{b}[/tex]
[tex]Period = \frac{2\pi }{\frac{1}{8}}[/tex]
Solving,
[tex]Period = 2\pi *8[/tex]
[tex]Period = 16\pi [/tex]
So, the period of given function is [tex]Period = 16\pi [/tex]
What's the area of a circle with radius 18 units?
A. 36π units2
B. 18π units2
C. 9π units2
D. 324π units2
Answers
option D is the answer.!!!!
Answer:
324π units² (Answer D)Step-by-step explanation:
The formula for the area of a circle of radius r is A = πr².
Here, the area is A = π(18 units)² = 324π units² (Answer D)
Let f(x)= cos(2x) +e^(-x). for what value of x on the interval (0,3) will f have the same instantaneous rate of change as the average rate of change of f over the interval?
Answers
[tex]f(x)[/tex] is continuous on [0, 3] and differentiable on (0, 3), so the mean value theorem applies here. It says that there is some [tex]c[/tex] in the open interval (0, 3) such that
[tex]f'(c)=\dfrac{f(3)-f(0)}{3-0}[/tex]
We have
[tex]f'(x)=-2\sin2x-e^{-x}[/tex]
so
[tex]-2\sin2c-e^{-c}=\dfrac{\cos6+e^{-3}-2}3\implies c\approx1.5418[/tex]
Evaluate the surface integral. s y ds, s is the helicoid with vector equation r(u, v) = u cos(v), u sin(v), v , 0 ≤ u ≤ 6, 0 ≤ v ≤ π.
Answers
Compute the surface element:
[tex]\mathrm dS=\|\vec r_u\times\vec r_v\|\,\mathrm du\,\mathrm dv[/tex]
[tex]\vec r(u,v)=(u\cos v,u\sin v,v)\implies\begin{cases}\vec r_u=(\cos v,\sin v,0)\\\vec r_v=(-u\sin v,u\cos v,1)\end{cases}[/tex]
[tex]\|\vec r_u\times\vec r_v\|=\sqrt{\sin^2v+(-\cos v)^2+u^2}=\sqrt{1+u^2}[/tex]
So the integral is
[tex]\displaystyle\iint_Sy\,\mathrm dS=\int_0^\pi\int_0^6u\sin v\sqrt{1+u^2}\,\mathrm du\,\mathrm dv[/tex]
[tex]=\displaystyle\left(\int_0^\pi\sin v\,\mathrm dv\right)\left(\int_0^6u\sqrt{1+u^2}\,\mathrm du\right)[/tex]
[tex]=\dfrac23(37^{3/2}-1)[/tex]
Solve the quadratic equation by factoring. 9x^2+24x+20=4
A. x=4/3
B. x=-3/4
C. x=4
D. x=-4/3
Answers
D. -4/3
Let's solve your equation step-by-step.
9x2+24x+20=4
Step 1: Subtract 4 from both sides.
9x2+24x+20−4=4−4
9x2+24x+16=0
Step 2: Factor left side of equation.
(3x+4)(3x+4)=0
Step 3: Set factors equal to 0.
3x+4=0 or 3x+4=0
x= −4 /3
Hope This helps
Answer:
D. x = [tex]-\frac{4}{3}[/tex]
Step-by-step explanation:
Step 1: Subtract 4 from both sides.
9x^2 + 24x + 20 − 4 = 4 − 4
9x^2 + 24x + 16 = 0
Step 2: Factor left side of equation.
(3x+4)(3x+4) = 0
Step 3: Set factors equal to 0.
3x+4=0 or 3x+4=0
x = [tex]-\frac{4}{3}[/tex]
I hope this helps your assignment! Anyways, have a fantastic day! :D
Lines a and b are parallel and lines e and f are parallel.
What is the value of x?
Answers
Answer:
the answer is 82
Step-by-step explanation:
The answer would be 82
If a certain negative number is multiplied by six, the result is the same as 20 less than the original number. What is the value of the original number?
Answers
The original negative number in the problem is found by setting up the equation 6x = x - 20 and solving for x, which results in the original number being -4.
Explanation:If a certain negative number is multiplied by six, the result is the same as 20 less than the original number. To solve for the original number, we can set up an equation based on the given condition. Let's assume the original number is x.
According to the problem, 6 times x equals x subtracted by 20:
6x = x - 20
To solve for x, we'll first move all terms involving x to one side of the equation by subtracting x from both sides:
6x - x = -20
This simplifies to:
5x = -20
Now, divide both sides by 5 to isolate x:
x = -20/5
x = -4
Therefore, the value of the original number is -4.
Chucky grabbed 121212 items in the grocery store that each had a different price and had a mean cost of about \$7.41$7.41dollar sign, 7, point, 41. One of the items was an entire wheel of cheese that cost \$39.99$39.99dollar sign, 39, point, 99. [Show data] \$1.29dollar sign, 1, point, 29 \$1.92dollar sign, 1, point, 92 \$3.19dollar sign, 3, point, 19 \$3.79dollar sign, 3, point, 79 \$3.99dollar sign, 3, point, 99 \$4.79dollar sign, 4, point, 79 \$5.19dollar sign, 5, point, 19 \$5.29dollar sign, 5, point, 29 \$5.49dollar sign, 5, point, 49 \$6.75dollar sign, 6, point, 75 \$7.19dollar sign, 7, point, 19 \$39.99dollar sign, 39, point, 99 Chucky then decided to put the wheel of cheese back and only buy the other 111111 items. How will removing the wheel of cheese affect the mean and median?
Answers
Answer:
Both the mean and median will decrease, but the mean will decrease more than the median.
Step-by-step explanation:
Removing the wheel of cheese will decrease the median a little bit, because the median shifts from between two data points to the lower of the two data points:
With the wheel of cheese, the median is the middle number, but there's no middle number in this data set! So, to find the median we take the mean of the two middle numbers, $4.79 and $5.19 which is $4.99.
Without the wheel of cheese,
Removing the wheel of cheese will decrease the mean significantly, because the total cost will decrease by $39.99, and the number of items decreases by only 1.
Answer: B
Step-by-step explanation:
40 points PLEASE HURRY!!!
Noya needs to determine the number of books that will fit into a box. If the box has a length of 18 inches and each book is 2/9 of an inch thick, which expression can be used to determine the number of books that will fit into the box?
A. 18 divided by 2/9
B. 18 x 2/9
C.2/9 divided by 18
D.9/2 divided by 18
Answers
Answer:
A. 18 divided by 2/9
Step-by-step explanation:
To determine the number of boos that will fit into a box that is 18 inches long, we divide the length of the box by the length of the books
18 inches divided by 2/9 inches per book
This tells us how many books
copy dot flip
18 * 9/2
81 books
Find the number of ways to listen to 4 different CDs from a selection of 15 cds?
A. 1355
B. 2730
C. 360,360
D. 32,760
Answers
Answer: letter a should be the correct answer
Step-by-step explanation:
D is the answer for this question
shelby's monthly periodic rate is 1.95%. what is the APR?
Answers
Monthly periodic rate = 1.95%
The APR is the annual periodic rate
The annual periodic rate is the monthly rate multiplied by 12, because there are 12 months in a year
APR = monthly periodic rate X number of months in a year
= 1.95% X 12
= 23.4%
By using the concept of Annual Periodic rate and Monthly periodic rate,
Annual periodic rate of Shelby = 23.4%
What is Annual Periodic Rate and Monthly periodic rate?
At first it is important to know about Periodic rate.
Periodic rate is the rate that can be charged on loans for a certain period.
If the periodic rate is calculated over a month then it is called monthly periodic rate.
If the periodic rate is calculated over a year then it is called Annual periodic rate.
Here,
Monthly periodic rate of Shelby = 1.95%
Annual periodic rate of Shelby = [tex]1.95 \times 12[/tex]
= 23.4%
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Which of the following conclusions can be made based on the scatterplot shown below?
A.) There is a positive correlation between plant growth and the time spent in light.
B.) There is a negative correlation between plant growth and the time spent in light.
C.) There is no correlation between plant growth and the time spent in light.
Answers
A) There is a positive correlation between plant growth and the time spent in light.
Hope this helps chu
Have a great day
The correlation coefficient helps us to know how strong is the relation between two variables. The correct option is A.
What is the correlation coefficient?The correlation coefficient helps us to know how strong is the relation between two variables. Its value is always between +1 to -1, where, the numerical value shows how strong is the relation between them and, the '+' or '-' sign shows whether the relationship is positive or negative.
1 indicates a strong positive relationship.-1 indicates a strong negative relationship.A result of zero indicates no relationship at all, therefore, independent variable.Since in the given scatterplot as the value of the percent of the time the plant is exposed to light is increased there is a simultaneous growth in the height of the plant in inches.
Therefore, For the given scatterplot the conclusion that can be made is that there is a positive correlation between plant growth and the time spent in light.
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The model represents an equation. What value of x makes the equation true?
A)
3
4
B)
9
2
C) −
3
4
D) −
9
2
Answers
Answer:
B) 9/2
Step-by-step explanation:
Subtracting 5x+7 from both sides leaves 2x = 9.
Dividing the 9 into two parts, we find ...
x = 9/2
Answer: 9/2 your welcome i got a 100% on the test by the way
Step-by-step explanation:
Hi I would really iike if someone helped me with this homework that I have I don't understand. an aquarium in the shape of a parallelepiped ABCDEFGH length AB = 80cm, width AD = 40cm, height DH = 50cm. a) we put HM = x cm give a frame for x b) give, according to x, the volume of water contained in the aquarium. This volume will be expressed in cm ^ 3 then in liters.
Answers
170^3 this is the answer just make sure you do you're work.
What is an equivalent form of the function f(x) = x^2 +8x+15 that reveals the zeros of the function? A. y= (x-3)^2-5 B. f(x) = (x-3)^2-5 C. f(x) = (x+3)(x+5)
Answers
ANSWER
C. f(x) = (x+3)(x+5)
EXPLANATION
The given function is
[tex]f(x) = x^2 +8x+15[/tex]
The factored form of the function reveals the zeros of the function.
From the factored form, we apply the zero product principle to get the zeros.
From the given options, the factored form of the polynomial is
[tex]f(x) = (x+3)(x+5)[/tex]
Help Plz ASAP!!!!!
Arianna's register shows a deposit on 9/5 in the amount of $310.25 an ATM withdrawal on 9/8 in the amount of $60.00 and check #149 on 9/14 in the amount of $240.67. No other Transactions were made
What should the balance be in Arianna register? Use this bank statement
A.-$32.63
B.$448.72
C.$587.88
D.877.82
Answers
Answer:
Step-by-step explanation:
Observe in the attached image that the final balance in your account on 9/1 is $578.30.
Then:
*On 9/6 he makes a payment of $140.30
*On 9/8, he makes a withdrawal from an ATM of $120
This means that:
$140.30 + $120 = $260.30
Then:
*On 9/18 you receive a deposit of $356.22
This means that:
$356.22
So, we have:
Beginning Balance: $578.30.
Total Withdrawals: $260.30
Total Deposits: $356.22
Final Balance = Beginning balance + Total Deposits - Total Withdrawals
Final Balance= $578.30 + $356.22 - $260.30
Final balance= $674.22
Then:
Date Description Debits(-) Credits(+) Balance
9/6 Auto Pay 140.30 438.00
9/8 ATM 120.0 318.00
9/18 Deposit 356.22 674.22
The balance in Arianna's register after the transactions will be $587.88. Then the correct option is C.
What is a bank transaction?
A monetary transfer is a record of money entering and exiting your account and is known as a bank transaction.
Arianna's register shows a deposit on 9/5 in the amount of $310.25 an ATM withdrawal on 9/8 in the amount of $60.00 and checks #149 on 9/14 in the amount of $240.67.
No other Transactions were made.
Then the balance be in Arianna's register will be
→ $ 578.37 + $ 310.25 - $ 60 - $ 240.67
→ $ 587.88
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Please help me out with this
Answers
Answer:
81.75 ft²
Step-by-step explanation:
The area (A) of a trapezoid is calculated using the formula
A = [tex]\frac{1}{2}[/tex] h (a + b)
where a, b are the parallel bases and h is the perpendicular height
Calculate h using the right triangle and the sine ratio
sin30° = [tex]\frac{opposite }{hypotenuse}[/tex] = [tex]\frac{h}{10}[/tex]
Multiply both sides by 10
10 × sin30° = h, thus
h = 5
a = 12 and b = 8.7 + 12 = 20.7, hence
A = [tex]\frac{1}{2}[/tex] × 5 × (12 + 20.7)
= 0.5 × 5 ×32.7
= 81.75 ft²