We need to find the probability of the card chosen at random is yellow or brown.
So, since there are 3 yellow cards and 7 brown cards, the total numbers of cards that are yellow or brown is:
[tex]3+7=10[/tex]Now, the probability that the chosen card is yellow or brown can be found by dividing the above value by the total number of cards in the box.
The total number of cards in the box is:
[tex]3+7+2=12[/tex]Thus, that probability is given by:
[tex]\frac{10}{12}=\frac{10\div2}{12\div2}=\frac{5}{6}[/tex]Therefore, the answer is:
[tex]\frac{5}{6}[/tex]Type the correct answer in each box.1020PX1150Parallel lines pand gare cut by two non-parallel lines, mand n, as shown in the figure.►gmnThe value of xisdegrees, and the value of y isdegrees.ResetNext
EXPLANATION
Given the parallel lines that are cutted by two non-parallel lines, m and n, the supplementary angle to 102 degrees is by the supplementary angles theorem 180-102= 78 degrees.
By the alternate interior angles theorem, the value of x is 78 degrees.
Also, by the corresponding angles theorem, the value of y is 115 degrees.
First use the Pythagorean theorem to find the exact length of the missing side. Then find the exact values of the six trigonometric functions for angle 0.
The trigonometric functions are given by the following formulas:
[tex]\begin{gathered} \sin \theta=\frac{a}{h} \\ \cos \theta=\frac{b}{h} \\ \tan \theta=\frac{a}{b} \\ \cot \theta=\frac{b}{a} \\ \sec \theta=\frac{h}{b} \\ \csc \theta=\frac{h}{a} \end{gathered}[/tex]Where we call a to the opposite leg to the angle θ (the side whose measure equals 20), b is the adjacent leg to angle θ (the side whose measure equals 21) and we call h to the hypotenuse (the larger side, whose measure equals 29).
By replacing 20 for a, 21 for b and 29 for h into the above formulas, we get:
[tex]\begin{gathered} \sin \theta=\frac{20}{29} \\ \cos \theta=\frac{21}{29} \\ \tan \theta=\frac{20}{21} \\ \csc \theta=\frac{29}{20} \\ \sec \theta=\frac{29}{21} \\ \cot \theta=\frac{21}{20} \end{gathered}[/tex]The area of a rectangular garden is 289 square feet. The garden is to be enclosed by a stone wall costing $22 per linear foot. The interior wall is to be constructed with brick costing $9 per linear foot. Express the cost C, to enclose the garden and add the interior wall as a function of x.
the area of the rectangular garden is 289 square ft
so
[tex]x\times y=289[/tex]so the value of y is 289/x
the outer perimeter of the garden is 2(x+y)
now perimeter is 2(x+289/x)
it is given that the outer wall cost 22 $ per linear foot
so the total cost is
[tex]\begin{gathered} 22\times2(x+\frac{289}{x}) \\ 22\times(2x+\frac{578}{x}) \end{gathered}[/tex]it is given that the cost of an interior wall is 9 $
and the length of the interior wall is x
the total cost of the interior wall is 9x
so the total cost of the wall is 9x +22 (2x + 578/x).
and the correct answer is 9x +22 (2x + 578/x). option B.
A board game that normally costs $30 is on sale for 25 percent off. What is the sale price of the game?
$22.50
$27.50
$32.50
$37.50
what is the answer to this? 3√5+15√5
what is the answer to this? 3√5+15√5
we have
3√5+15√5=18√5
answer is 18√5O A. 1376 square inchesO B. 672 square inchesO C. 1562 square inchesO D. 936 square inches
The seat back cushion is a cuboid. The surafce area can be calculated below
[tex]\begin{gathered} l=26\text{ inches} \\ h=5\text{ inches} \\ w=18\text{ inches} \\ \text{surface area=2(}lw+wh+hl\text{)} \\ \text{surface area=}2(26\times18+18\times5+5\times26) \\ \text{surface area=}2(468+90+130) \\ \text{surface area=}2\times688 \\ \text{surface area}=1376inches^2 \end{gathered}[/tex]7. Simplify(6x + y)s
6xs + ys
Explanations:The given expression is:
(6x + y)s
This can be simplified by simplying expanding the brackets
The equation then becomes:
6xs + ys
Answer:
6xs + ys
Step-by-step explanation:
Do they have the same value? Is +3 equal to -3 and -10 equal to +10? Why?
+3 and -3 do not have the same value
+10 and -10 do not have the same value
Explanation:+3 is a positive number while -3 is a negative number
+3 ≠ -3 (Since one is positive and the other is negative)
The difference between +3 and -3 = 3 - (-3) = 6
Therefore, +3 and -3 do not have the same value
+10 is a positive number while -10 is a negative number
+10 ≠ -10 (Since one is positive and the other is negative)
The difference between +10 and -10 = 10 - (-10) = 20
Therefore, +10 and -10 do not have the same value
decide whether the events are independent or dependent and explain your answer.-drawing a ball from a lottery machine, not replacing it, and then drawing a second ball.
If the probability of an event is unaffected by other events, it is called an independent event. If the probability of an event is affected by other events, then it is called a dependent event.
A ball is drawn from a lottery machine. Then, a second ball is drawn without replacing the first ball. Let T be the number of balls in the lottery machine initially. Before the first ball is drawn, the number of balls in the machine is T. At the time the second ball is drawn, the number of balls in the machine is T-1. From T-1 balls, the second ball is drawn. So, the event of drawing the second ball is affected by the event of drawing the first ball.
Therefore, the event of drawing a ball from a lottery machine, not replacing it, and then drawing a second ball is a dependent event.
find 2 numbers if their ratio is 9:11 and their difference is 6 the numbers can be _, _ or _, _ HELP ASAP
Answer:
27: 33
you could also do -27 and -33 ig
Step-by-step explanation:
That's the only one possible.
Answer:
The only two numbers that your ratio is 9:11 and their differences is 6 are:
33 and 27
Step-by-step explanation:
9a = 11b Eq. 1
a - b = 6 Eq. 2
From Eq. 2:
a = 6 + b Eq. 3
Replacing Eq. 3 in Eq. 1:
9(6+b) = 11b
9*6 + 9*b = 11b
54 + 9b = 11b
54 = 11b - 9b
54 = 2b
54/2 = b
27 = b
From Eq. 3:
a = 6 + 27
a = 33
Check:
From Eq. 1:
9*33 = 11*27 = 297
A circular pool measures 12 feet across. One cubic yard of concrete is to be used to create a circular border of uniform width around the pool. If the border is to have a depth of 6 inches, how wide will the border be?
SOLUTION:
Step 1:
In this question, we are given the following:
A circular pool measures 12 feet across.
One cubic yard of concrete is to be used to create a circular border of uniform width around the pool.
If the border is to have a depth of 6 inches, how wide will the border be?
Step 2:
From the question, we can see that:
[tex]6\text{ inches = 0. 5 feet}[/tex][tex]1\text{ cubic yard = 3 ft x 3ft x 3ft = }27ft^3[/tex][tex]\begin{gathered} \text{Let the radius of the pool = ( 6+x ) feet} \\ \text{Let the width of the concrete that is used to } \\ \text{create the circular border = 6 feet} \end{gathered}[/tex][tex]\text{Let the depth of the border = 6 inches = }\frac{6}{12}=\text{ 0. 5 inches}[/tex]Step 3:
[tex]\begin{gathered} U\sin g\text{ } \\ \pi R^2h\text{ - }\pi r^2\text{ h = 27} \\ \pi(6+x)^2\text{ 0. 5 - }\pi(6)^2\text{ 0. 5 = 27} \\ \text{0. 5}\pi(x^2\text{ + 12x + 36 - 36 ) = 27} \\ 0.\text{ 5 }\pi(x^2\text{ + 12 x) = 27} \\ \text{Divide both sides by 0. 5 }\pi\text{ , we have that:} \end{gathered}[/tex][tex]x^2\text{ + 12 x - (}\frac{27}{0.\text{ 5}\pi})=\text{ 0}[/tex]Solving this, we have that:
CONCLUSION:
From the calculations above, we can see that the value of the x:
( which is the width of the border ) = 1. 293 feet
(correct to 3 decimal places)
This figure shows two similar polygons; DEFG∼TUVS. Find the value of x.
According to the question, both polygons are similar. It means you can use proportions to find the value of x.
[tex]\frac{DE}{TU}=\frac{EF}{UV}[/tex]Replace for the given values in the picture
[tex]\begin{gathered} \frac{x}{6}=\frac{4}{12} \\ x=\frac{4}{12}\cdot6 \\ x=2 \end{gathered}[/tex]x has a value of 2.
Write a recursive formula for the following sequence. You are welcome to submit an image of handwritten work. If you choose to type then use the following notation to indicate terms; a_n and a_(n-1). To earn full credit be sure to share all work/calculations and thinking.a_n = { \frac{3}{5}, \frac{1}{10}, \frac{1}{60}, \frac{1}{360} }
Answer:
[tex]a_n=a_{n-1}\left(\frac{1}{6}\right)[/tex][tex]a_n=\frac{3}{5}\left(\frac{1}{6}\right){}^{n-1}[/tex]
Explanation:
we can see for the fractions with 1 as the numerator that the denominator is multiplied by 6 and the numerator remains the same, that corresponds to multiply the previous fraction by 1/6 and when verifying with the first fraction we observe that applies for all the terms.
i inserted a picture of the questioncan you state whether the answer is A, B, C OR D
Looking at the triangles, they are both right triangles. They have congruent legs = 12. They have congruent acute angles of 45 degerees. Thus, they are congruent triangles. The answer is True
x = 3y for y how should we solve it
If x=3y is the equation then y = x/3.
What is Equation?Two or more expressions with an Equal sign is called as Equation.
The given expression x equal to three y.
Here x and y are two variables.
The value of x is three times of y.
The value of y is x over three. If we know the value of x we can substitute in place of x and we can calculate it.
Divide both sides by 3.
y=x/3.
Hence the value of y is x/3.
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PLEASE HELP! *not a test, just a math practice that I don't understand.
1) Let's analyze those statements according to the Parallelism Postulates/Theorems.
8) If m∠4 = 50º then m∠6 =50º
Angles ∠4 and ∠6 are Alternate Interior angles and Alternate Interior angles are always congruent
So m∠4 ≅ m∠6
9) If m∠4 = 50, then m∠8 =50º
Angles ∠4 and ∠8 are Corresponding angles and Corresponding angles are always congruent
10) If m∠4 = 50º, then m∠5 =130º
Angles ∠4 and ∠5 are Collateral angles and Collateral angles are always supplementary. So
≅
Graph the function f(x) = 4 sin(-2x) on the graph below
Answer:
Explanation:
Here, we want to plot the graph of f(x)
The general equation of a sine graph is:
[tex]y\text{ = A sin (Bx + C) + D}[/tex]where A is the amplitude of the curve
B is -2
C is 0
D is 0
Mathematically, the period of the graph and B are related as follows:
[tex]\begin{gathered} \text{Period = }\frac{2\pi}{|B|} \\ \\ Period\text{ = }\frac{2\pi}{2} \\ \\ \text{Period = }\pi \end{gathered}[/tex]What this means is that the distance between two peaks on the graph is pi
We have the plot as follows:
A net of arectangular pyramidis shown. Therectangular base haslength 24 cm andwidth 21 cm. Thenet of the pyramidhas length 69.2 cmand width 64.6 cm.Find the surfacearea of the pyramid.
Solution
The Image will be of help
To find x
[tex]\begin{gathered} x+24+x=69.2 \\ 2x+24=69.2 \\ 2x=69.2-24 \\ 2x=45.2 \\ x=\frac{45.2}{2} \\ x=22.6 \end{gathered}[/tex]To find y
[tex]\begin{gathered} y+21+y=64.6 \\ 2y+21=64.6 \\ 2y=64.6-21 \\ 2y=43.6 \\ y=\frac{43.6}{2} \\ y=21.8 \end{gathered}[/tex]The diagram below will help us to find the Surface Area of the Pyramid
The surface area is
[tex]SurfaceArea=A_1+2A_2+2A_3[/tex]To find A1
[tex]A_1=24\times21=504[/tex]To find A2
[tex]\begin{gathered} A_2=\frac{1}{2}b\times h \\ 2A_2=b\times h \\ 2A_2=21\times22.6 \\ 2A_2=474.6 \end{gathered}[/tex]To find A3
[tex]\begin{gathered} A_3=\frac{1}{2}bh \\ 2A_3=b\times h \\ 2A_3=24\times21.8 \\ 2A_3=523.2 \end{gathered}[/tex]The surface Area
[tex]\begin{gathered} SurfaceArea=A_1+2A_2+2A_3 \\ SurfaceArea=504+474.6+523.2 \\ SurfaceArea=1501.8cm^2 \end{gathered}[/tex]Thus,
[tex]SurfaceArea=1501.8cm^2[/tex]
[tex]x \geqslant - 2[/tex]
PLEASE HELP!!
A)
B)
C)
D)
Answer:
B
Step-by-step explanation:
[tex]x\geq -2[/tex] means that [tex]x[/tex] can be all values that are greater than -2, and the line under the inequality sign adds that [tex]x[/tex] can be equal to it as well.
Since B represents all values of [tex]x[/tex] that are greater than -2 along with -2 itself due to the closed circle, it is the correct answer.
Answer:
it is c i took the test i hope this helps
which expression could be substituted for x in the second equation to find the value of y?
Substitution
We have the system of equations:
x + 2y = 20
2x - 3y = -1
To solve it with the substitution method, we need to solve the first equation for x and substitute it in the second equation.
Subtracting 2y to the first equation:
x = -2y + 20
This expression corresponds to choice B.
What is the area of the shaded region if the radius of the circle is 6 in.
Then, the area of 1/4 of the circle is:
[tex]\begin{gathered} A=\text{ }\frac{\theta}{360}\text{ x }\pi r^2 \\ A=\text{ }\frac{90}{360}\text{ x }\pi r^2 \\ A\text{ = }\frac{1}{4}\pi\text{ 6}^2 \\ A=\text{ 9}\pi \\ \\ \end{gathered}[/tex]The area of the triangle is:
[tex]\begin{gathered} A=\text{ }\frac{b\text{ x h }}{2} \\ A\text{ = }\frac{6\text{ x 6}}{2} \\ A=\text{ 18in}^2 \end{gathered}[/tex]The area of the shaded region is the area of 1/4 of the circle minus the area of the triangle:
[tex]\begin{gathered} A\text{ = 9}\pi\text{ - 18 in}^2 \\ A=\text{ 28.27in}^2\text{ - 18in}^2 \\ A=\text{ 10.27in}^2 \end{gathered}[/tex]The number of skateboards that can be produced by a company can be represented by the function f(h) = 325h, where h is the number of hours. The total manufacturing cost for b skateboards is represented by the function g(b) = 0.008b2 + 8b + 100. Which function shows the total manufacturing cost of skateboards as a function of the number of hours? g(f(h)) = 325h2 + 80h + 100 g(f(h)) = 3425h + 100 g(f(h)) = 845h2 + 2,600h + 100 g(f(h)) = 2.6h2 + 2,600h + 100
The function which shows the total manufacturing cost of skateboards as a function of the number of hours is; g(f(h)) = 845h2 + 2,600h + 100.
Which function shows the manufacturing cost as a function of number of hours?It follows from the task content that the function which shows the manufacturing cost as a function of the number of hours be determined.
Since, the number of skateboards is given in terms of hours as; f(h) = 325h and;
The manufacturing cost, g is given in terms of the number of skateboards, b manufactured;
The function instance which represents the manufacturing cost as a function of hours is; g(f(h)).
Therefore, we have; g(f(h)) = 0.008(325h)² + 8(325h) + 100.
Hence, the correct function is; g(f(h)) = 845h2 + 2,600h + 100.
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Evaluate the function: g(x)=-x+4Find: g(b-3)
The given function is:
[tex]g(x)=-x+4[/tex]Value of :
[tex]g(b-3)=?[/tex][tex]\begin{gathered} g(x)=-x+4 \\ x=b-3 \\ g(b-3)=-(b-3)+4 \\ g(b-3)=-b+3+4 \\ g(b-3)=7-b \end{gathered}[/tex]so the g(b-3) is 7-b.
An online company is advertising a mixer on sale for 25 percent off the original price for 260.99. What is the sale price for the mixer . Round your answer to the nearest cent , if necessary.
$195.74
1) We can find out the sale price for the mixer, by writing out an equation:
In the discount factor 1 stands for 100% and 25% =0.25
2) So we can calculate it then this way:
[tex]\begin{gathered} 260.99(1-0.25)= \\ 260.99\text{ (0.75)=}195.74 \\ \end{gathered}[/tex]Note that we have rounded it off to the nearest cent 195.7425 to 195.74 since the last digit "4" is lesser than 5, we round it down.
3) So the price of that mixer, with a discount of 25% (off) is $195.74
Alternatively, we can find that price by setting a proportion:
0.25 = 1/4
Writing out the ratios we have:
260.99 --------- 1
x ---------------- 1/4
Cross multiplying it we have:
260.99 x 1/4 = x
x=65.2475
Subtracting that value 25% (65.2475) from 260.99 we have:
260.99 - 65.2475 =195.7425 ≈ 195.74
Find the exact value of the expression. No decimal answers. Show all work.Hint: Use an identity to expand the expression.
Given the expression:
[tex]\cos (\frac{\pi}{4}+\frac{\pi}{6})[/tex]You can expand it by using the following Identity:
[tex]\cos \mleft(A+B\mright)\equiv cos(A)cos(B)-sin(A)sin(B)[/tex]You can identify that, in this case:
[tex]\begin{gathered} A=\frac{\pi}{4} \\ \\ B=\frac{\pi}{6} \end{gathered}[/tex]Then, you can expand it as follows:
[tex]\cos (\frac{\pi}{4}+\frac{\pi}{6})=cos(\frac{\pi}{4})cos(\frac{\pi}{6})-sin(\frac{\pi}{4})sin(\frac{\pi}{6})[/tex]By definition:
[tex]\cos (\frac{\pi}{4})=\frac{\sqrt[]{2}}{2}[/tex][tex]\cos (\frac{\pi}{6})=\frac{\sqrt[]{3}}{2}[/tex][tex]\sin (\frac{\pi}{4})=\frac{\sqrt[]{2}}{2}[/tex][tex]\sin (\frac{\pi}{6})=\frac{1}{2}[/tex]Then, you can substitute values:
[tex]=(\frac{\sqrt[]{2}}{2})(\frac{\sqrt[]{3}}{2})-(\frac{\sqrt[]{2}}{2})(\frac{1}{2})[/tex]Simplifying, you get:
[tex]\begin{gathered} =(\frac{\sqrt[]{2}}{2})(\frac{\sqrt[]{3}}{2})-(\frac{\sqrt[]{2}}{2})(\frac{1}{2}) \\ \\ =\frac{\sqrt[]{6}}{4}-\frac{\sqrt[]{2}}{4} \end{gathered}[/tex][tex]=\frac{\sqrt[]{6}-\sqrt[]{2}}{4}[/tex]Hence, the answer is:
[tex]\frac{\sqrt[]{6}-\sqrt[]{2}}{4}[/tex]If there are 3 possible outcomes for event A, 5 possible outcomes for event B, and 2 possible outcomes for event C, how many possible outcomes are there for event A & event B & event C? Note that these three events are independent of each other. The outcome of one event does not impact the outcome of the other events.
Possible outcomes for events A and events B and events C which are independent of each other is equal to 3/100.
As given in the question,
Total number of outcomes = 10
Possible outcomes of event A =3
P(A) =3/10
Possible outcome of event B =5
P(B) =5/10
Possible outcome of event C =2
P(C)=2/10
A, B, C are independent of each other
P(A∩B∩C) = P(A) × P(B) × P(C)
= (3/10) × (5/10) × (2/10)
= 3/100
Therefore, possible outcomes for events A and events B and events C which are independent of each other is equal to 3/100.
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the x intercept of a functions is called?
In this case, the answer is very simple:
x
I have the answers for the first two but now I'm just confused
which of the following terms best describes a group of equations in which at least one equation is nonlinear, all of the equations have the same variables, and all of the equations are used together to solve a problem?a) solution of nonlinear equationb) graph of nonlinear equationc) graph of linear equationsd) system of nonlinear equations
Solution
- The correct answer is "A system of nonlinear equations"
- This is because the definition of a system of nonlinear equations is is a system of two or more equations in two or more variables containing at least one equation that is not linear.
Final Answer
OPTION D
hi i dont understand this question, can u do it step by step?
Problem #2
Given the diagram of the statement, we have:
From the diagram, we see that we have two triangles:
Triangle 1 or △ADP, with:
• angle ,θ,,
,• hypotenuse ,h = AP,,
,• adjacent cathetus, ac = AD = x cm.
,• opposite cathetus ,oc = DP,.
Triangle 2 or △OZP, with:
• angle θ,
,• hypotenuse, h = OP = 4 cm,,
,• adjacent cathetus, ac = ZP = AP/2,.
(a) △ADP: sides and area
Formula 1) From geometry, we know that for right triangles Pitagoras Theorem states:
[tex]h^2=ac^2+oc^2.[/tex]Where h is the hypotenuse, ac is the adjacent cathetus and oc is the opposite cathetus.
Formula 2) From trigonometry, we have the following trigonometric relation for right triangles:
[tex]\cos \theta=\frac{ac}{h}.[/tex]Where:
• θ is the angle,
,• h is the hypotenuse,
,• ac is the adjacent cathetus.
(1) Replacing the data of Triangle 1 in Formulas 1 and 2, we have:
[tex]\begin{gathered} AP^2=AD^2+DP^2\Rightarrow DP=\sqrt[]{AP^2-AD^2}=\sqrt[]{AP^2-x^2\cdot cm^2}\text{.} \\ \cos \theta=\frac{AD}{AP}=\frac{x\cdot cm}{AP}\text{.} \end{gathered}[/tex](2) Replacing the data of Triangle 2 in Formula 2, we have:
[tex]\cos \theta=\frac{ZP}{OP}=\frac{AP/2}{4cm}.[/tex](3) Equalling the right side of the equations with cos θ in (1) and (2), we get:
[tex]\frac{x\cdot cm}{AP}=\frac{AP/2}{4cm}.[/tex]Solving for AP², we get:
[tex]\begin{gathered} x\cdot cm=\frac{AP^2}{8cm}, \\ AP^2=8x\cdot cm^2\text{.} \end{gathered}[/tex](4) Replacing the expression of AP² in the equation for DP in (1), we have the equation for side DP in terms of x:
[tex]DP^{}=\sqrt[]{8x\cdot cm^2-x^2\cdot cm^2}=\sqrt[]{x\cdot(8-x)}\cdot cm\text{.}[/tex](ii) The area of a triangle is given by:
[tex]S=\frac{1}{2}\cdot base\cdot height.[/tex]In the case of triangle △ADP, we have:
• base = DP,
,• height = AD.
Replacing the values of DP and AD in the formula for S, we get:
[tex]S=\frac{1}{2}\cdot DP\cdot AD=\frac{1}{2}\cdot(\sqrt[]{x\cdot(8-x)}\cdot cm)\cdot(x\cdot cm)=\frac{x}{2}\cdot\sqrt[]{x\cdot(8-x)}\cdot cm^2.[/tex](b) Maximum value of S
We must find the maximum value of S in terms of x. To do that, we compute the first derivative of S(x):
[tex]\begin{gathered} S^{\prime}(x)=\frac{dS}{dx}=\frac{1}{2}\cdot\sqrt[]{x\cdot(8-x)}\cdot cm^2+\frac{x}{2}\cdot\frac{1}{2}\cdot\frac{8-2x}{\sqrt{x\cdot(8-x)}}\cdot cm^2 \\ =\frac{1}{2}\cdot\sqrt[]{x\cdot(8-x)}\cdot cm^2+\frac{x}{2}\cdot\frac{(4-x^{})}{\cdot\sqrt[]{x\cdot(8-x)}}\cdot cm^2 \\ =\frac{1}{2}\cdot\frac{x\cdot(8-x)+x\cdot(4-x)}{\sqrt[]{x\cdot(8-x)}}\cdot cm^2 \\ =\frac{x\cdot(6-x)}{\sqrt[]{x\cdot(8-x)}}\cdot cm^2\text{.} \end{gathered}[/tex]Now, we equal to zero the last equation and solve for x, we get:
[tex]S^{\prime}(x)=\frac{x\cdot(6-x)}{\sqrt[]{x\cdot(8-x)}}\cdot cm^2=0\Rightarrow x=6.[/tex]We have found that the value x = 6 maximizes the area S(x). Replacing x = 6 in S(x), we get the maximum area:
[tex]S(6)=\frac{6}{2}\cdot\sqrt[]{6\cdot(8-6)}\cdot cm^2=3\cdot\sqrt[]{12}\cdot cm^2=6\cdot\sqrt[]{3}\cdot cm^2.[/tex](c) Rate of change
We know that the length AD = x cm decreases at a rate of 1/√3 cm/s, so we have:
[tex]\frac{d(AD)}{dt}=\frac{d(x\cdot cm)}{dt}=\frac{dx}{dt}\cdot cm=-\frac{1}{\sqrt[]{3}}\cdot\frac{cm}{s}\Rightarrow\frac{dx}{dt}=-\frac{1}{\sqrt[]{3}}\cdot\frac{1}{s}\text{.}[/tex]The rate of change of the area S(x) is given by:
[tex]\frac{dS}{dt}=\frac{dS}{dx}\cdot\frac{dx}{dt}\text{.}[/tex]Where we have applied the chain rule for differentiation.
Replacing the expression obtained in (b) for dS/dx and the result obtained for dx/dt, we get:
[tex]\frac{dS}{dt}(x)=(\frac{x\cdot(6-x)}{\sqrt[]{x\cdot(8-x)}}\cdot cm^2\text{)}\cdot(-\frac{1}{\sqrt[]{3}}\cdot\frac{1}{s}\text{)}[/tex]Finally, we evaluate the last expression for x = 2, we get:
[tex]\frac{dS}{dt}(2)=(\frac{2\cdot(6-2)}{\sqrt[]{2\cdot(8-2)}}\cdot cm^2\text{)}\cdot(-\frac{1}{\sqrt[]{3}}\cdot\frac{1}{s})=-\frac{8}{\sqrt[]{12}}\cdot\frac{1}{\sqrt[]{3}}\cdot\frac{cm^2}{s}=-\frac{8}{\sqrt[]{36}}\cdot\frac{cm^2}{s}=-\frac{8}{6}\cdot\frac{cm^2}{s}=-\frac{4}{3}\cdot\frac{cm^2}{s}.[/tex]So the rate of change of the area of △ADP is -4/3 cm²/s.
Answers
(a)
• (i), Side DP in terms of x:
[tex]DP(x)=\sqrt[]{x\cdot(8-x)}\cdot cm\text{.}[/tex]• (ii), Area of ADP in terms of x:
[tex]S(x)=\frac{x}{2}\cdot\sqrt[]{x\cdot(8-x)}\cdot cm^2.[/tex](b) The maximum value of S is 6√3 cm².
(c) The rate of change of the area of △ADP is -4/3 cm²/s when x = 2.