Solve the following inequality: 6p - 15 < 33
Answers
WE are to solve an inequality, so we proceed to isolate the variable "p" on one side of the inequality symbol:
6 p - 15 < 33
we add 15 to both sides:
6 p < 33 + 15
6 p < 48
now divide both sides by 6 (notice that since 6 is a positive number, the division doesn't change the direction of the inequality)
p < 48/6
p < 8
So we need to highlight on the number line, the line that starts at "8" and goes all the way to the left (to minus infinity), and make sure that at the point "8" you draw an "empty" circle to indicate that the number 8 itself is NOT included in your set of solutions.
Related Questions
O A. 1376 square inchesO B. 672 square inchesO C. 1562 square inchesO D. 936 square inches
Answers
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]help meeeeeeeeee pleaseee !!!!!
Answers
The composition will be:
(g o h)(x) = 5*√x
By evaluating in x = 0, we get:
(g o h)(0) = 0
How to evaluate the composition?Here we have the two functions:
g(x) = 5x
h(x) = √x
And we want to get the composition:
(g o h)(x) = g( h(x))
So we need to evaluate g(x) in h(x), we will get:
g( h(x)) = 5*h(x) = 5*√x
And now we want to evaluate this in x = 0, we will et:
(g o h)(0) = 5*√0 = 0
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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.
Answers
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]i inserted a picture of the questioncan you state whether the answer is A, B, C OR D
Answers
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
Find the exact value of the expression. No decimal answers. Show all work.Hint: Use an identity to expand the expression.
Answers
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]Write standard form for the equation of the line: y = 1/2x - 5*
Answers
Explanation
the standard form of a line is in the form Ax + By = C where A is a positive integer, and B, and C are integers
so, we need to write in this form
[tex]Ax+By=C[/tex]Step 1
subtrac 1/2x in both sides
[tex]\begin{gathered} y=\frac{1}{2}x-5 \\ \\ y-\frac{1}{2}x=\frac{1}{2}x-5-\frac{1}{2}x \\ \\ y-\frac{1}{2}x=-5 \\ \text{reorder} \\ -\frac{1}{2}x+y=-5 \end{gathered}[/tex]I hope this helps you
4. Which of the following represent the distance
formula? Select all that apply.
A d = √(x₁-x₂)² + (y₁ − y₂)²
B d = √(x₂− ×₂)² + (⁄₂ − y,}²
C d = √(x₂+x₂)² + (y₂ + y,)²
D d=√√₂-X₁1² + VY₂ − Y₁1²
Answers
A appears to be the only correct answer
a^2+b^2=c^2
you are solving for c when finding distance, so (a^2 + b^2) must be square rooted, as a whole, not separately
and a = (x1-x2)
and b = (y1-y2)
you can flip the 1 and 2 but you have to flip for both x and y
like x1-x2 means you have to do y1-y2
like x2-x1 means you have to do y2-y1
so both above are correct as long as the order of 1 and 2 stays the same for both x and y
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.
Answers
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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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} }
Answers
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.
Preston drove to his new college and then back home.Round trip he traveled 642 miles. Preston drives aHonda Civic and gets 38 miles for every gallon of gas. IfPreston needs to make 15 round trips a year how muchwill it cost him in gas assuming the price of gas stays at$2.48 a gallon for all his trips?$Round all answers to the nearest hundredthsDo not put a label, just the numeric value
Answers
1) Gathering the data
Preston
642 miles
38 miles/gallon
15 round trips
1 gallon = $2.48
2) Considering that each round trip consists of 642 miles
So Preston in 15 roundtrips is going to make
15 x 642 miles =9,630 miles
His car gets 38 miles per gallon. So we can write a proportion for that:
38 miles ---------1 gallon
9,630 miles ----- x
Cross multiplying it:
38x = 9,630 Divide by 38
x =9630/38
x=253.42 gallons
Finally, let's set another proportion to find out the cost of it
1 gallon -------------- $2.48
253.42 -------------- y
y= 253.42 x 2.48
y=628.4816
3) Rounding off to the nearest hundredth
$628. 48 That's how much Preston will spend.
x = 3y for y how should we solve it
Answers
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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Can anyone help? I’ve asked this same question 6 times!
Answers
Answer: 54080
Since the first number cannot be 0 or 1, there would be only 8 possible numbers for the first number. For the second number, we can now have all 10 numbers.
The number of different combinations of numbers would then be:
[tex]8\times10=80[/tex]Then, for the first letter, we have 26 possible letters, as well as the second letter. The number of different combinations of letters would then be:
[tex]26\times26=676[/tex]So, for a license plate that has 2 numbers and 2 letters, where the first number cannot be 0 or 1, there would be:
[tex]8\times10\times26\times26=54080[/tex]Question 2-22
A cake is cut into 12 equal slices. After 3 days Jake has eaten 5 slices. What is his wealty rate of eating the cale
5
36
.
B
TH
cakesliveek
er
9
Answers
Using the concept of Fraction, the weekly rate of Jake eating the cake is 11.2.
What is Fraction?Fraction represents parts of a whole or group of objects. A fraction consists of two parts. The numerator is the number at the beginning of the line. It specifies the number of equal parts taken from the whole or collection. The number below the line is the denominator. It shows the total number of equal parts into which the whole is divided or the total number of identical objects in a collection.
We know that,
The cake is cut into 12 equal slices.
After 3 days Jake eats 5 slices then,
For 1 day = [tex]\frac{5}{3}[/tex]
= 1.6
Then for 7 days,
1.6 × 7 = 11.2
Hence, Jake's weekly rate of eating the cake is 11.2.
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The complete question would be
'A cake is cut into 12 equal slices. After 3 days Jake has eaten 5 slices. What is his weekly rate of eating the cake?'
Graph the function f(x) = 4 sin(-2x) on the graph below
Answers
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:
Do they have the same value? Is +3 equal to -3 and -10 equal to +10? Why?
Answers
+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
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
Answers
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
7. Simplify(6x + y)s
Answers
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:
Calculate Sample Variance for the following data collection: 10, 11, 12, 13, 14,18.
Answers
The Variance of a set of data is defined as the average of the square of the deviation from the mean.
The first step is to calculate the mean of the data.
[tex]\frac{10+11+12+13+14+18}{6}=13[/tex]Now we take the difference from the mean, square it, and then average the result.
[tex]\frac{(10-13)^2+(11-13^2)+(12-13)^2+(13-13)^2+(14-13)^2+(18-13)^2}{6}[/tex][tex]\Rightarrow\frac{9+4+1+0+1+25}{6}[/tex][tex]\Rightarrow6.67[/tex]Hence, the variance of the data is 6.7 (rounded to the nearest tenth)
What is the area of the shaded region if the radius of the circle is 6 in.
Answers
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]Which problems can be solved using the equation 3 X 6 = |? Circle all the correct answers. A Cleo has 3 times as many bananas as Arianna has. Cleo has 6 bananas. A How many bananas does Arianna have? B Dylan has 6 oranges. Jane has 3 times as many oranges as Dylan. В How many oranges does Jane have? C Liam has 3 cherries. Brian has 6 times as many cherries as Liam. How many cherries does Brian have? D Tina has 3 more peaches than Charlie. Charlie has 6 peaches. How many peaches does Tina have? E Jaclyn has 6 times as many grapes as Kaitlyn. Kaitlyn has 3 grapes. How many grapes does Jaclyn have?
Answers
A. Cleo (C) has 3 times as many bananas as Arianna (A) has. Cleo has 6 bananas. How many bananas does Arianna have? Equation:
[tex]\begin{gathered} 3A=C \\ \\ C=6 \\ \\ 3A=6 \\ \\ ?=\frac{6}{3} \end{gathered}[/tex]It doesn't correspond to the given equation
_______________
B. Dylan (D) has 6 oranges. Jane (J) has 3 times as many oranges as Dylan. How many oranges does Jane have? Equation:
[tex]\begin{gathered} D=6 \\ J=3D \\ \\ J=3\times D \\ \\ \text{?}=3\times6 \end{gathered}[/tex]It correspond to the given equation
___________
C. Liam (L) has 3 cherries. Brian (B) has 6 times as many cherries as Liam. How many cherries does Brian have? Equation:
[tex]\begin{gathered} L=3 \\ B=6L \\ \\ \text{?}=6\times3 \end{gathered}[/tex]It correspond to the given equation
_______________
Tina (T) has 3 more peaches than Charlie (C). Charlie has 6 peaches. How many peaches does Tina have? Equation:
[tex]\begin{gathered} T=3+C \\ C=6 \\ \\ T=3+6 \\ \\ \text{?}=3+6 \end{gathered}[/tex]It doesn't correspond to the given equation
___________________
E. Jaclyn (J) has 6 times as many grapes as Kaitlyn (K). Kaitlyn has 3 grapes. How many grapes does Jaclyn have? Equation:
[tex]\begin{gathered} J=6K \\ K=3 \\ \\ J=6\times3 \\ \\ \text{?}=6\times3 \end{gathered}[/tex]It correspond to the given equation
____________
Answer: B, C and EThe 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
Answers
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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Hello! I need some guidance please. Having trouble with which graph is correct
Answers
Given:
[tex]y\ge3x+3[/tex]Required:
to show which graph is correct for the inequality.
Explanation:
Given graph is correct for the equation.
Required answer:
The given graph is correct.
12. Suppose you roll a pair of six-sided dice.(a) What is the probability that the sum of the numbers on your dice is exactly 4? (b) What is the probability that the sum of the numbers on your dice is at most 2? (c) What is the probability that the sum of the numbers on your dice is at least 12?
Answers
Probability is computed as follows:
[tex]\text{probability}=\frac{\text{ number of favorable outcomes}}{\text{ total number of outcomes}}[/tex]When rolling a pair of six-sided dice, the total number of outcomes is 36 (= 6x6)
(a) number of favorable outcomes: 3 (dice: 1 and 3, 2 and 2, 3 and 1)
Then, the probability that the sum of the numbers on your dice is exactly 4 is:
[tex]\text{probability }=\frac{3}{36}[/tex](b) number of favorable outcomes: 1 (dice: 1 and 1)
Then, the probability that the sum of the numbers on your dice is at most 2 is:
[tex]\text{probability }=\frac{1}{36}[/tex](c) number of favorable outcomes: 1 (dice: 6 and 6)
Then, the probability that the sum of the numbers on your dice is at least 12 is:
[tex]\text{probability }=\frac{1}{36}[/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.
Answers
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.
StatusRecovery8Help ResourcessAABC ~ AXYZFind the missing side length, s.B.3 65А&Х-ZCross multiplySE ][?] = [ ]153s
Answers
Since triangles ABC and XYZ are similar, the ratio between their corresponding sides is constant; thus,
[tex]\begin{gathered} \frac{AB}{XY}=\frac{BC}{YZ} \\ \Rightarrow\frac{3}{5}=\frac{6}{s} \end{gathered}[/tex]Solving for s,
[tex]\begin{gathered} \frac{3}{5}=\frac{6}{s} \\ \Rightarrow\frac{3}{5}\cdot s=\frac{6}{s}\cdot s \\ \Rightarrow\frac{3s}{5}=6 \\ \Rightarrow\frac{3s}{5}\cdot5=6\cdot5 \\ \Rightarrow3s=30 \\ \Rightarrow s=\frac{30}{3} \\ \Rightarrow s=10 \end{gathered}[/tex]Thus, the result of the cross multiplication is 3s=30 and the answer is s=10
find 2 numbers if their ratio is 9:11 and their difference is 6 the numbers can be _, _ or _, _ HELP ASAP
Answers
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
[tex]x \geqslant - 2[/tex]
PLEASE HELP!!
A)
B)
C)
D)
Answers
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
I have the answers for the first two but now I'm just confused
Answers
hi i dont understand this question, can u do it step by step?
Answers
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.
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?
Answers
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)