The minimum diameter for a hyperbolic cooling tower is 76 feet, which occurs at a height of 173 feet. The top of the cooling tower has a diameter of 93 feet, and the total height of the tower is 250 feet. Write the equation for the hyperbola that models the sides of the cooling tower assuming that the center of the hyperbola occurs at the height for which the diameter is least.Round your a and b values to the nearest hundredth if necessary.
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STEP - BY - STEP EXPLANATION
What to find?
The equation for the hyperbola that models the sides of the cooling tower assuming that the center of the hyperbola occurs at the height for which the diameter is least.
Given:
Minimum diameter = 76 feet
Height = 173 feet
Diameter of the top of cooling tower = 93 feet
Total height of tower = 250 feet
Consider the general hyperbolic formula below:
[tex]\frac{x^2}{a^2}-\frac{y^2}{b^2}=1[/tex]But;
2a = 76
⇒ a = 38
x=93/2 =46.5
y=250 - 173 =77
Substitute the values into the formula above and determine the value of b.
[tex]\frac{(46.5)^2}{38^2}-\frac{77^2}{b^2}=1[/tex][tex]b=109.18[/tex]Now substitute the values a= 38 and b=109.18 into the general formula
[tex]\frac{x^2}{38^2}-\frac{y^2}{109.18^2}=1[/tex]ANSWER
[tex]\frac{x^2}{38^2}-\frac{y^2}{109.18^2}=1[/tex]
Related Questions
???can anyone please help???
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Because he wants to mantain a constant rate over the first hours, we can see that on the first hour he should travel 26 miles.
How many miles should he travel in the first hour?The total race is 150 miles, if we discount the last 20 miles (that Jackson will bike as fast as he can) we get:
150mi - 20mi = 130mi
We know that he travels these 130 miles at a constant rate over 5 hours, so the distance that he moves each hour (an particularly the first hour) is given by the quotient between the distance and the time.
R = 130mi/5h = 26mi/h
So he should travel 26 miles in the first hour.
Learn more about constant rates:
https://brainly.com/question/25598021
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The table below, from the Statistical Abstract of the United States, showsamusement park attendance at the top 15 amusement parks for given years. Findthe equation of the line that best fit the data.Year (x) | ,Amusement Park Attendance at Top 15 Amusement Parks (in thousands) (y)2009 | 107,3482010 | 109,3212011 | 112,5092012 | 116,4202013 | 119,951
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Year (x) | Attendance (in thousands) (y)
2009 | 107,348
2010 | 109,321
2011 | 112,509
2012 | 116,420
2013 | 119,951
Using Excel the next equation is obtained:
[tex]y=3230.5x-6\cdot10^6[/tex]PLS HELP ME I BEG OF YPU PLS PLS
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Answer: Y= 2x + 4
Step-by-step explanation:
The line is going up which means it is positive, giving it a positive slope of 2x. It intersects with the Y-axis at (0, 4), giving it a positive Y-int of 4. You can use rise over run to find the slope; pick two easy points and count how many units up and over it is to the next point!
A county fair sells adult admission passes, child admission passes, and ride tickets. One family paid $29 for two adult passes,three child passes, and nine ride tickets. Another family paid $19 for one adult pass, two child passes, and eight ride tickets. A third family paid $51 for three adult passes,five child passes, and twenty one ride tickets. Find the individual costs of an adult pass,a child pass, and a ride ticket? Show all work
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x= adult passes
y= child passes
z=ride tickets
the first family:
[tex]2x+3y+9z=29\text{ (1)}[/tex]the sencond family
[tex]x+2y+8z=19\text{ (2)}[/tex]third family
[tex]3x+5y+21z=51\text{ (3)}[/tex]now we have the 3 equations, and we can solve x, y and z
for the equation of the second family we have:
[tex]x=-2y-8z+19\text{ (4)}[/tex]reeplace the new equation(4) in (1), we have:
[tex]2(-2y-8z+19)+3y+9z=29[/tex][tex]-4y-16z+38+3y+9z=29[/tex][tex]y+7z=9\text{ (5)}[/tex]reeplace (4) in (3)
[tex]3(-2y-8z+19)+5y+21z=51[/tex][tex]-6y-24z+57+5y+21z=51[/tex][tex]-y-3z=-6[/tex][tex]y+3z=6\text{ (6)}[/tex]with 5 and 6, we have a 2x2 equation
that we can solve easier
solving 5 and 6, we have:
[tex]\begin{gathered} y+7z=9 \\ y=9-7z\text{ (7)} \end{gathered}[/tex]reeplace 7 in 6
[tex]\begin{gathered} 9-7z+3z=6 \\ 4z=3 \\ z=\frac{3}{4}=0.75 \end{gathered}[/tex]now we find y, reeplace z in (7)
[tex]\begin{gathered} y=9-7(0.75) \\ y=9-5.25 \\ y=3.75 \end{gathered}[/tex]and finally we can find x, reeplacing y and z in (4)
[tex]\begin{gathered} x=-2y-8z+19 \\ x=-2(3.75)-8(0.75)+19 \\ x=-7.5-6+19 \\ x=5.5 \end{gathered}[/tex]Jackson has a points card for a movie theater.He receives 55 rewards points just for signing upHe earned 12.5 points for each visit to the movie theaterHe needs at least 210 points for a free movie ticketWrite and solve an inequality which can be used to determine x, the number of visits Jackson can make to earn his first free movie ticket
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Given data:
The given reward is 55.
The point earned by visit is 12.5.
The number of point for the movie ticket 210.
The given expression for the inequality is,
55+12.5x ≥ 210
12.5x ≥ 155
x ≥12.4
Thus, the minimum number of visit is 13.
Suppose that the functions p and q are defined as follows.p(x) = -2x-1q(x)=x²+1Find the following.(q*p)(-2)=(p*q)(-2)=
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Answer:
(a)10
(b)-11
Explanation:
Given the function p(x) and q(x) defined as follows:
[tex]\begin{gathered} p(x)=-2x-1 \\ q(x)=x^2+1 \end{gathered}[/tex]Part A
[tex]\begin{gathered} (q\circ p)(x)=q(p(x)) \\ =(-2x-1)^2+1 \\ (q\circ p)(-2)=(-2(-2)-1)^2+1 \\ =(4-1)^2+1 \\ =3^2+1 \\ (q\circ p)(-2)=10 \end{gathered}[/tex]Part B
[tex]\begin{gathered} (p\circ q)(x)=p(q(x)) \\ =-2(x^2+1)-1 \\ (p\circ q)(-2)=-2((-2)^2+1)-1 \\ =-2(4+1)-1 \\ =-2(5)-1=-10-1 \\ (p\circ q)(-2)=-11 \end{gathered}[/tex]Find the slope of the line.
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I’m struggling with my homework assignment, can anyone help me?
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We can find the inverse function writing f(x) as y in the original function and then changing x with y and isolating y again, so
[tex]y=\sqrt[]{x}-5[/tex]Changing y with x we have
[tex]x=\sqrt[]{y}-5[/tex]Now we must get y on one side of the equation again, then
[tex]\begin{gathered} x=\sqrt[]{y}-5 \\ x+5=\sqrt[]{y} \\ (\sqrt[]{y})^2=(x+5)^2 \\ y=(x+5)^2 \end{gathered}[/tex]The domain of the inverse function in the image of the original function f, the image of f is x ≥ -5, then the domain of the inverse of will be x ≥ -5, so our answer is
[tex]f^{-1}(x)=(x+5)^2,x\ge-5[/tex]138°12 CSolve for the area of the sector, to the nearest tenth.14.528.90 173.4452.4
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Given Data:
The central angle is, θ = 138.
The radius is, r = 12 cm.
The area of the sector can be calculated as,
[tex]A=\frac{\theta}{360}\times\pi\times r^2[/tex]Substituting the values, the area can be calculated as,
[tex]A=\frac{138}{360}\times3.14\times12^2=173.4[/tex]In a canoe race, Team A is traveling 6 miles per hour and is 2 miles ahead of Team B. Team B is also traveling 6 miles per hour. The teams continue traveling at their current rates for the remainder of the race. Using d for distance (in miles) and t for time (in hours), write a system of linear equations that represent this situation.Equation for Team A:Equation for Team B:Will team B catch up to Team Ao Yeso No
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First of all, we need to remember the equation of the speed and time:
distance = speed x time
Now, for the first team (team A):
[tex]\begin{gathered} x=6\frac{miles}{hour}\cdot t+2; \\ x\text{ - distance} \\ t\text{ - time} \end{gathered}[/tex]After that, for the second team (team B):
[tex]x=6\frac{miles}{hour}\cdot t[/tex]Finally, the team B never catch up to team A!
Nancy needs at least 1000 gigabytes of storage to take pictures and videos on her upcoming vacation. She checks and finds that she has 105 GB available on her phone. She plans on buying additional memory cards to get the rest of the storage she needs. The cheapest memory cards she can find each hold 256 GB and cost $10. She wants to spend as little money as possible and still get the storage she needs. Let C represent the number of memory cards that Nancy buys. 1) Which inequality describes this scenario? Choose 1 answer: 105 + 100 < 1000 105 + 100 > 1000 105 + 2560 < 1000 105 + 256C 1000
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The total GB is 1000
The available GB is 105 GB.
Each 256 GB and cost $10.
Let C be the number of memory cards, then we have,
[tex]105+256C\ge1000[/tex]Thus, option D is correct.
find the slope of the line. (-3,0), (2,2), (7,4), (12,6)
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find the slope of the line.
(-3,0), (2,2), (7,4), (12,6)
To find the slope we need two points
so
we take
(-3,0), (2,2)
so
m=(2-0)/(2+3)
m=2/5
Verify with the other two points
(7,4), (12,6)
m=(6-4)/(12-7)
m=2/5 ----> is ok
therefore
the slope is 2/5Options for the first box: inverse and direct Options: 275, 50, 5,000, 13,750 Options for the third box: $275.00, $137.50, $550.00
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• The proportional situation represents an Inverse Variation
,• The constant of variation, k = 13750
,• If all 100 students participate in the fundraiser, each will contribute $137.5
Explanation:From the given information, we notice that the more students involved in the fundraiser, the less the amount each student needs to contribute.
This is an INVERSE VARIATION
Let s represent the number of students who participated in the fundraiser, and t represents the amount needed to be contributed by each student, we have:
[tex]\begin{gathered} s\propto\frac{1}{t} \\ \\ \Rightarrow s=\frac{k}{t} \\ \\ OR \\ st=k \end{gathered}[/tex]To find k, we use the information that s = 50 when t = 275
So,
[tex]\begin{gathered} k=50\times275 \\ =13750 \end{gathered}[/tex]From the above, we have the formula:
[tex]st=13750[/tex]If 100 students participate in the fundraiser, we have:
[tex]\begin{gathered} 100t=13750 \\ t=\frac{13750}{100}=137.5 \end{gathered}[/tex]Each student needs to contribute $137.5
Unsure of this one, I need it explained with the answer
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ANSWER
k = 1 or 21
STEP-BY-STEP EXPLANATION:
According to the question, we were given the below trigonometric function
[tex]\sec ^2x\text{ - 22tanx + 20 = 0}[/tex]Recall that, we have trigonometric identity which is written below
[tex]\sin ^2\theta+cos^2\theta\text{ = 1}[/tex][tex]\text{Divide through by }\cos ^2\theta[/tex][tex]\begin{gathered} \frac{\sin^2\theta}{\cos^2\theta}\text{ + }\frac{cos^2\theta}{\cos^2\theta}\text{ =}\frac{1}{\cos ^2\theta} \\ \tan ^2\theta+1=sec^2\theta \\ \text{Let x = }\theta \\ \tan ^2x+1=sec^2x \end{gathered}[/tex]The next thing is to rewrite the equation
[tex]\begin{gathered} \text{ since sec}^2x=tan^2x\text{ + 1} \\ \text{Hence,} \\ \tan ^2x\text{ + 1 - 22tanx + 20 = 0} \\ \text{Let k = tanx} \\ k^2\text{ + 1 -22k + 20 = 0} \\ \text{Collect the like terms} \\ k^2\text{ - 22k + 21 = 0} \end{gathered}[/tex]The next thing is to find the value of P by factorizing the above equation.
Recall that, the standard form of the quadratic function is given as
[tex]ax^2\text{ + bx + c = 0}[/tex]Let
a = 1
b = -22
c = 21
The next thing is to find the value of ac
[tex]\begin{gathered} ac\text{ = 1 }\cdot\text{ 2}1 \\ ac\text{ = 2}1 \end{gathered}[/tex][tex]\begin{gathered} k^2\text{ - k -21k + 21 =0} \\ k(k\text{ -1) -21(}k\text{- 1) = 0} \\ (k\text{ -1) (k -21) = 0} \\ k\text{ -1 = 0 or =k - 21 = 0} \\ k\text{ = 1 or k = 22} \end{gathered}[/tex]Hence, the value of k is 1 or 21
The area of a rectangle is x^2+14x+24. What is the length when the width is x+2?
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we have that:
[tex]\frac{x^2+14x+24}{x+2}=\frac{(x+2)(x+12)}{x+2}=x+12[/tex]so the length is x+12
round each number to the nearest ten, hundred, and thousand5,999
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SOLUTIONS
Round each number to the nearest ten, hundred, and thousand
5,999
[tex]5999=6000\text{ \lparen nearest thousand\rparen}[/tex][tex]5999=6000\text{ \lparen nearest ten\rparen}[/tex][tex]5999=6000\text{ \lparen nearest hundred\rparen}[/tex]please help with this problem I have a test in a few minutes which be about this kind of topic but I don't Understand
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We want to find f(2) for the following function
f(x) = 2x² + 3x
This means that we want to find the value of f(x) when x = 2. So, we replace all the x by 2:
f(x) = 2x² + 3x
f(2) = 2(2)² + 3 · 2
Since 2² = 4 and 3 · 2 = 6 then
f(2) = 2(2)² + 3 · 2
f(2) = 2 · 4 + 6
f(2) = 8 + 6
f(2) = 14
Answer: f(2) = 14Simplify the following polynomials. All final answers must be in standard form!-3x(4x + 12)
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Given the expression:
[tex]-3x(4x\text{ + 12)}[/tex]Let's simplify the expression by multiplying -3x to each of the terms inside the parenthesis.
We get,
[tex]-3x(4x\text{ + 12)}[/tex][tex](-3x)(4x)\text{ + (12)}(-3x)[/tex][tex](-12x^2)\text{ + (-36x)}[/tex][tex]-12x^2\text{ - 36x}[/tex]Therefore, the simplified form of the given expression is -12x^2 - 36x.
the length of the longer leg of a right triangle is 3 ft more than three times the length of the shorter leg. the length of the hypotenuse is 4 ft more than three times the length of the shorter leg. find the side lengths of the triangle.
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with the pythagorean theorem
[tex]\begin{gathered} (4+3x)^2=x^2+(3+3x)^2 \\ 16+24x+9x^2=x^2+9+18x+9x^2 \\ 16+24x+9x^2=10x^2+18x+9 \\ 16+24x+9x^2-9=10x^2+18x+9-9 \\ 9x^2+24x+7=10x^2+18x \\ 9x^2+24x+7-18x=10x^2+18x-18x \\ 9x^2+6x+7=10x^2 \\ 9x^2+6x+7-10x^2=10x^2-10x^2 \\ -x^2+6x+7=0 \end{gathered}[/tex]using the formula of the quadratic equation
[tex]\begin{gathered} x_{1,\: 2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a} \\ x1=\frac{-6+\sqrt{6^2-4\left(-1\right)\cdot\:7}}{2\left(-1\right)}=-1 \\ x2=\frac{-6-\sqrt{6^2-4\left(-1\right)\cdot\:7}}{2\left(-1\right)}=7 \end{gathered}[/tex]the length cannot be negative, therefore x=7
length of the shorter leg is: 7ft
length of the longer leg is: 3+3(7)= 24ft
length of the hypotenuse is: 4+3(7)= 25ft
"10 more than one-fourteenth of some number, w" can be expressed algebraically as
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To express the statement algebraically, we will break down the sentence into phrases or words we can interprete numerically:
the unknown number = w
one fourteenth = 1/14
one fourteenth of some number w:
[tex]\begin{gathered} =\frac{1}{14}\times w \\ =\text{ }\frac{w}{14} \end{gathered}[/tex]10 more: it means we will be adding 10 to the algebraic expression after
10 more than one-fourteenth of some number w will be:
[tex]=10\text{ + }\frac{w}{14}[/tex]In ΔKLM, the measure of ∠M =90°, the measure of ∠K=,86°, and MK = 86 feet. Find the length of KL to the nearest tenth of a foot.
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ΔKLM is a right triangle, the following measures are known:
∠M= 90º
∠K=86º
MK= 86 feet
The side KL is the hypotenuse of the right triangle. To determine its measure you have to apply the trigonometric ratios. In this case, we know one of the angles of the triangle and the length of the side next to the angle, the trigonometric ratio that relates the adjacent side and the hypotenuse is the cosine:
[tex]\cos \theta=\frac{adjacent}{hypotenuse}[/tex]From this expression, you can calculate the measure of the hypotenuse.
[tex]\cos 86=\frac{86}{x}[/tex]-Multiply both sides of the expression by x to take the term out of the denominators place:
[tex]\begin{gathered} x\cos 86=x\cdot\frac{86}{x} \\ x\cos 86=86 \end{gathered}[/tex]-Divide both sides of the expression by cos 86 to reach the value of x:
[tex]\begin{gathered} x\cdot\frac{\cos86}{\cos86}=\frac{86}{\cos 86} \\ x=\frac{86}{\cos 86} \\ x=1232.86 \\ x\approx1232.9ft \end{gathered}[/tex]The length of KL is equal to 1232.9ft
Use calculus to find the dimensions of a rectangle with area of 196 square-feet that has the smallest perimeter.
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Explanation
In the question, we are given that the area of the rectangle is;
[tex]\text{Area}=196\text{ square fe}et[/tex]Recall that the area and perimeter of a rectangle are given by the formulas below.
[tex]\begin{gathered} \text{Area = Lenth x Width = L}\times W \\ \text{Perimeter = 2(L+W)} \end{gathered}[/tex]From the area of the rectangle, we can isolate the variable of the width.
[tex]\begin{gathered} \text{Area}=\text{ L x W} \\ W=\frac{\text{Area}}{L} \\ W=\frac{196}{L} \end{gathered}[/tex]Therefore, the formula for the perimeter is transformed to give;
[tex]\begin{gathered} \text{Perimeter = 2( L + }\frac{\text{196}}{L}) \\ \text{Simplifying the expression gives;} \\ P=2(\frac{L^2+196}{L}) \\ P=\frac{2L^2+392^{}}{L} \\ P=2L+392L^{-1} \end{gathered}[/tex]Recall, via the rules of differentiation
[tex]\begin{gathered} \text{for y = x}^n \\ \frac{dy}{dx}=nx^{n-1} \end{gathered}[/tex]Therefore,
[tex]\begin{gathered} \frac{dP}{dL}=2-392L^{-2}^{} \\ \text{But }\frac{dP}{dL}=0 \\ 0=2-392L^{-2} \\ 392^{}L^{-2}=2 \\ \frac{1}{L^2}=\frac{2}{392}^{} \\ L^2=\frac{392}{2} \\ L^2=196 \\ L=\sqrt[]{196} \\ L=14 \end{gathered}[/tex]Since
[tex]W=\frac{196}{L}=\frac{196}{14}=14[/tex]Answer: Length = 14 and Width = 14
Is the following pair of vectors Parallel, Perpendicular/Orthogonal or Neither?m = < 1 , 5 > n = < 3 , 15 >
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1) To find out we need to calculate the dot product of those two vectors
[tex]\begin{gathered} m\cdot n=\mleft\langle1,5\mright\rangle\cdot\mleft\langle3,15\mright\rangle=1\cdot3+5\cdot15=3+75=78 \\ \end{gathered}[/tex]Since these vectors have a dot product different than zero, then they are not Orthogonal.
2) Let's now check if they are perpendicular, calculating the norm of each one and the angle between them:
[tex]\begin{gathered} \mleft\|m\mright\|=\sqrt[]{1^2+5^2}=\sqrt[]{26} \\ \|n\|=\sqrt[]{3^2+15^2}=\sqrt[]{9+225}=\sqrt[]{234} \end{gathered}[/tex]And finally the angle theta between them:
[tex]\begin{gathered} \theta=\cos ^{-1}(\frac{u\cdot v}{\|m\|\cdot\|n\|}) \\ \theta=\cos ^{-1}(\frac{78}{\sqrt[]{26}\cdot\sqrt[]{234}}) \\ \theta=0 \end{gathered}[/tex]3) Since the angle is 0, these vectors are parallel since parallel vectors for 0º or 180º
Solve for the height h in this right triangle. Show all steps and round your answer to thenearest hundredth.h39°875 feet
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Step 1
Name all sides
Side facing given angle is the opposite = h
Side facing right angleis the hypotanuse
The other side is the adjacent = 875 feet
[tex]\begin{gathered} \tan \theta\text{ = }\frac{opposite}{\text{adjacent}} \\ \theta\text{ = 39} \end{gathered}[/tex][tex]\begin{gathered} \tan 39\text{ = }\frac{h}{875} \\ 0.809\text{ = }\frac{h}{875} \\ \text{Cross multiple} \\ h\text{ = 0.809 x 875} \\ h\text{ = 708.561} \\ h\text{ = 708.56 } \end{gathered}[/tex]Use the graph to find the appropriate solutions to the equation
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The first thing that you should notice is that you have two functions:
[tex]f(x)=\text{ -}2|x\text{ -}3|+1;\text{ and }g(x)=\text{ -}2\sqrt{x\text{ -}1}[/tex]Then their respective graphs are given by the blue and the red functions in your picture. Now, the main idea, is to know the points where they are equal and in that way, you will have the initial equation as follows:
[tex]f(x)=\text{ -}2|x\text{ - }3|+1=\text{ -}2\sqrt{x\text{ - }1}\text{ }=g(x)[/tex]The equation will be satisfied in terms of their graphs at the points where they intersect themselves, at the x-coordinate.
The intersection points A and B at the x-coordinate are the answer for the equation. Then
[tex]x\approx1.7\text{ or }x\approx5.7[/tex]Question 5Points 3A model rocket is projected straight upward from the ground level according to theheight equation h =-16f2 + 144t, t> 0, where h is the height in feet and t is the time inseconds. At what time is the height of the rocket maximum and what is that height?
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Solution
Step 1:
Write the equation:
[tex]h\text{ = -16t}^2\text{ + 144t}[/tex]Step 2
[tex]At\text{ maximum height, }\frac{dh}{dt}\text{ = 0}[/tex]Step 3:
[tex]\begin{gathered} h\text{ = -16t}^2\text{ + 144t} \\ \\ \frac{dh}{dt}\text{ = -32t + 144} \\ \\ 32t\text{ = 144} \\ t\text{ = }\frac{144}{32} \\ t\text{ = 4.5} \end{gathered}[/tex]Step 4
Substitute t = 4.5 into the height equation.
[tex]\begin{gathered} h\text{ = -16 }\times\text{ 4.5}^2\text{ + 144 }\times\text{ 4.5} \\ h\text{ = -324 + 648} \\ \text{h = 324} \end{gathered}[/tex]Hello, may I please have help with this problem. Thank you.
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Hello! We can solve this exercise using proportionality.
Let's look at the triangles:
In the smallest, there are two sides with measurement which equals 16.
In the biggest, there are the same sides but with another measurement: 20.
Knowing that we know that the biggest triangle follows the same structure as the smallest, but a teeny bit bigger, right?
So, as we can say that they follow the same proportionality, let's equal them:
[tex]\begin{gathered} \frac{16}{20}=\frac{24}{n} \\ \\ \text{multiplying accross, we will get:} \\ 16.n=24.20 \\ 16n=480 \\ n=\frac{480}{16} \\ n=30 \end{gathered}[/tex]So, n = 30.
Another way:
we know that the same side before measured 16 and now measures 20, so we can write the proportion: 16/20.
If we simplify this fraction we will get 4/5, or in decimal, 0.8.
Now, we will divide the previous measure of the long side by this obtained proportion:
[tex]\frac{24}{0.8}=30[/tex]Find f(5) for f(x)= 2(2)*O A. 32B. 2O C. 4D. 8
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Given function is
[tex]2^x[/tex]Fo
What is 1584 in terms of pi?
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Multiply. Write the result in standard form.(2 + 1)(3 − 5)
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To multiply, we'll use all the terms in the 1st bracket and multiply all the terms in the 2nd bracket;
[tex](2x+1)(3x-5)=6x^2-10x+3x-5=6x^2-7x-5[/tex]So, the required expression is 6x^2 - 7x -5