Answer:
(a) 144 ft
(b) 2 s
(c) 5 s
(d) 128 ft
Step-by-step explanation:
[tex]h(t) = -16t^2 + 64t + 80[/tex]
Part (a)
To find the turning point of the parabola:
1. differentiate the function:
[tex]\implies h'(t) = -32t + 64[/tex]
2. Set it to zero and solve for t:
[tex]\implies h'(t) =0[/tex]
[tex]\implies -32t + 64=0[/tex]
[tex]\implies 32t = 64[/tex]
[tex]\implies t=2[/tex]
3. Input found value of t back into the function and solve for h:
[tex]\implies h(2) = -16(2)^2 + 64(2) + 80=144[/tex]
Therefore, the maximum height reached by the ball is 144 ft
Part (b)
As found in part (a), the maximum height is reached when t = 2, so the ball reached its maximum height at 2 seconds.
Part (c)
The ball will hit the ground when h(t) = 0
1. Set the function to zero
[tex]\implies h(t)=0[/tex]
[tex]\implies -16t^2 + 64t + 80=0[/tex]
2. Solve for t:
Divide both sides by 16
[tex]\implies -t^2 + 4t + 5=0[/tex]
[tex]\implies t^2 - 4t - 5=0[/tex]
Factor:
[tex]\implies t^2 +t-5t - 5=0[/tex]
[tex]\implies t(t+1)-5(t+1)=0[/tex]
[tex]\implies (t+1)(t-5)=0[/tex]
Therefore:
[tex](t+1)=0 \implies t=-1[/tex]
[tex](t-5)=0 \implies t=5[/tex]
As time is positive, t = 5
So the ball will hit the ground at 5 seconds.
Part (d)
To find the height of the ball after 1 second, input t = 1 into the function and solve for h:
[tex]\implies h(1) = -16(1)^2 + 64(1) + 80=128[/tex]
Therefore, the height of the ball after 1 second is 128 ft
--------------------------------------------------------------------------------------
Parts (a) & (b) - alternative method
Rewrite the function in vertex form by completing the square.
Set the function to zero:
[tex]\implies-16t^2 + 64t + 80=0[/tex]
Subtract 144 from both sides:
[tex]\implies -16t^2+64t-64=-144[/tex]
Factor out -16:
[tex]\implies -16(t^2 - 4t +4)=-144[/tex]
Factor expression in brackets:
[tex]\implies -16(t-2)^2=-144[/tex]
Add 144 to both sides:
[tex]\implies -16(t-2)^2+144=0[/tex]
Therefore:
[tex]\implies h(t)=-16(t-2)^2+144[/tex]
The maximum height is the y-value of the vertex.
The vertex is (2, 144) so the maximum height is 144 ft.
The time the ball reached the max height is the x-value of vertex.
So the time then the ball reached the maximum height is 2 s
Which transformations are needed to change the parent cosine function to y = 0.35 cosine (8 (x minus startfraction pi over 4 endfraction))? vertical stretch of 0.35, horizontal stretch to a period of 16 pi, phase shift of startfraction pi over 4 endfraction units to the right vertical compression of 0.35, horizontal compression to a period of 4 pi, phase shift of startfraction pi over 4 endfraction units to the left vertical compression of 0.35, horizontal compression to a period of startfraction pi over 4 endfraction, phase shift of startfraction pi over 4 endfraction units to the right vertical stretch of 0.35, horizontal stretch to a period of startfraction pi over 4 endfraction, phase shift of startfraction pi over 4 endfraction units to the right
The transformations that are needed to change the parent cosine function to y = 0.35×cos(8(x-π/4)) are:
vertical stretch of 0.35horizontal compression of period of [tex]\pi/4[/tex]phase shift of [tex]\pi/4[/tex] to rightHow does transformation of a function happens?The transformation of a function may involve any change.
Usually, these can be shift horizontally (by transforming inputs) or vertically (by transforming output), stretching (multiplying outputs or inputs) etc.
If the original function is [tex]y = f(x)[/tex], assuming horizontal axis is input axis and vertical is for outputs, then:
Horizontal shift (also called phase shift): Left shift by c units: [tex]y = f(x+c)[/tex]earlier)Right shift by c units: [tex]y = f(x-c)[/tex]output, but c units late)Vertical shift:Up by d units: [tex]y = f(x) + d[/tex]Down by d units: [tex]y = f(x) - d[/tex]Stretching:Vertical stretch by a factor k: [tex]y = k \times f(x)[/tex]Horizontal stretch by a factor k: [tex]y = f(\dfrac{x}{k})[/tex]For this case, we're specified that:
y = cos(x) (the parent cosine function) was transformed to [tex]y = 0.35\cos(8(x-\pi/4))[/tex]
We can see its vertical stretch by 0.35, right shift by [tex]\pi/4[/tex]horizontal stretch by 1/8
Period of cos(x) is of [tex]2\pi[/tex] length. But 1.8 stretching makes its period shrink to [tex]2\pi/8 = \pi/4[/tex]
Thus, the transformations that are needed to change the parent cosine function to y = 0.35×cos(8(x-π/4)) are:
vertical stretch of 0.35horizontal compression to period of [tex]\pi/4[/tex] (which means period of cosine is shrunk to [tex]\pi/4[/tex] which originally was [tex]2\pi[/tex] )phase shift of [tex]\pi/4[/tex] to rightLearn more about transformation of functions here:
https://brainly.com/question/17006186
Answer: vertical compression of 0.35, horizontal compression to a period of StartFraction pi Over 4 EndFraction, phase shift of StartFraction pi Over 4 EndFraction units to the right
Step-by-step explanation:
18 less than the quotient of a number and -5 is greater than 3
Step-by-step explanation:
x = "a number"
x/-5 - 18 > 3
A parallelogram with a right angle that is not a rectangle.
Answer:
One is a rhombus.
Step-by-step explanation:
. SCULPTURE An artist creates two metal sculptures in the shape of regular octagons. A side of the larger sculpture is 3.5 times longer than a side of the smaller octagon. The area of the smaller octagon is 19.28 square inches.
a. What is the area of the larger octagon?
b. The artist is going to pack the sculptures in a cylindrical box to take them to an art show. To the nearest inch, what is the diameter of the smallest box he can use? Explain your reasoning. (Hint: Find the radius of the larger octagon.)
By using what we know about octagons, we will see that:
a) The area is 244.89 in^2b) The diameter is 18.58 inWhat is the area of the larger octagon?
For an octagon of side length S, the area is:
A = 2*(1 + √2)*S^2
In this case, the area of the smaller octagon is:
19.28 in^2 = 2*(1 + √2)*S^2
Solving for S we get:
S = √( 19.28 in^2/(2*(1 + √2))) = 2.03 in
If we apply an enlargement of a factor 3.5, the new side length is:
S' = 3.5*2.03 in = 7.11 in
Then the area of the larger octagon is:
A' = 2*(1 + √2)*(7.11 in)^2 = 244.89 in^2
b) For an octagon of side length S, the radius is:
R = S*√(1 + 1/√2)
Then the radius of the larger octagon is:
R' = (7.11 in)*√(1 + 1/√2) = 9.29 in
Then the diameter of the box must be:
D = 2*R' = 2*9.29 in = 18.58 in
If you want to learn more about octagons, you can read:
https://brainly.com/question/1592456
Please Help! Which of the following methods is not used to prove triangles are congruent?
Answer:
AAA is not a congruency postulate because it could only prove that triangles are similar, not congruent.
Step-by-step explanation:
You have decided to purchase a car for $22,346. 16. The credit union requires a 10% down payment and will finance the balance with a 5. 4% annual interest loan for 36 months. The sales tax in your city is 7. 6%, and the license and title charges are $125. 13. Determine the amount that the credit union will finance you for. Round your answer to the nearest cent. A. $24,179. 10 c. $21,761. 20 b. $24,169. 60 d. $21,752. 64.
The amount that the credit union will finance is $24169.60
We have given that the cost of the car is $22,346. 16
and the credit union required 10% down payment
and sale tax=7.6%
The license and title charges are $125. 13.
We have to determine that the amount that the credit union will
finance Now we have sale tax 7.6% for $22,346. 16
Therefore, we get
[tex]\frac{7.6}{100} \times 22,346. 16=1698.31[/tex]
What is the amount that the credit union finance ?credit union finance=cost of car+sales tax+the license and title charges.
[tex]=22346.16+1698.31+12.13\\=24169.60[/tex]
Therefore, the amount that the credit union will finance is $24169.60
To learn more about the credit union will finance visit:
https://brainly.com/question/15641576
Parisian sewer #1 contains 9 fewer gigantic rats—rats as big as cats, some might say—than parisian sewer #2. parisian sewer #2 contains 5 fewer gigantic rats than parisian sewer #3. if there are 56 gigantic rats in all three sewers combined, how many rats are in sewer #1?
There are different kinds of math problem. There will be 11 rats in sewer #1.
What are word problem?The term word problems is known to be problems that are associated with a story, math, etc. They are known to often vary in terms of technicality.
Lets take
sewer #1 = a
sewer #2 = b
sewer #3 = c
Note that A=B-9
So then you would have:
A=B-9
B=C- 5
A+B+C=56
Then you have to do a substitution so as to find C:
(B- 9) + (C-5) + C = 56
{ (C- 5)-9} + (C-5) + C = 56
3C - 19 = 56
3C = 75
B = C- 5
B = 25 - 5
Therefore, B = 20
A = B - 9
= 25 - 9
=11
Therefore, there are are 11 rats in sewer #1
Learn more about Word Problems from
https://brainly.com/question/21405634
Identify the solution of the inequality |4x − 2| > 14 and the graph that represents it.
The solution to the inequality |4x − 2| > 14 is x > 4 and x < -3
What is an inequality?
An inequality is an expression that shows the non equal comparison of two or more numbers and variables.
Given the inequality:
|4x − 2| > 14
Hence:
4x - 2 > 14 or -(4x - 2) > 14
x > 4 or x < -3
The solution to the inequality |4x − 2| > 14 is x > 4 and x < -3
Find out more on inequality at: https://brainly.com/question/24372553
-4-(3) =
3+(-5) =
4+6=
-5-9=
-4+(-9) =
Step-by-step explanation:
-4-(3)=-7
step two
3+(-5)=-2
step three
4+6=10
step four
-5-9=-14
step five
-4+(-9)=-13
Answer:
a) -4 - ( 3) = - 7
b) 3 + ( -5) = 3 - 5 = -2
c) 4 + 6 = 10
d) -5 -9 = - 14
e) -4 + ( -9) = -4 -9 = - 13...
You make commission working at Footlocker. On Saturday you sold
$1,500 worth of merchandise and made $120 in commission. On
Sunday you sold $1,300 worth of merchandise. If you made the same percent commission,how much did you earn on Sunday?
A. $8
B. $86. 66
C. $104
D. $1,039. 99
Answer:
C. $104
Step-by-step explanation:
First, we need to figure out what percent $120 is of $1500.
120 ÷ 1500 = 0.08 = 8%. So, $120 is 8% of $1500, and this is the percentage that'll be used to calculate Sunday's commission.
0.08 (8%) × 1300 = 104
Thus, you earned $104 on Sunday.
20 points!!!Solid metal support poles in the form of right
cylinders are made out of metal with a density of 5.2
g/cm². This metal can be purchased for $0.35 per
kilogram. Calculate the cost of a utility pole with a
radius of 19 cm and a height of 620 cm. Round your
answer to the nearest cent. (Note: the diagram is not
drawn to scale)
Density is mass per unit volume. The cost of the metals from the calculations performed is $1280.
What is density?Density is defined as the ratio of the mass of an object to its volume. We have the following information;
Mass of the metal = ?
Density of the metal = 5.2 g/cm3
height of the metal = 620 cm
radius of the metal = 19 cm
Volume of a cylinder = πr^2h = 3.142 * (19)^2 * 620 cm = 703242cm^3
Mass = density * volume = 5.2 g/cm3 * 703242cm^3
= 3657 Kg
If the cost is $0.35 per kilogram, then 3657 Kg costs; 3657 Kg * $0.35
= $1280
Learn more about volume: https://brainly.com/question/1578538
The diagram shows a plan for the rectangular house with an attached porch. The combined area of the house and the porch is 733ft2. What is the value of x? Show your work. The numbers are 59ft, 13ft, 6 ft, 7ft, 9ft is the porch, x ft
The value of x in the diagram is 8 feet that makes the combined area of the house and the porch to be 733 ft²
How to calculate area?
Combined area = big rectangle + smaller rectangle - triangle
Hence:
733 = (13 * (59 - x)) + (x * 9) - 0.5(9 - 7) * (x - 6)
733 = 767 - 13x + 9x - x + 6
x = 8 feet
The value of x in the diagram is 8 feet that makes the combined area of the house and the porch to be 733 ft²
Find out more on area at: https://brainly.com/question/25292087
A flat-bottomed ice cream waffle cup is shaped like a 72 mm tall cone with a 42 mm diameter opening at the top, but with the bottom 24 mm of the cone replaced by a flat waffle bottom with a 14 mm diameter. to the nearest square millimeter, what area of waffle does the cone have?
a.) 4002
b.)4398
c.)4552
d.)4157
The area of waffle in the cone is the amount of space covered by the waffle
The cone has 5630 square millimeter of waffle
How to determine the surface area?
The surface area of a cone is calculated using:
A = πr * (r+[tex]\sqrt{[/tex](h^2 + r^2))
For the complete 72 mm tall cone, we have:
Height (h) = 72 mm
Radius (r) = 42 mm/2 = 21 mm
So, the surface area is:
A = πr * (r+[tex]\sqrt{[/tex](h^2 + r^2))
A = 3.142 * 21 * (21+[tex]\sqrt{[/tex](72^2 + 21^2))
Evaluate
A = 3.142 * 21 * (21+75)
A = 6334.272
When the cone is cut at the top, we have:
Height (h) = 24 mm
Radius (r) = 14 mm/2 = 7 mm
So, the surface area is:
A = πr * (r+[tex]\sqrt{[/tex](h^2 + r^2))
A = 3.142 * 7 * (7+[tex]\sqrt{[/tex](24^2 + 7^2))
Evaluate
A = 3.142 * 7 * (7 + 25)
A = 703.808
Calculate the difference (d) between the areas
d = 6334.272 - 2.5*703.808
Evaluate
d = 5630.464
Approximate
d = 5630
Hence, the cone has 5630 square millimeter of waffle
Read more about surface area at:
https://brainly.com/question/6613758
Please answer soon
(-2 • -4)^2 = ?
Answer:
64
Step-by-step explanation:
(-2(-4))^2
8^2
64
hope that helps
Rob goes to a store and buys a camera originally priced at $30. The store is running a discount of 18%. The sales tax rate is 7.4%. What is the final price of the camera?
pls help asap
Answer:
$26.42
Step-by-step explanation:
P = 30 * 0.82 * 1.074 = 26.42
an unknown number is increased by -3. the result is -8
find the unknown number
Answer:
The unknown number is -5.
Step-by-step explanation:
-3 + -5 = -8
-8 - -3 = -5
-8 - -5 = -3
Tom knows that in his school 10 out of every 85 students are left-handed.
There are 391 students in Tom's school.
how many students in Tom's school are left handed?
Answer:
46 are left handed
Step-by-step explanation:
divide 10 by 85 and you get 0.11764705882352941176470588235294 then multiply that by 391 and you get 46
Fine the area of the circle. Round to the nearest tenth. Use 3.14 for pie.
Answer:
the answer is 379.9m²
Step-by-step explanation:
half of 22 is ll time 3.14 equals 379.94. to the nearest tenth is 379.9
y = x + 2
5x - 4y = -3
Solving systems by equations
Answer: x = 5, y = 7
Step-by-step explanation:
First, put the variables together/rearrange:
y - x = 2 & 5x - 4y = -3
Cancel out variable x by multiply the first equation by 5:
5y - 5x = 10 & 5x - 4y = -3
Add equations:
y = 7
Then determine x by plugging y back into the first equation y = x + 2:
x = 5
Which expression is equivalent to (o)(5)?
Of(5) g(5)
Of(5) + g(5)
O5f(5)
05g(5)
Determine the value of y, if x is −8. y=x 2 +4
Step-by-step explanation:
given: x = -8,
putting value of x in given equation,
y=(-8)2+4
y= -16+4
y=-12
4. Let f(x) = x2 - 81. Find f'(x). (1 point)
O
+ Vx+81
09x
w
1
x2 -81
81
Here we are given that ;
[tex]{:\implies \quad \sf f(x)=x^{2}-81}[/tex]
Differentiating both sides w.r.t.x ;
[tex]{:\implies \quad \sf f^{\prime}(x)=\dfrac{d}{dx}(x^{2}-81)}[/tex]
[tex]{:\implies \quad \sf f^{\prime}(x)=\dfrac{d}{dx}(x^{2})-\dfrac{d}{dx}(81)}[/tex]
[tex]{:\implies \quad \sf f^{\prime}(x)=2x^{2-1}-0}[/tex]
[tex]{:\implies \quad \bf \therefore \quad \underline{\underline{f^{\prime}(x)=2x}}}[/tex]
This is the required answer
paul says x/y can result in either positive or negative quotient. is paul correct? explain why or why not.
Answer:
Dividing by same signs give a positive answer.
Dividing by different signs give a negative answer.
Step-by-step explanation:
Quotient is just a term for the answer given by a division problem.
positive/positive = positive
positive/negative = negative
negative/positive = negative
negative/negative = positive
I’m not entirely sure with this question, anyone can please help?
Step-by-step explanation:
3x + 4 = 72 - x
3x - x = 72 - 4
2x = 68
x = 39
What is the distance between (4, 6) and (4, -1) ?
7 units
5 units
4 units
I don't know
The distance between points (4,6) and (4,-1) is 7 units.
Given The coordinates of the end points of line (4,6) and (4,-1).
Distance is the measurement about how far things are present on the surface. It is usually calculated in some units known as km, m, etc.
The distance between two points can be calculated as under:
Distance=[tex]\sqrt{(x2-x1)^{2} +(y2-y1)^{2} }[/tex] where the coordinates are (x1,y1),(x2,y2).
To calculate the distance we need to put the values of coordinates in the formula:
Distance between (4,6) and (4,-1)=[tex]\sqrt{(4-4)^{2} +(-1-6)^{2} }[/tex]
=[tex]\sqrt{(0)^{2}+(-7)^{2} }[/tex]
=[tex]\sqrt{0+49}[/tex]
=[tex]\sqrt{49}[/tex]
=7
Hence the distance between (4,6) and (4,-1) is 7 units.
Learn more about distance at https://brainly.com/question/2854969
#SPJ10
Answer:
it would be 7 units :)
Step-by-step explanation: cuz im smart and said so
You pick a card at random.
3 4 5 6
What is P(not 6)?
Write your answer as a fraction or whole number.
Answer:3 or 3/10
Step-by-step explanation:
Which expression represents 4 fifths t(negative 10 t)? –8t 46 over 5 46 over 5 –8t2
Answer:
is this a multiple choice question? if so can you retype it out to show the separate answer choices because im having trouble reading your question.
Step-by-step explanation:
Answer:
–8t^2 or DStep-by-step explanation:
Took the test and got it right.
Your brand new car costs 16000 dollars. If sales tax is 8%, how much will it truly cost
with sales tax?
Sales tax amount:-
8% of 16000$0.08(16000)8(160)1280$Price:-
16000+128017280$Answer:
$17,280
Step-by-step explanation:
Given
Cost of car = $16,000Sales tax = 8%Solving :
When sales tax is added to the cost, the final cost will be 8% more than 16,000 or 108% of 16,00016,000 x 1.08$17,280 is the actual price of the car with taxSimplest radical form please !!Need answers asap
Answer:
11) [tex]5\sqrt{7}[/tex] yd
12) [tex]2\sqrt{10}[/tex] cm
13) 9 ft
14) [tex]\sqrt{95}[/tex] cm
15) [tex]5\sqrt{5}[/tex] km
16) [tex]\sqrt{39}[/tex] yd
someone, please help me out with these two problems, it's trig.
Answer:
2nd problem
The sum of the angles of the triangle is 180°.
B= 90°