PLEASE HELP MATH CIRCLES

Answer:

16/29

Step-by-step explanation:

P(A∪B) = P(A) + P(B) - P(A∩B)

P(basketball or baseball) = P(basketball) + P(baseball) - P(both)

= (13/29) + (7/29) - (4/29)

= 16/29

The probability that a randomly chosen student plays either sport is 16/29.

Answer:

see below

Step-by-step explanation:

Choose a couple of values for x. Figure out the corresponding values for y. Plot those points and draw a line through them.

Let's choose x=0 and x=4. Then the corresponding y-values are ...

y = 2·0 = 0 . . . . . point (x, y) = (0, 0)

y = 2·4 = 8 . . . . . point (x, y) = (4, 8)

These are graphed below.

Answer:

35 years

Step-by-step explanation:

We have been given an exponential decay function that models the weight of the bones of an extinct dinosaur;

[tex]f(x)=10e^{-0.02x}[/tex]

The initial weight of the bones is;

substitute x with 0 in the function, f(0) = 10 grams

We are required to determine the number of years it will take for the bones to be less than 5 grams. The solution can be achieved either analytically or graphically. I obtained the graph of the function from desmos graphing tool as shown in the attachment below.

From the graph, the bones will weigh 5 grams after approximately 34.65 years. This implies that it will take 35 years for the bones to be less than 5 grams.

Answer:

12

Step-by-step explanation:

SOH CAH TOA reminds you that ...

Sin = Opposite/Hypotenuse

so ...

sin(45°) = (6√2)/x

Your memory of trig functions tells you sin(45°) = 1/√2, so we have ...

1/√2 = (6√2)/x

Multiplying by (√2)x gives ...

x = 6(√2)^2 = 6·2

x = 12

_____

You can simply recognize that this is an isosceles right triangle, so the hypotenuse (x) is √2 times the leg length:

x = (6√2)·√2 = 6·2 = 12

Answer:

[tex]A = 144\ ft[/tex]

Step-by-step explanation:

The area of a triangle is:

[tex]A = 0.5b*h[/tex]

Where b is the base of the triangle and h is the height

In this case we know the hypotenuse of the triangle and the angle B.

Then we can use the sine of the angle to find the side opposite the angle

By definition we know that

[tex]sin (\theta) = \frac{opposite}{hypotenuse}[/tex]

In this case hypotenuse = 24

opposite = b

Then:

[tex]sin (45) = \frac{b}{24}[/tex]

[tex]b= 24*sin(45)[/tex]

[tex]b=12\sqrt{2}[/tex]

Now

[tex]cos(\theta) = \frac{adjacent}{hypotenuse}[/tex]

adjacent = a = h

[tex]cos(45) = \frac{h}{24}[/tex]

[tex]h = 24*cos(45)\\\\h=12\sqrt{2}[/tex]

Then the area is:

[tex]A = 0.5*12\sqrt{2}(12\sqrt{2})\\\\A=144\ ft[/tex]

ANSWER

[tex]Area = 144 {ft}^{2} [/tex]

EXPLANATION

We use the sine ratio to find the missing side.

[tex] \sin(45 \degree) = \frac{AC}{24} [/tex]

[tex]24\sin(45 \degree) = AC[/tex]

[tex]AC = 24 \times \frac{ \sqrt{2} }{2} [/tex]

[tex]AC = 12 \sqrt{2} ft[/tex]

The triangle is a right isosceles triangle.

This implies that,

AC=BC=12√2 ft.

The area of the triangle is:

[tex]Area = \frac{1}{2} bh[/tex]

We substitute the values to get,

[tex]Area = \frac{1}{2} \times 12 \sqrt{2} \times 12 \sqrt{2} [/tex]

[tex]Area = 144 {ft}^{2} [/tex]

Answer:

309 ft

Step-by-step explanation:

In order to solve this I have to assume that the sling shot is ground level.  Since you did not provide an initial height, without making the assumption that it is 0, we cannot solve the problem at all.

The standard form of a quadratic function is

[tex]f(x)=ax^2+bx+c[/tex]

c is the initial height for which we are going to sub in a 0.  Given 2 points, we are going to plug in the y and the x, one point each into 2 quadratic functions, to find the model.  The first coordinate is (1, 121):

[tex]121=a(1)^2+b(1)+0[/tex] and 121 = a + b

The second coordinate is (2, 224):

[tex]224=a(2)^2+b(2)+0[/tex] and 224 = 4a + 2b

Solve the first equation for a:

a = 121 - b

and sub it in for a in the second equation:

224 = 4(121 - b) + 2b and

224 = 484 - 4b + 2b and

-260 = -2b so b = 130.

Now we can sub that in for b and solve for a:

a = 121 - 130 so a = -9.

The equation then that models the motion is

[tex]f(x)=-9x^2+130x[/tex]

Now that we know that, all we have to do now is to find f(3):

[tex]f(3)=-9(3)^2+130(3)[/tex] and

f(3) = 309 ft

____________________________________________________

Answer:

Your answer would be x + y = 1

____________________________________________________

Step-by-step explanation:

In this scenario, we know that the line of CD passes through the coordinates (0,1), and would also be parallel to the equation x + y = 3.

When two lines are parallel, that means that their slopes are equal.

The slope of the line must be:

[tex]x + y = 3[/tex]

Move the x to the other side by subtracting

[tex]y= -x + 3[/tex]

The slope for the equation would be -1, since there is a invisible one after the equal sign. When there's no other number there, it would be 1.

The slope of the line CD would be -1.

Now, we would need to plug in -1 into the equation, to find the standard form.

[tex](y - 1) = m(x - 0)\\\\(y-1)=-1(x)\\\\x+y=1[/tex]

[tex]x + y = 1[/tex] should be your FINAL answer.

____________________________________________________

Answer:

(−∞,−2) and (0,4)

Step-by-step explanation:

The function is positive when it is above the x-axis.  This is when x is less than -2 and when x is between 0 and 4.

This question is asking about the intervals in which a given function is positive. Without knowing the exact function, one would typically evaluate the function in the given intervals to decide whether the result is positive or negative.

The question is asking in which intervals a given mathematical function is positive. Without more specific information about the function, it is impossible to definitively say which of the intervals the function is positive in. Normally, you would evaluate the function at multiple points within the given intervals and analyze the output to determine if the function is positive or negative within those ranges.

For example, if you had the function f(x) = x^2 - 3x + 2, you could plug in a couple of values in the intervals (−∞,−2), (0,4), (4,∞), (−1.5,−1), (2,2.5), (−2, 0) and see whether the output is positive.

Again, without the specific function this exercise is theoretical and is based on your understanding of intervals and their relationship with function values.

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Answer:

  150π in³ ≈ 471.24 in³

Step-by-step explanation:

The formula for the volume of a cylinder is ...

  V = πr²h

The radius is half the diameter, so you have a volume of ...

  V = π(5 in)²(6 in) = 150π in³ ≈ 471.24 in³

Answer:

Equation 1: r = -5 * cos theta

Equation 2: r = 1 – ( 4 * sin theta  )

Step-by-step explanation:

Graph 1:

This graph is a circle along negative x- axis.

General equation for graph:

R = a cos theta          ∴ a =  diameter of circle

From given graph, it is included that:

a = -5  

a/2 = -2.5 (center of circle)

Equation 1: r = -5 cos theta

Graph 2:

This graph is an inner-loop limacon.

The inner-loop limacon is in the downward direction along the negative y-axis

The general equation for the graph will be :

r = a – b sin theta  

a will represent x – intercept, from graph it is included that:

a = { +1, -1  }

For inner-loop on y-axis, b - a = 3     ………….1  

For outer-loop on y-axis, a + b = 5    …………2

Adding both 1 and 2 to find values of a and b

b – a = 3

a + b = 5

2b     = 8     ⇒      b = 4

Putting value of b in 2

a + 4 = 5       ⇒       a = 1

substituting values of a and b in general equation:

Equation 2: r = 1 – 4 sin theta

To find the points where the tangent is horizontal or vertical on the given curve, we find the slope, set it equal to zero or undefined, and solve for t. Then substitute the values of t in the equations to find the corresponding points on the curve.

To find the points on the curve where the tangent is horizontal or vertical, we need to find the slope of the curve and determine when it is zero or undefined. For the given curve x = t^3 - 3t, y = t^2 - 4, we can find the slope dy/dx, set it equal to zero or undefined, and solve for t. Once we have the values of t, we can substitute them back into the equations x = t^3 - 3t and y = t^2 - 4 to find the corresponding points on the curve.

To find the horizontal tangent, we set dy/dx equal to zero:

dy/dx = (dy/dt) / (dx/dt) = (2t) / (3t^2 - 3) = 0

Setting the numerator equal to zero, 2t = 0, we find t = 0. Substituting t = 0 back into the equations x = t^3 - 3t and y = t^2 - 4, we get the point (0, -4).

To find the vertical tangent, we set dx/dt equal to zero:

dx/dt = 3t^2 - 3 = 0

Solving for t, we find t = ±1. Substituting t = 1 and t = -1 back into the equations x = t^3 - 3t and y = t^2 - 4, we get the points (2, -3) and (-2, -3) respectively.

Therefore, the points on the curve where the tangent is horizontal or vertical are (0, -4), (2, -3), and (-2, -3).

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The points on the curve defined by x = t^3 - 3t and y = t^2 - 4 where the tangent is horizontal or vertical are (-2, 3), (0, -4), and (2, 3).

In the subject of Mathematics, specifically calculus, the question is seeking the points on the curve defined by the parametric equations x = t^3 - 3t and y = t^2 - 4 where the tangent is horizontal or vertical. This means we are looking for the values of t where the derivative dy/dx equals 0 (horizontal tangent) or is undefined (vertical tangent).

First, we need to calculate the derivatives dx/dt and dy/dt. dx/dt = 3t^2 - 3 and dy/dt = 2t. Then we can find the overall derivative dy/dx = (dy/dt)/(dx/dt).

For a horizontal tangent, dy/dx = 0, meaning the numerator of our derivative equation must be zero: dy/dt = 2t = 0. This gives us t = 0.

For a vertical tangent, dy/dx is undefined, meaning the denominator of our derivative equation must be zero: dx/dt = 3t^2 - 3 =0. Solving this equation gives us t = -1, 1.

Substitute t = -1, 0, and 1 into x = t^3 - 3t and y = t^2 - 4 to get the points in the (x, y) format. This results in the points: (-2, 3), (0, -4), and (2, 3).

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Answer:

(36,000,000 / 573) - (38,000,000 / 845) = 17856.8 ≈ 17857 people per square mile

Answer:

17857 people per mile²

Step-by-step explanation:

Area of Tokyo, Japan = 845 square miles

Population Tokyo = 38 million

Area of Delhi, India = 573 square miles

Population of Deli = 36 million

Population density of Tokyo = [tex]\frac{38}{845}[/tex] = 0.04497 millions/miles²

Population density of Delhi = [tex]\frac{36}{573}[/tex] = 0.062827 millions/miles²

Difference in population density of Delhi and Tokyo = 0.062827 - 0.04497

                      = 0.17857 million per mile²

or 17857 people per mile² is the answer.

Answer:

6.25

Step-by-step explanation:

I find it convenient to use technology to compute the mean absolute deviation. (see below)

Answer:

(MAD) Mean Absolute Deviation: 6.25

Step-by-step explanation:

Mean: 23 + 28 + 16 + 25 + 18 + 31 + 14 + 37 = 192/8 = 24

24 - 23 = 1

24 - 28 = 4

24 - 16 = 8

24 - 25 = 1

24 - 18 = 6

24 - 31 = 7

24 - 14 = 10

24 - 37 = 13

Mean Absolute Deviation (MAD): 1 + 4 + 8 + 1 + 6 + 7 + 10 + 13 = 50/8 = 6.25

Answer:

75

Step-by-step explanation:

"given that it's a junior" means to only look at juniors.

From the table, under junior, there are 2  males and 6 females. 2 + 6 = 8. The total number of juniors is 8.

p(female given junior) = 6/8 = 3/4 = 0.75 = 75%

Answer: 75

Answer:

75

Step-by-step explanation:

"given that it's a junior" means to only look at juniors.

From the table, under junior, there are 2  males and 6 females. 2 + 6 = 8. The total number of juniors is 8.

p(female given junior) = 6/8 = 3/4 = 0.75 = 75%

Answer: 75

Answer:

Step-by-step explanation:

The mnemonic SOH CAH TOA reminds you that ...

  Sin = Opposite/Hypotenuse

In ΔAHD, the side opposite angle DAH is DH, and the hypotenuse is AD, so we have ...

  sin(∠DAH) = DH/AD = 4/8

  ∠DAH = arcsin(4/8) = 30°

That makes ΔAHD a 30°-60°-90° triangle, so the side lengths have the ratios 1 : √3 : 2.

∠CAB = 2·30° = 60°, so ΔABC is also a 30°-60°-90° triangle having the same ratios of side lengths.

In short, ...

  AH = √3·DH = 4√3

  AC = 2·AH = 8√3

  AB = 2·AC = 16√3

  BC = √3·AC = 8·(√3)² = 24

Answer:

theoretical is 15 each

experimental is Red: 13 Blue: 14 Yellow:18 Green:15

Step-by-step explanation:

the theoretical probability is what should statisticly happen when you do it so if there are for outcomes with an equal chance of occurring then 1 out of every 4 or 1/4 of the time each one should happen so divide 60 by 4 and you get 15

the experimental probability is what happens when someone actually spins it 60 times and in your scenario

RESULTS HERE

Red: 13 Blue: 14 Yellow:18 Green:15

is what happened so that is the experimental probability

Answer:

  6 is in the "hundreds" place

Step-by-step explanation:

The value of the 6 can be found by setting the other digits to zero:

  00,600.00 = 600

The 6 represents six hundred, hence is in the hundreds place.

Answer:

  88.49 units²

Step-by-step explanation:

Use the formula for the area of a sector.

  A = (1/2)r²·θ

where θ is the central angle of the sector in radians, and r is the radius.

Here, the central angle of the sector is 360°-300° = 60° = π/3 radians. Then the area is ...

  A = (1/2)(13)²(π/3) = 169π/6 ≈ 88.49 . . . . units²

To find the area of a sector, use the formula A = (θ/360) × πr². Plug in the provided values for the central angle and the radius. The final answer should carry the same number of significant figures as the radius provided.

To find the area of the sector (A), we will use the formula: A = (θ/360) × πr², where θ represents the sector's central angle in degrees and r the radius of the circle. Suppose you are given that the central angle (θ) is 90° (or π/2 in radians) and the radius (r) is 0.0500 m, as suggested in the provided information.

Plugging these values into the formula, we get A = (90/360) × 3.14(0.0500 m)² = 7.85 × 10-3 m² rounded to two decimal places. Even though the output from the calculator is a number with more digits, [1.11] , we need to make sure our final answer is limited to two significant figures to match the given radius value.

If the radius of the circle was given as 0.800 m (or 80.0 cm), then going through the same process produces an area of 1.26 m² for a one meter length along the curve of the mirror, for instance.

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[tex]5\frac{3}{7}[/tex] Kilometers of thread.

The key to solve this problem is using the rule of three.

We have to change mixed number to improper fraction in order to solve the problem.

A mixed number is a number formed by an integer and a proper fraction (one whose quotient is less than 1).

An improper fraction  is one whose denominator is less than its numerator.

To change a mixed number to an improper fraction:

1. Multiply the whole number by the denominator and add to the numerator.

2. The denominator of the mixed number is unchanged.

It takes [tex]2\frac{1}{4}[/tex] kilometers of thread to make [tex]3\frac{1}{2}[/tex] boxes of shirts. How many kilometers of thread would it take to make 8 boxes?

We need to change [tex]2\frac{1}{4}[/tex] and [tex]3\frac{1}{2}[/tex] to an improper franctions:

[tex]2\frac{1}{4}=\frac{(2)(4)+1}{4}=\frac{9}{4}[/tex]

[tex]3\frac{1}{2}=\frac{(3)(2)+1}{2}=\frac{7}{2}[/tex]

To calculate how many kilometers of thread would it take to make 8 boxes, we use the rule of three:

9/4 Km of thread -------------> 7/2 boxes of shirts

                    x     <-------------  8 boxes of shirts

[tex]x = \frac{(\frac{9}{4})(8)}{\frac{7}{2}}= \frac{19}{\frac{7}{2}}\\x=\frac{38}{7}[/tex]

Convert the improper fraction 38/7 to a mixed number:

1. Divide the numerator by the denominator.

38÷7 = 5 and a remainder of 3

2.  5 become the whole number, the remainder is the numerator, and the denominator is unchanged.

38/7 = 5 3/7

It would take 5 3/7 kilometers of thread make 8 boxes of shirts.

Answer:

D, y = cos(x -π/2)

Step-by-step explanation:

When the cosine function is shifted right by π/2 units, it looks like the sine function. That is what we have here. To shift f(x) to the right, replace x by x-(amount of shift). Here, this means the graph is described by ...

y = 2cos(x -π/2)

_____

The vertical scale factor is 2 on the graph and in all answer choices.

(r^-7)^6 = r^-1 = 1/r

Therefore the answer is D. 1/r

Let me know if you have any questions.

Answer:

B. 1/r^42

Step-by-step explanation:

(r^-7)^6= r^-7*6= r^-42.

As a positive exponent: 1/r^42

Answer:

A) 525,500

B) decreasing by 0.995% per year

C) 430,243

D) After 20 years, the population can be expected to be about 20% smaller.

E) 2009

Step-by-step explanation:

A) t=0 represents the year 2000, so put 0 where t is in the expression and evaluate it. Of course, e^0 = 1, so the y-value is 525.5 thousand, or 525,500.

__

B) Each year, the population is multiplied by e^-0.01 ≈ 0.99004983, or about 1 - 0.995%. That is, the population is decreasing by 0.995% per year.

__

C) t represents the number of years since 2000, so the year 2020 is represented by t=20. Put that value in the equation and do the arithmetic.

y = 525.5·e^(-0.01·20) = 525.5·e^-0.2 ≈ 430.243 . . . . thousands

The population in 2020 is predicted to be 430,243.

__

D) The decrease is about 1% per year, so a rough estimate of the decrease over 20 years is 20%. The population of about 500,000 will decrease by about 100,000 in that time period, so will be about 400,000. The value we calculated is in that ballpark. (The actual decrease is about 18.13%; or about 95.2 thousand.)

__

E) Your working shows the general idea, but you need to remember the numbers in the equation are thousands:

480 = 525.5·e^(-0.01t)

0.913416 = e^(-0.01t) . . . . divide by 525.5

ln(0.913416) = -0.01·t . . . . take the natural log

-100ln(0.913416) = t ≈ 9.06

The population will be 480 thousand after 9 .06 years, in the year 2009.

Answer:

B. 0.43, -0.43

Step-by-step explanation:

The given equation is

[tex]-2=3-7\sqrt[5]{x^2}[/tex]

Combine similar terms to get:

[tex]-2-3=-7\sqrt[5]{x^2}[/tex]

[tex]-5=-7\sqrt[5]{x^2}[/tex]

[tex]\sqrt[5]{x^2}=\frac{5}{7}[/tex]

[tex]x^2=(\frac{5}{7})^5[/tex]

[tex]x^2=\frac{3125}{16807}[/tex]

[tex]x=\pm \sqrt{\frac{3125}{16807}}[/tex]

[tex]x=\pm 0.43[/tex]

[tex]x=0.43[/tex] or [tex]x=-0.43[/tex]

The correct answer is B.

Answer:

B

Step-by-step explanation:

First subtract 3 from the equation:

[tex]-2-3=3-7\sqrt[5]{x^2}-3\\ \\-5=-7\sqrt[5]{x^2}[/tex]

Now divide the equation by -7:

[tex]\sqrt[5]{x^2}=\dfrac{5}{7}[/tex]

Now raise the equation to the 5th power:

[tex]x^2=\left(\dfrac{5}{7}\right)^5[/tex]

Take square root:

[tex]x=\pm \sqrt{\left(\dfrac{5}{7}\right)^5} =\pm \dfrac{25}{49}\sqrt{\dfrac{5}{7}} \\ \\x_1\approx 0.43\\ \\x_2\approx -0.43[/tex]

Answer:

  (a) max P(A∩B) = 1/3; min P(A∩B) = 1/12

  (b) max P(A∪B) = 1; min P(A∪B) = 3/4

Step-by-step explanation:

Let the universal set be the numbers 1–12, U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, each with probability 1/12.

Let event A be any of the numbers 1–9, {1, 2, 3, 4, 5, 6, 7, 8, 9}. If a number is chosen at random from U, the probability of event A is 9/12 = 3/4.

a1) Let event B be any of the numbers 1–4, {1, 2, 3, 4}. If a number is chosen at random from U, the probability of event B is 4/12 = 1/3.

The set A∩B is the numbers 1–4, {1, 2, 3, 4}, so the probability of that event is also 4/12 = 1/3.

In general the maximum value of P(A∩B) will be min(P(A), P(B)). Here, that is min(3/4, 1/3) = 1/3.

__

a2) Let event B be any of the numbers 9–12, {9, 10, 11, 12}. If a number is chosen at random from U, the probability of event B is 4/12 = 1/3. The set A∩B is the number {9}, so the probability of that event is 1/12.

In general, the minimum value of P(A∩B) is max(0, P(A) +P(B) -1). Here, that is max(0, 3/4 +1/3 -1) = 1/12.

__

b1) Let event B be defined as in (a2), the numbers 9–12. Then A∪B is the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, which is equal to the universal set, U. That is, the probability of event A∪B when drawing a number from U is 1.

In general, the maximum value of P(A∪B) is min(1, P(A)+P(B)). Here, that is min(1, 3/4+1/3) = 1.

__

b2) Let event B be defined as in (a1), the numbers 1–4. Then A∪B is the set {1, 2, 3, 4, 5, 6, 7, 8, 9}. If a number is chosen at random from U, the probability of event A∪B is 9/12 = 3/4.

In general, the minimum value of P(A∪B) is max(P(A), P(B)). Here, that is max(3/4, 1/3) = 3/4.

The value of f(-3) for the function f(x) = -6x + 3 is 21. The value of g(3) for the function g(x) = 3x + 21x - 3 is 66.

To find the value of f(-3), you substitute -3 in place of x in the function f(x) = -6x + 3. You get f(-3) = -6(-3) + 3 = 18 + 3 = 21.

To find the value of g(3), we substitute 3 in place of x in the function g(x) = 3x + 21x - 3. This gives g(3)= 3(3) + 21*(3)-3 = 9 + 57 = 66.

So, f(-3) = 21 and g(3) = 66.

To find the value of f(–3) and g(3), we need to substitute the given values into the respective functions.

For f(x) = –6x + 3, substituting x = –3 into the function, we get:

f(–3) = –6(–3) + 3 = 18 + 3 = 21

For g(x) = 3x + 21x – 3, substituting x = 3 into the function, we get:

g(3) = 3(3) + 21(3) – 3 = 9 + 63 – 3 = 69

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For this case we have the following system of equations:

[tex]8x-9y = -122\\-8x-6y = -28[/tex]

To solve, we add both equations:

[tex]8x-8x-9y-6y = -122-28\\-15y = -150\\y = \frac {-150} {- 15}\\y = 10[/tex]

We find the value of "x":

[tex]8x = -122 + 9y\\x = \frac {-122 + 9y} {8}\\x = \frac {-122 + 9 (10)} {8}\\x = \frac {-122 + 90} {8}\\x = \frac {-32} {8}\\x = -4[/tex]

The solution is (-4,10)

ANswer:

Option C

Answer:

No.

Step-by-step explanation:

First, subtract 32-5= 27

Then multiply 5 * 4=20

since 20 is less than 27, she will not have enough.

Answer:

No

Step-by-step explanation:

She already has 5 and the total is 32 so she has to make 28 friendship bracelets.

Bracelets in packages of 4 and she buys 5 packages which is 20 so no, she doesn't have enough bracelets if she buys 5 packages.

Answer:

  24%

Step-by-step explanation:

2610 of the 10730 students are graduates. The probability of choosing a graduate at random from all students is ...

  2610/10730 × 100% ≈ 24.324% ≈ 24%

Answer:

Part 1) The statement is false

Part 2) The statement is false

Part 3) The statement is true

Step-by-step explanation:

Let

h(t)-----> the height of an object launched to the air

t ----> the time in seconds after the object is launched

we have

[tex]h(t)=-16t^{2} +72t[/tex]

Verify each statement

case 1) The factored form of the equation is h(t)=-16(t-4.5)

The statement is false

Because

The factored form is equal to

[tex]h(t)=-16t(t-4.5)[/tex]  

case 2) The object will hit the ground at t=72 seconds

The statement is false

Because

we know that

The object will hit the ground when h(t)=0

substitute in the equation and solve for t

[tex]0=-16t(t-4.5)[/tex]  

so

[tex](t-4.5)=0[/tex]  

[tex]t=4.5\ sec[/tex]  

case 3) The t-value for the maximum of the function is 2.25

The statement is true

Because

Convert the quadratic equation in vertex form

[tex]h(t)=-16t^{2} +72t[/tex]

[tex]h(t)=-16(t^{2} -4.5t)[/tex]

[tex]h(t)-81=-16(t^{2} -4.5t+2.25^{2})[/tex]

[tex]h(t)-81=-16(t-2.25)^{2}[/tex]

[tex]h(t)=-16(t-2.25)^{2}+81[/tex] ---> quadratic equation in vertex form

The vertex is a maximum

The vertex is the point (2.25,81)

To solve this problem, we need to use the z-score formula to standardize the values and then look up the corresponding probabilities in the standard normal distribution table.

To solve this problem, we need to use the z-score formula to standardize the values and then look up the corresponding probabilities in the standard normal distribution table. The z-score formula is given by (X - μ) / σ, where X is the given value, μ is the mean, and σ is the standard deviation. Here are the calculations for each question:

First, we need to calculate the z-score for 61 hours:

z = (61 - 56) / 3.3 = 1.52

Next, we can look up the probability corresponding to a z-score of 1.52 in the standard normal distribution table. The probability of getting a value greater than 1.52 is approximately 0.0655, or 6.55%.

First, we need to calculate the z-score for 53 hours:

z = (53 - 56) / 3.3 = -0.9091

Next, we can look up the probability corresponding to a z-score of -0.9091 in the standard normal distribution table. The probability of getting a value less than or equal to -0.9091 is approximately 0.1814, or 18.14%.

First, we need to calculate the z-scores for 57 hours and 62 hours:

z1 = (57 - 56) / 3.3 = 0.303

z2 = (62 - 56) / 3.3 = 1.82

Next, we can look up the probabilities corresponding to z1 and z2 in the standard normal distribution table. The probability of getting a value between z1 and z2 is approximately 0.1988, or 19.88%.

First, we need to calculate the z-score for 46 hours:

z = (46 - 56) / 3.3 = -3.03

Next, we can look up the probability corresponding to a z-score of -3.03 in the standard normal distribution table. The probability of getting a value less than -3.03 is approximately 0.00123, or 0.123%.

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The question involves the computation and interpretation of Z-scores in a normally distributed data, which in this case is the lifetime of light bulbs. The probabilities are found by calculating the Z-scores and then looking up these scores in a Z-table or using a calculator. About 6.4% of bulbs will last more than 61 hours, 18.1% will last 53 hours or less, approximately 34.1% will last between 57 and 62 hours, and only about 0.1% will last less than 46 hours.

The question is about using the properties of a normal distribution to find probabilities related to the lifetime of light bulbs. To do this, we use the mean and standard deviation to compute Z-scores, which give us the number of standard deviations away from the mean a certain value is.

(a) To find the proportion of light bulbs that will last more than 61 hours, we calculate the Z-score for 61 hours: Z = (61 - 56)/3.3 = 1.52. We look this Z-score up in a Z-score table or use a calculator to find that the probability of getting a Z-score of 1.52 is about 0.064. Therefore, about 6.4% of light bulbs will last more than 61 hours.

(b) For finding the proportion of light bulbs that will last 53 hours or less, we calculate the Z-score for 53 hours: Z = (53 - 56)/3.3 = -0.91. Looking this up, we find that about 18.1% of light bulbs will last less than or equal to 53 hours.

(c) To find the proportion of light bulbs that will last between 57 and 62 hours, we calculate the Z-scores and find the probabilities for both, then subtract the smaller from the larger. The Z-score for 57 hours is 0.30 (probability about 37.5%) and for 62 hours is 1.82 (probability about 3.4%). Thus, about 34.1% of all light bulbs will last between 57 and 62 hours.

(d) Finally, to find the probability that a light bulb lasts less than 46 hours, we again calculate the Z-score: Z = (46 - 56)/3.3 = -3.03. This Z-score is quite small, suggesting this is unlikely: indeed, only about 0.1% of all light bulbs last less than 46 hours.

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