Answer:
Events E and F are not independent if the probability of event E occurring is affecting the probability of event F occurring.
Step-by-step explanation:
Two events are independent when the probability of one event occurring has no connection with that of the other event.
Example, when you toss a coin and roll a six sided die, the probability of getting a head or a tail has no connection with the probability of getting any number face.A real life example will be the probability going to the mall and owning a cat at home.These two have no influence on one another.
Mathematically independent events can be calculated as;
P(E∩F)=P(E)-P(F)
Which of the following does not describe a rigid motion transformation?
The transformation which do not describe a rigid motion transformation is:
Option: C
C. dilating a figure by a scale factor of 1/4
Step-by-step explanation:Rigid motion transformation is a transformation in which the shape and size of the figure is preserved i.e. it remains the same.
A)
Translating a figure 5 units right.
We know that in the translation transformation the shape and size of the figure remains the same only the location of points are changed.
B)
Rotating a figure 90 degrees.
In rotation the shape and size is preserved.
Hence it is a rigid transformation.
C)
dilating a figure by a scale factor of 1/4
This is not a rigid transformation because the size of the figure is changed.
since the scale factor is less than 1.
Hence, the transformation is a reduction of the original figure.
D)
reflecting a figure across the x-axis.
The reflection is also a rigid transformation.
since it preserves the shape and size of the object.
Answer:
The correct answer is C.
(Help!!will give Brainest, if correct)
A system of inequalities can be used to determine the depth of a toy, in meters, in a pool depending on the time, in seconds, since it was dropped. Which constraint could
be part of the scenario?
A)The pool is 1 meter deep.
B)The pool is 2 meters deep.
C)The toy falls at a rate of at least a 1/2
meter per second.
D)The toy sinks at a rate of no more than a
1/2 meter per second
In the given scenario regarding a toy sinking in a pool, the constraints related to the depth of the pool and the rate of sinking of the toy could be part of the scenario, which illustrates a system of inequality related to the depth of the toy in the pool over time.
Explanation:The question requires understanding of the constraints in a scenario that involve a system of inequalities related to the depth of a toy in a pool over time. In this case, the toy is falling, or sinking, into the pool. Therefore, the scenario will involve the depth of the pool and the rate at which the toy is falling.
Firstly, The depth of the pool is important because the toy cannot sink deeper than the pool is. Therefore, both constraints A) The pool is 1 meter deep and B) The pool is 2 meters deep could both be part of the scenario, depending on the actual depth of the pool involved.
Secondly, The rate at which the toy is falling (sinking) is also important. Both constraints C) The toy falls at a rate of at least a 1/2 meter per second and D) The toy sinks at a rate of no more than a 1/2 meter per second could be part of the scenario, depending on the actual sinking rate of the toy.
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The constraint that could be part of the scenario is that the D) toy sinks at a rate of no more than a 1/2 meter per second.
Explanation:A system of inequalities is a set of two or more inequalities involving the same variables. The solution to the system is the set of values that satisfy all the inequalities simultaneously. Graphically, the solution represents the overlapping region of the individual inequalities on a coordinate plane.
The constraint that could be part of the scenario is that the toy sinks at a rate of no more than a 1/2 meter per second. This constraint ensures that the depth of the toy in the pool does not change too rapidly. If the toy sank at a faster rate than 1/2 meter per second, it would quickly reach the bottom of the pool.
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What is the surface area of a sphere with a radius of 12 units?
A. 144pi units
B. 576 units2
C. 576pi units
D. 288pi units2
Answer:
C
Step-by-step explanation:
The surface area (A) of a sphere is calculated as
A = 4πr² ← r is the radius, here r = 12, hence
A = 4π × 12²
= 4π × 144 = 576π units² → C
The surface area of the sphere is 576 square units. The correct option is B.
What is surface area?The space occupied by any two-dimensional figure in a plane is called the area. The area of the outer surface of any body is called as the surface area.
The surface area (A) of a sphere is calculated as
A = 4πr² ← r is the radius, here r = 12, hence
A = 4π × 12²
A= 4π × 144
A = 576π units²
Hence, the correct option is B.
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Graph the equation below
Answer:
See picture.
In the picture I graphed (0,1) and then graphed (1,3).
I connected the points with a straight-edge.
Step-by-step explanation:
This question is asking us to use slope-intercept form of a line to answer it.
Slope-intercept form is y=mx+b where m is the slope and b is the y-intercept.
Your equation is y=2x+1 so our m=2 and b=1.
So our y-intercept is 1. This is the first point we graph.
The slope is 2 or as a fraction 2/1. Recall that slope=rise/run. So this tells us after we plot (0,1) we need to go up 2 units and right 1 unit to get one more point to graph. This point will be (0+1,1+2) or just (1,3).
I will draw a graph also to show you this:
Answer:
Graph Attached Below
Step-by-step explanation:
Hello!
To graph a line, we just need any two points that belong to that line.
We know the y-intercept, (0,1), given in the equation itself. We can plot that point as our first point.
The second point can be found by using the slope. The slope is 2/1, and we can go up 2 units and to the right 1 unit to find the second point.
The second point is (1,3).
the radius of Earth is 6400 km. the distance of the moon from the Earth's surface is 380000 km. find the radius of the moon which subtends an angle of 20'at the centre of the earth
Answer:
1100 km
Step-by-step explanation:
The problem doesn't clearly state whether the 380,000 km is from the Earth's surface to the moon's center, or to the moon's surface. Since we'll be rounding to 2 significant figures, it's not enough to make a difference, so I'll assume it's to the moon's center.
Draw circle representing the moon and earth. Draw tangent lines from the earth's center to the edge of the moon. The angle between these lines is the subtended angle. Now draw a line representing the moon's radius from the center of the moon to the point where the tangent line intersects. Notice this forms a right angle.
(See attached diagram)
Using trigonometry:
sin(θ/2) = r / (R + h)
r = (R + h) sin(θ/2)
Given that R = 6400 km, h = 380,000 km, and θ = 20' = 1/3 degrees:
r = (6400 + 380000) sin(1/6 °)
r = 1100
The moon's radius is 1100 km.
The points (3, 24) and (7, 56) represent points of a function where y, the number of photographs, varies directly with x, the number of pages in an album. Which statement describes another point on the graph of this function?
A 50-page photo album holds 400 photographs.
An 80-page photo album holds 560 photographs.
A 100-page photo album holds 8,000 photographs.
A 900-page photo album holds 8,400 photographs.
Answer:
Option A 50-page photo album holds 400 photographs.
Step-by-step explanation:
we know that
A relationship between two variables, x, and y, represent a proportional variation if it can be expressed in the form [tex]y/x=k[/tex] or [tex]y=kx[/tex]
step 1
Find the value of k
For the point (3,24)
x=3,y=24
k=y/x
k=24/3=8
The equation is equal to
y=8x
step 2
Verify each statement
case A) 50-page photo album holds 400 photographs.
For x=50
substitute in the equation
y=8(50)=400 -----> is correct
case B) 80-page photo album holds 560 photographs.
For x=80
substitute in the equation
y=8(80)=640 -----> is not correct
case C) 100-page photo album holds 8,000 photographs.
For x=100
substitute in the equation
y=8(100)=800 -----> is not correct
case D) 900-page photo album holds 8,400 photographs.
For x=900
substitute in the equation
y=8(900)=7,200 -----> is not correct
Answer:
Yes ! The correct answer is A.) 50-page photo album holds 400 photographs.
Step-by-step explanation:
I did the Unit Test and i got it correct.
Write the equation of the line that passes though (-3,5) and (2,10) in slope intercept form
Answer:
see explanation
Step-by-step explanation:
The equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
Calculate m using the slope formula
m = (y₂ - y₁ ) / (x₂ - x₁ )
with (x₁, y₁ ) = (- 3, 5) and (x₂, y₂ ) = (2, 10)
m = [tex]\frac{10-5}{2+3}[/tex] = [tex]\frac{5}{7}[/tex], hence
y = [tex]\frac{5}{7}[/tex] x + c ← is the partial equation of the line
To find c substitute either of the 2 points into the partial equation
Using (2, 10), then
10 = [tex]\frac{10}{7}[/tex] + c ⇒ c = 10 - [tex]\frac{10}{7}[/tex] = [tex]\frac{60}{7}[/tex]
y = [tex]\frac{5}{7}[/tex] x + [tex]\frac{60}{7}[/tex] ← in slope- intercept form
Ms. Lawton is redecorating her office. She has a choice of 4 colors of paint, 2 kinds of curtains, and 5 colors of carpet. How many different ways are there to redecorate?
Answer:
40 combinations
Step-by-step explanation:
We just multiply the outcomes.
4*2*5=40
There are 40 different ways to redecorate.
4 colors of paint * 2 kinds of curtains * 5 colors of carpet
= 40 outcomes.
What is an outcome in probability?In the possibility principle, an outcome is a probable result of a test or trial. every possible final result of a selected experiment is precise, and one-of-a-kind results are together one of a kind (most effective final results will occur on each trial of the experiment).
To locate the total wide variety of consequences for two or greater occasions, multiply the variety of effects for every occasion collectively. that is referred to as the product rule for counting because it involves multiplying to discover a product.
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What’s is the cubic feet of 24 by 72 by 18
Answer:
I have to assume that the 24, the 72, and the 18 are feet.
If those are the dimensions of a rectangular prism, like a box, a crate,
or a humongous block of ice, then
Volume = (length) x (width) x (height) =
(24-ft) x (72-ft) x (18-ft) = 31,104 cubic ft
If those numbers are inches, then here's the easy way to handle it:
24 inches = 2 ft
72 inches = 6 ft
18 inches = 1.5 ft
Volume = (length) x (width) x (height) =
(2ft) x (6ft) x (1.5ft) = 18 cubic feet
which of the following is an equation of a line parallel to the equation [tex]y=\frac{1}{2} x+1[/tex]
Answer:
Any equation of a line having a slope of 1/2.
Step-by-step explanation:
It appears that the question gives answer choices and this problem could have many solutions, but parallel lines have equal slopes. The slope of this line is 1/2 since it is in y=mx+b form where m is the slope. So a few examples of equations of lines parallel to the given equation would be y = 1/2x - 5 or y = 1/2x + 10 etc.
b(n)=1(−2)^n-1
What is the fourth term in the sequence?
Answer:
-8
Step-by-step explanation:
If you want to find the fourth term in the sequence, or b(4), then you just plug in the value 4 for n, resulting in
b(4)=1(-2)^(4-1).
Simplifying, we get
b(4)=1(-2)^3
=1(-2*-2*-2)
=1(-8)
=-8
Bill has 6 markers in a backpack. One of them is purple and one is blue. Find the probability Bill will reach into the backpack without looking and grab the purple marker and then reach in a second time and grab the blue marker. Express your answer as a fraction in simplest form.
Answer:
Step-by-step explanation:
His chances of getting the purple one are 1/6
His chances of getting the blue one after drawing the purple one and not replacing are 1/5
The probability of both happening are 1/6 * 1/5 = 1/30
Joey bought 10 cupcakes and has four friends. He wants to have nothing left, but he wants everyone including him to have the same amount of cupcakes. How many cupcakes can each person get?
we use divsion, 10 divided by 5 is 2, everyone get 2 yumy cupcakes.
Which formula below gives the average rate of change of the function z(x) = -6x + 2 + 3 on the interval -1 ≤ x ≤ 2 ?
Answer:
The average rate of change is -6
Step-by-step explanation:
The formula for average rate of change is
[tex]=\frac{f(x_2)-f(x_1)}{x_2-x_1}[/tex]
where
x₁=-1 and x₂=2
In this case we replace f(x) with z(x)
Substitute values of x₁ and x₂ in the formula given the function as
[tex]z(x)=-6x+2+3\\z(x)=-6x+5\\\\\\=\frac{z(x_2)-z(x_1)}{x_2-x_1} \\\\\\z(x_1)=z(-1)=-6*-1+5=6+5=11\\\\\\z(x_2)=z(2)=-6*2+5=-12+5=-7\\\\\\x_2-x_1=2--1=3\\\\Rateofchange=\frac{-7-11}{3} =\frac{-18}{3} =-6[/tex]
Drag the tiles to the correct boxes to complete the pairs. Not all tiles will be used.
Answer:
1. mode
2. median
3. mean
Rewrite the polynomial 4x^2-7x^3-x^6+3x^4-1 in descending order, using coefficients of 0 for any missing
terms.
Which term is first second third, etc., when the polynomial is written in this way?
Match the term number to the appropriate term.
Term 7
Select an option
Term 6
Select an option
Term 5
Select an option
Term 4
Select an option
Term 3
Select an option
Term 2
Select an option
Term 1
Select an optio
Answer:
The polyester in standard form with zero placeholders is
-x^6+0x^5+3x^4+-7x^3+4x^2+0x+-1
7th term -1
6th term 0x
5th term 4x^2
4th term -7x^3
3rd term 3x^4
2nd term 0x^5
1st term -x^6
Step-by-step explanation:
We want to write 4x^2-7x^3-x^6+3x^4-1 in descending order means want the exponents to being counting down.
6 is the highest exponent so lets rearrange this so that tern comes first:
-x^6+.....
5 is the highest but we are missing so now we have:
-x^6+0x^5+....
4 is the next highest and I see +3x^4 in our polynomial so that comes next:
-x^6+0x^5+3x^4+...
3 is the next highest and I see -7x^3 so that term comes next:
-x^6+0x^5+3x^4+-7x^3+....
2 is the next highest and I see +4x^2 so that term comes next:
-x^6+0x^5+3x^4+-7x^3+4x^2+...
1 is the next highest and we are missing an x term so the next term is +0x.....
-x^6+0x^5+3x^4+-7x^3+4x^2+0x+....
0 is the next highest and will be the last term we write. This is the constant tern, the term without the variable and that is -1.
-x^6+0x^5+3x^4+-7x^3+4x^2+0x+-1
Now I guess you are reading this from left to right when numbering the terms as first, second, and so on...
1st term -x^6
2nd term 0x^5
3rd term 3x^4
4th tern -7x^3
5th term 4x^2
6th term 0x
7th term -1
What is the value of the digit 8 in the number 56,782,010,000?
Answers: Ten millions
Find (f-g)(x) of the problem
Answer:
[tex]3x^2-2x-6[/tex]
Step-by-step explanation:
[tex](f-g)(x)[/tex]
[tex]f(x)-g(x)[/tex]
So plug in your functions:
[tex][3x^2-2]-[2x+4][/tex]
Distribute:
[tex]3x^2-2-2x-4[/tex]
Combine like terms:
[tex]3x^2-2x-6[/tex] There was only one pair of like terms -2 and -4.
Use the interactive to graph a line with a slope of zero and passing through the 0,4
Answer:
You need to draw a horizontal line that goes through the point (0,4).
Step-by-step explanation:
Slope of zero means it does not go up or down. It looks like this ↔, but stretched out to the two sides of the graph.
A line with a slope of zero is flat and does not rise or fall. To graph this line, draw a horizontal line through the given y-coordinate.
Explanation:A line with a slope of zero means that the line is flat and has no rise or fall. To graph a line with a slope of zero passing through the point (0,4), you would draw a horizontal line through the y-coordinate 4. Since the slope is zero, the line will not change in the vertical direction.
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The figures are similar. Find the area. The area of △ △ A B C is 15 square cm. The height of △ △ A B C is 5 cm and the height of △ △ D E F is 13 cm. Find the area of △ △ D E F . Round to the nearest square cm if necessary.
Answer:
The area of triangle DEF is [tex]282\ cm^{2}[/tex]
Step-by-step explanation:
step 1
Find the scale factor
we know that
If two figures are similar, then the ratio of its corresponding sides is proportional and this ratio is called the scale factor
To find the scale factor, divide the height of triangle DEF by the height of triangle ABC
Let
z ------> the scale factor
[tex]z=\frac{13}{5}[/tex]
step 2
Find the area of triangle DEF
we know that
If two figures are similar, then the ratio of its areas is equal to the scale factor squared
Let
z -----> the scale factor
x ----> the area of triangle DEF
y -----> the area of triangle ABC
[tex]z^{2}=\frac{x}{y}[/tex]
we have
[tex]z=\frac{13}{5}[/tex]
[tex]y=15\ cm^{2}[/tex]
substitute and solve for x
[tex](\frac{13}{5})^{2}=\frac{x}{15}[/tex]
[tex]x=(\frac{169}{25})(15)[/tex]
[tex]x=282\ cm^{2}[/tex]
The spinner of the compass is two congruent isosceles triangles connected by their bases as shown in the diagram. The base of each of these triangles is 2 centimeters and the legs are 5 centimeters.
If the metal used to construct the spinner costs $13.25 per square centimeter, how much will it cost to make this part of the compass? Round to the nearest cent.
Cost =
Answer:
fff
Step-by-step explanation:
By pythagorean theorum we calculate the height:
[tex]\sqrt{5^2-1^2}[/tex]
=[tex]\sqrt{24}[/tex] = height of triangle
area of triangle = base * height
area = [tex]2*\sqrt{24}[/tex]
There are two triangles so:
[tex]2*2*\sqrt{24}=4\sqrt{24}[/tex]
Multiply this by 13.25 to get total cost:
=$259.65
Answer:
$ 129.82 because it is to the nearest cent
what is the inverse operation of F(x)=2x+3
Answer:
[tex]f^{-1}[/tex](x) = [tex]\frac{x-3}{2}[/tex]
Step-by-step explanation:
Let y = f(x) and rearrange making x the subject
y = 2x + 3 ( subtract 3 from both sides )
y - 3 = 2x ( divide both sides by 2 )
[tex]\frac{y-3}{2}[/tex] = x
Change y back into terms of x, hence
[tex]f^{-1}[/tex](x ) = [tex]\frac{x-3}{2}[/tex]
Keep in mind that f(x) means the same thing as y so...
y = 2x + 3
To find the inverse operation you switch the places of x and y like so...
x = 2y + 3
Now you must solve for y. To start off you must subtract 3 to both sides.
x - 3 = 2y + 3 - 3
x - 3 = 2y
Now divide 2 to both sides to completely isolate y
(x - 3) / 2 = 2y / 2
[tex]\frac{1}{2} x - \frac{3}{2}[/tex] = y
OR
[tex]\frac{x}{2} -\frac{3}{2}[/tex] = y
Hope this helped!
~Just a girl in love with Shawn Mendes
Help with this question thanks
Answer:
37.7 = h
Step-by-step explanation:
Area of a rectangle is base * height
A =bh
Substitute what you know
1059.37 = 28.1 * h
Divide each side by 28.1
1059.37/28.1 = 28.1 * h/28.1
37.7 = h
3.
Determine the angle PMA of the lampshade.
15 cm-
25 cm
-25 cm.-
M
Answer:
∠PMA=55°
Step-by-step explanation:
see the attached figure with the letter X to better understand the problem
we know that
In the right triangle PXM
PX=LA=25 cm
XM=AM-15/2=25-7.5=17.5 cm
∠PMA=∠PMX
tan(∠PMA)=(PX)/(XM)
tan(∠PMA)=(25)/(17.5)
∠PMA=arctan((25)/(17.5))=55°
The quotient of 6 cubed and 4 squared
ANSWER
13.5
EXPLANATION
We write 6 cubed as 6³
We write 4 squared as 4²
The quotient of 6 cubed and 4 squared is written as
[tex] \frac{ {6}^{3} }{ {4}^{2} } [/tex]
We expand to obtain:
[tex] \frac{6 \times 6 \times 6}{4 \times 4} [/tex]
Cancel out the common factors to get,
[tex] \frac{3 \times 3 \times 3}{2} [/tex]
This simplifies to
[tex] \frac{27}{2} = 13.5[/tex]
What is the area of the composite figure
A 88 B 80 C 110 D 112
Answer: 88
Step-by-step explanation:
(8x2)=16/2=8
8x6=48
8x4=32
48+32+8=88
Answer:
112
Step-by-step explanation:
The product of two consecutive positive integers is 342. Represent in the above situation in the form of quadratic equation. Find the numbet also.
Answer:
18 and 19
Step-by-step explanation:
Consecutive positive integers have a difference of 1 between them
let n and n + 1 be the 2 integers, then
n(n + 1) = 342, that is
n² + n = 342 ( subtract 342 from both sides )
n² + n - 342 = 0 ← quadratic equation in standard form
(n + 19)(n - 18) = 0 ← in factored form
Equate each factor to zero and solve for n
n + 19 = 0 ⇒ n = - 19
n - 18 = 0 ⇒ n = 18
However, n > 0 ⇒ n = 18 and n + 1 = 18 + 1 = 19
The 2 integers are 18 and 19
Determine the slant asymptote for the function f(x)=3x^2-4x + 5/ x-3
Answer:
y=3x+5
Step-by-step explanation:
To determine the slant asymptote you must perform polynomial division and also the degree of the numerator must be one greater than degree of the denominator for it to even exist. (We do have that here by the way since degree of the top is 2 and the degree of the bottom is 1).
Let's begin the division process:
I'm using synthetic. You can also use long.
3 goes on the outside because we are dividing by x-3
3 | 3 -4 5
| 9 15
----------------------
3 5 20
So [tex]\frac{3x^2-4x+5}{x-3}=3x+5+\frac{20}{x-3}[/tex]
So the slant aymptote is y=3x-5 since 20/(x-3) approaches 0 as x approaches infinity.
To find the slant asymptote of the given function, perform long division to divide the numerator by the denominator. As x approaches infinity, the slant asymptote is 3x + 5.
Explanation:To find the slant asymptote of the function f(x) = (3x^2 - 4x + 5) / (x - 3), we need to determine what happens as x approaches positive or negative infinity. We can do this by dividing the numerator polynomial by the denominator polynomial using long division.
Performing long division, we get:
3x^2 - 4x + 5 / (x - 3) = 3x + 5 + (8 / (x - 3))
As x approaches infinity, the 8 / (x - 3) term becomes negligible and we are left with the slant asymptote:
f(x) = 3x + 5
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ABCD is an isosceles trapezoid with AD II BC, mZB = 60°, and mZC = (3x +15) Solve for x.
A 15
B. 25
C. 35
D. 60
Answer:
A 15
Step-by-step explanation:
In an isosceles trapezoid, each pair of base angles is congruent.
In this isosceles trapezoid, angles B and C are congruent, so their measures are equal.
m<B = m<C
60 = 3x + 15
Subtract 15 from both sides.
45 = 3x
Divide both sides by 3.
15 = x
x = 15
Answer: The correct option is (A) 15.
Step-by-step explanation: As shown in the attached figure below, ABCD is an isosceles trapezoid with AD parallel to BC.
Also, m∠B = 60° and ∠C = (3x +15)°.
We are to find the value of x.
Since ABCD is an isosceles trapezoid with AB and CD as the two legs, so we have
[tex]AB=CD\\\\\Rightarrow m\angle C=m\angle B~~~~~~~~~~~~~~~~~~~[\textup{Angles opposite to equal legs are equal}]\\\\\Rightarrow (3x+15)^\circ=60^\circ\\\\\Rightarrow 3x+15=60\\\\\Rightarrow 3x=60-15\\\\\Rightarrow 3x=45\\\\\Rightarrow x=\dfrac{45}{3}\\\\\Rightarrow x=15.[/tex]
Thus, the required value of x is 15.
Option (A) is CORRECT.
8. A basketball with a diameter of 9.5 inches
is packaged in a cubic box measuring
9.5 inches on each edge. Determine the
volume of the empty space in the box.
The correct answer is [tex]\(\boxed{\frac{1919}{16} - \frac{481\pi}{128}}\)[/tex] cubic inches.
To determine the volume of the empty space in the box, we need to calculate the volume of the box and the volume of the basketball, and then subtract the volume of the basketball from the volume of the box.
First, let's calculate the volume of the cubic box. Since the edge of the cube is given as 9.5 inches, the volume of the cube [tex]\( V_{box} \)[/tex] is the edge length cubed:
[tex]\[ V_{box} = 9.5^3 \][/tex]
[tex]\[ V_{box} = \left(\frac{95}{10}\right)^3 \][/tex]
[tex]\[ V_{box} = \frac{95^3}{10^3} \][/tex]
[tex]\[ V_{box} = \frac{95^3}{1000} \][/tex]
[tex]\[ V_{box} = \frac{857375}{1000} \][/tex]
[tex]\[ V_{box} = \frac{1919}{4} \][/tex] cubic inches.
Next, we calculate the volume of the basketball. The basketball is a sphere with a diameter of 9.5 inches, so its radius [tex]\( r \)[/tex] is half of that:
[tex]\[ r = \frac{9.5}{2} \][/tex]
[tex]\[ r = \frac{95}{20} \][/tex]
[tex]\[ r = \frac{19}{4} \] inches.[/tex]
The volume of a sphere [tex]\( V_{sphere} \)[/tex] is given by the formula:
[tex]\[ V_{sphere} = \frac{4}{3}\pi r^3 \][/tex]
[tex]\[ V_{sphere} = \frac{4}{3}\pi \left(\frac{19}{4}\right)^3 \][/tex]
[tex]\[ V_{sphere} = \frac{4}{3}\pi \left(\frac{19^3}{4^3}\right) \][/tex]
[tex]\[ V_{sphere} = \frac{4}{3}\pi \left(\frac{6859}{64}\right) \][/tex]
[tex]\[ V_{sphere} = \frac{481\pi}{128} \] cubic inches.[/tex]
Now, we subtract the volume of the basketball from the volume of the box to find the volume of the empty space [tex]\( V_{empty} \):[/tex]
[tex]\[ V_{empty} = V_{box} - V_{sphere} \][/tex]
[tex]\[ V_{empty} = \frac{1919}{4} - \frac{481\pi}{128} \] cubic inches.[/tex]
To simplify the expression, we can multiply the terms by a common denominator of 128 to combine them:
[tex]\[ V_{empty} = \frac{1919 \times 32}{128} - \frac{481\pi}{128} \][/tex]
[tex]\[ V_{empty} = \frac{61408}{128} - \frac{481\pi}{128} \][/tex]
[tex]\[ V_{empty} = \frac{61408 - 481\pi}{128} \] cubic inches.[/tex]
However, we notice that the denominator of 128 can be simplified by dividing both numerator and denominator by 4, which gives us the final
[tex]\[ V_{empty} = \frac{1919}{16} - \frac{481\pi}{128} \][/tex] cubic inches.
Therefore, the volume of the empty space in the box is [tex]\(\boxed{\frac{1919}{16} - \frac{481\pi}{128}}\)[/tex] cubic inches.