Answers
if x = 0, y = 3
ordered pair (0,3)
Related Questions
A photograph has a length that is 6 inches longer than its width, x. So its area is given by the expression x(x+6) square inches. If the area of the photograph is 112 square inches, what is the width of the photograph?
Answers
112 = x^2 + 6x
x^2 + 6x - 112 = 0
(x - 8)(x + 14) = 0
x - 8 = 0
x = 8
x + 14 = 0
x = -14....not this one because it is negative
x = 8 <===width = 8 inches
L = x + 6
L = 8 + 6
L = 14 ....length = 14
The width of the photograph is 8 inches
let
width of the photograph = x
length of the photograph = 6 + x
Area = x(6 + x)
112 = x(6 + x)
112 = 6x + x²
x² + 6x - 112 = 0
x² - 8x + 14x - 112 = 0
x(x - 8) + 14(x - 8) = 0
(x + 14)(x - 8)
x = -14 or 8
x = 8
x cannot be negative so we use only 8.
width of the photograph = 8 inches
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Use cylindrical coordinates. find the volume of the solid that is enclosed by the cone z = x2 + y2 and the sphere x2 + y2 + z2 = 72.
Answers
Step 1: The intersection curve is only at the origin.
Step 2: Set up the volume integral over the sphere region.
Step 3:Evaluate the integral to find the volume: [tex]\(V = 216\pi\sqrt{2}\).[/tex]
To find the volume of the solid enclosed by the cone [tex]\(z = x^2 + y^2\)[/tex] and the sphere[tex]\(x^2 + y^2 + z^2 = 72\),[/tex] we'll first find the intersection curve of the cone and the sphere in cylindrical coordinates, then set up the triple integral to find the volume.
Step 1: Finding the intersection curve:
The cone equation in cylindrical coordinates becomes [tex]\(z = r^2\)[/tex] and the sphere equation remains the same as [tex]\(r^2 + z^2 = 72\).[/tex]
To find the intersection, we set these two equations equal to each other:
[tex]\[r^2 = r^2 + z^2\][/tex]
Substitute [tex]\(z = r^2\)[/tex] from the cone equation:
[tex]\[r^2 = r^2 + (r^2)^2\][/tex]
[tex]\[r^2 = r^2 + r^4\][/tex]
[tex]\[r^4 = 0\][/tex]
From this, we see that the only solution is (r = 0), which corresponds to the point at the origin.
Step 2: Setting up the integral for volume:
We'll integrate over the region where the cone lies within the sphere, which is the entire sphere. In cylindrical coordinates, the limits of integration are [tex]\(0 \leq r \leq 6\)[/tex] (since the sphere has radius [tex]\(\sqrt{72} = 6\)) and \(0 \leq \theta \leq 2\pi\).[/tex]
The limits for \(z\) are from the cone to the sphere, so it's from [tex]\(r^2\) to \(\sqrt{72-r^2}\).[/tex]
Thus, the volume integral is:
[tex]\[V = \iiint_{E} r \, dz \, dr \, d\theta\][/tex]
Where (E) is the region enclosed by the cone and the sphere.
Step 3: Evaluate the integral:
[tex]\[V = \int_{0}^{2\pi} \int_{0}^{6} \int_{r^2}^{\sqrt{72-r^2}} r \, dz \, dr \, d\theta\][/tex]
Let's evaluate this integral step by step:
[tex]\[V = \int_{0}^{2\pi} \int_{0}^{6} (r\sqrt{72-r^2} - r^3) \, dr \, d\theta\][/tex]
[tex]\[V = \int_{0}^{2\pi} \left[-\frac{1}{4}(72-r^2)^{3/2} - \frac{1}{4}r^4\right]_{0}^{6} \, d\theta\][/tex]
[tex]\[V = \int_{0}^{2\pi} \left[-\frac{1}{4}(0)^{3/2} - \frac{1}{4}(6^4) - \left(-\frac{1}{4}(72)^{3/2} - \frac{1}{4}(0)^4\right)\right] \, d\theta\][/tex]
[tex]\[V = \int_{0}^{2\pi} \left[\frac{1}{4}(72)^{3/2}\right] \, d\theta\][/tex]
[tex]\[V = \frac{1}{4}(72)^{3/2} \int_{0}^{2\pi} 1 \, d\theta\][/tex]
[tex]\[V = \frac{1}{4}(72)^{3/2} \cdot 2\pi\][/tex]
[tex]\[V = 36\pi \sqrt{72}\][/tex]
[tex]\[V = 36\pi \times 6\sqrt{2}\][/tex]
[tex]\[V = 216\pi\sqrt{2}\][/tex]
So, the volume of the solid enclosed by the cone and the sphere is [tex]\(216\pi\sqrt{2}\).[/tex]
Children of age 3-6 years require 25% less sleep per day than infants, who need 16hours. How much slee do the 3-6 year olds need?
Answers
16 - 0.25(16) = 16 - 4 = 12 hrs <=
16 - 4 = 12
So therefore, 3-6 year old aneed 12 hours of sleep.
Or you could solve it this way:
100 - 0.25 = 0.75
16 × 0.75 = 12
Either of the two ways you would get 12
A car rents for $180 per week plus $0.75 per mile. Find the rental cost for a two-week trip of 500 miles for a group of three people.
Answers
Answer:
$735
Step-by-step explanation:
$180/week * 2 weeks = $360 for 2 weeks
$0.75/mile * 500 miles = $375
$360 + $375 = $735 for 2 week trip of 50 miles
What is the slope of a line that is perpendicular to the line whose equation is 0.5x−5y=9
Answers
0.5x-5y=9
-5y=-0.5x+9
y=0.1x-9/5
The slope is 0.1
To find a perpendicular slope, find the negative reciprocal.
The reciprocal of 0.1 is 10
The negative of 10 is -10
Final answer: -10
To find the slope of a line perpendicular to the line defined by the equation 0.5x−5y=9, first convert the equation into slope-intercept form to determine the slope of the original line. The slope of the line perpendicular to this is its negative reciprocal. For this specific problem, the slope of the line perpendicular to the given line is -10.
Explanation:Firstly, rearrange the given equation, 0.5x−5y=9, into the slope-intercept format, i.e. y=mx+b. Here, 'm' is the slope of the line. To illustrate this rearrangement: 5y = 0.5x - 9, then divide through by 5 to get y = 0.1x - 1.8. Thus, the slope of the given line is 0.1.
Lines that are perpendicular to each other have slopes that are negative reciprocals of each other. In this case, the slope of the given line is 0.1, so the negative reciprocal (which represents the slope of the line perpendicular to this line) is -1/0.1, which equals -10. This is the slope of the line perpendicular to the given line.
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What is the future value of $20 a week for 10 years at 6 percent interest? assume the first payment occurs at the end of this week?
Answers
52 wk x 10yr=520 wks
520x20=10,400x 0.06% interest=624
10,400+624=
$11,024
A team of 10 players is to be selected from a class of 6 girls and 7 boys. Match each scenario to its probability. You have to drag the sentences to match probabilities.
Possible probabilities are:
0.07
0.12
0.44
0.49
Options for answers are:
The probability that a randomly chosen team includes all 6 girls in the class.
The probability that a randomly chosen team has 3 girls and 7 boys.
The probability that a randomly chosen team has either 4 or 6 boys.
The probability that a randomly chosen team has 5 girls and 5 boys.
Answers
Answer:
0.07 -The probability that a randomly chosen team has 3 girls and 7 boys.
0.12 -The probability that a randomly chosen team includes all 6 girls in the class.
0.44 -The probability that a randomly chosen team has 5 girls and 5 boys.
0.49-The probability that a randomly chosen team has either 4 or 6 boys.
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When a team of 10 players is to be selected from a class of 6 girls and 7 boys the probability of each scenario occurring is:
0.07 = The probability that a randomly chosen team has 3 girls and 7 boys.
0.12 = The probability that a randomly chosen team includes all 6 girls in the class.
0.44 = The probability that a randomly chosen team has 5 girls and 5 boys.
0.49 = The probability that a randomly chosen team has either 4 or 6 boys.
What is probability?
In simple words, it refers to a numerical guess of how likely an event is to occur. Hence, when it comes to probability we believe there is no guarantee the event would occur.
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A ball of radius 0.200 m rolls with a constant linear speed of 3.00 m/s along a horizontal table. the ball rolls off the edge and falls a vertical distance of 2.08 m before hitting the floor. what is the angular displacement of the ball while the ball is in the air
Answers
The angular displacement of the ball while it's in the air is 9.77 radians.
Here's how we can find it:
Time in the air: First, we need to find the time the ball spends in the air. We can use the following kinematic equation for vertical motion:
h = 1/2 * g * t^2
where:
h is the vertical distance (2.08 m)
g is the acceleration due to gravity (9.81 m/s²)
t is the time in the air
Solving for t, we get:
t = sqrt(2 * h / g) = sqrt(2 * 2.08 m / 9.81 m/s²) ≈ 1.43 s
Angular velocity: The angular velocity (ω) of the ball is related to its linear velocity (v) and radius (r) by the following equation:
ω = v / r
where:
v is the linear velocity (3.00 m/s)
r is the radius (0.200 m)
Plugging in the values, we get:
ω = 3.00 m/s / 0.200 m = 15 rad/s
Angular displacement: Finally, the angular displacement (θ) of the ball is the product of its angular velocity (ω) and the time (t) it spends in the air:
θ = ω * t = 15 rad/s * 1.43 s ≈ 21.45 rad ≈ 9.77 rad (rounded to three significant figures)
Therefore, the angular displacement of the ball while in the air is approximately 9.77 radians.
EASY 5 POINTS!!! This box has the same length, width, and height as the 2 basketballs. Which expression gives the volume of the empty space encircling the basketballs inside the box?
Answers
1. We know it's a difference, therefore we need to subtract the volume of the balls from the volume of the box. B is wrong.
2. The box is 3dimensional so we must have 3 linear measurements multiplied together in it's volume calculation. D is wrong.
3. The sphere volume formula uses radius cubed (not diameter). C is wrong
A is the only one left and must be correct.
To check:
Volume of the box: [tex] V=lwh = 48 \times 24 \times 24 = 48 \times 24^{2}[/tex]
Volume of two balls: [tex] V = 2(\frac{4}{3} \pi r^{3}) = 2(\frac{4}{3} \pi \times 12^{3})[/tex]
Total volume = box - balls: [tex]V_{total}=48 \times 24^{2} - 2(\frac{4}{3} \pi \times 12^{3})[/tex]
This matches A, just as we thought.
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If Ying Zyiyu covers the first 200 miles of a trip going 50mph. and the last hundred going 40mph, what was his average speed for the entire trip?
Answers
the last 100 miles were done at 40mph, how long did that take? well, 100/40, or 2 hours and a half.
so.. Ying covered 300 miles in 4+2.5 hours or 300/6.5 is his average speed.
Answer:
46.15
Step-by-step explanation:
I do RSM :)
Find the x and y intercepts of 10x-15y=30
Answers
So the coordinate of that point is (x, 0) , that is, y must be 0 since the point is on the x-axis.
letting y=0, we find:
10x-15y=30
10x=30
x=3
With the same logic, letting x=0, we find the y-intercept:
10x-15y=30
-15y=30
y=-2
Answer:
x- intercept : 3 [point (3, 0)]
y- intercept : -2 [point (0, -2)]
A new car dealer offered to sell her demonstrator model for 90% of the retail price. If the sale price was $9,750, what was the retail price?
Answers
90% = 9,750
9750/90 = 1%
1% = 108.3
100% = 10,833.3
original retail price = $10,833.30
Let x be a random variable giving the number of aces in a random draw of 4 cards from an ordinary deck of 52 cards. construct a table showing the probability distribution of x
Answers
The provided table outlines the probability distribution for drawing varying numbers of aces in a random draw of 4 cards from a standard 52-card deck, using combinatorial probability calculations.
Explanation:To answer your question, we need to consider the different possibilities for pulling aces in a draw of four cards. There are 4 aces in a standard deck of 52 cards, and here's a table showing the probability distribution for each outcome:
x = 0 - this represents no aces drawn. The number of ways to choose no aces from 4 aces and 4 non-aces from 48 non-aces is (4 choose 0)*(48 choose 4). So the probability P(X=0) = (comb(4, 0)*comb(48, 4))/comb(52,4)x = 1 - one ace is drawn. The probability P(X=1) = (comb(4, 1)*comb(48, 3))/comb(52,4).x = 2 - two aces are drawn. The probability P(X=2) = (comb(4, 2)*comb(48, 2))/comb(52,4).x = 3 - three aces are drawn. The probability P(X=3) = (comb(4, 3)*comb(48, 1))/comb(52,4).x = 4 - all four aces are drawn, and the probability P(X=4) = (comb(4, 4)*comb(48, 0))/comb(52,4).Please note that 'comb(a, b)' in this case represents 'a choose b', which is a way to compute combinations in probability.
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An artisan is creating a circular street mural for an art festival. The mural is going to be 50 feet wide. One sector of the mural spans 38 degrees. What is the area of the sector to the nearest square foot?
Answers
Solve the following system of equations.
7x -8y= -19
-2x +5y =0
x=
y=
Answers
x= -5
y= -2
PLEASE HELP ME!!! NOT GOOD AT MATH.
Answers
L = length = 8 mm
W = width = 6 mm
H = height = 4 mm
TSA = 2*(L*W + W*H + L*H)
TSA = 2*(8*6 + 6*4 + 8*4)
TSA = 208
The units are "square millimeters" often written as sq mm or "mm^2"
Answer: 208
--------------------------------
Part B)
From part A, the total surface area (TSA) was 208 square mm. If it costs $3 per square mm, then that means the total cost C is
C = (cost per sq mm)*(total surface area)
C = 3*208
C = 624
The units for the cost are in dollars. This result is assuming that there isn't any waste.
Answer: 624
--------------------------------
Part C)
Volume = Length*Width*Height
V = L*W*H
V = 8*6*4
V = 192
The units are in cubic millimeters. The notation is sometimes "cubic mm" or "mm^3"
Answer: 192
Where would we find the point (–10,0)? A. Quadrant III B. y-axis C. Quadrant II D. x-axis
Answers
Harry rolls a number cube what is the probability that he will roll an even number or a number greater than four
Answers
P(E OR >4)=(3/6)+(2/6)
P(E OR >4)=5/6
The quotient of 8.4x10^9 and a number n results in 5.6 x 10^27 What is the value of n?
Answers
so 8.4x10^9/5.6 x 10^27=n
n=1.5x10^-18
Answer:
[tex]1.5\times 10^{-18}[/tex]
Step-by-step explanation:
We have been given that the quotient of [tex]8.4\times 10^9[/tex] and a number n results in [tex]5.6\times 10^{27}[/tex]
To find the value of n we will write our given information in an equation as:
[tex]\frac{8.4\times 10^9}{n}=5.6\times 10^{27}[/tex]
[tex]\frac{8.4\times 10^9}{5.6\times 10^{27}}=n[/tex]
Using quotient rule of exponents [tex]\frac{a^m}{a^n}=a^{m-n}[/tex] we will get,
[tex]\frac{8.4\times 10^{9-27}}{5.6}=n[/tex]
[tex]\frac{8.4\times 10^{-18}}{5.6}=n[/tex]
[tex]1.5\times 10^{-18}=n[/tex]
Therefore, the value of n is [tex]1.5\times 10^{-18}[/tex].
A rectangle or shaped swimming pool has a perimeter of 104 feet and is 4 feet longer than 3 times its width. Find its dimensions.
Answers
______
| |
| |
| | [tex]x[/tex]
| |
| |
|_____ |
(i'm using meters instead of feet because I'm more used to it)
The perimeter of the rectangle is [tex]2x+2y[/tex]
The perimeter is 104m
∴[tex]2y+2x=104[/tex]
------------------------------------------------------------------------
It's 4 feet longer than 3 times its width.
[tex]3y=x+4[/tex]
[tex]3y-x=4[/tex]
[tex]6y-2x=8[/tex]
[tex]2y+2x=104[/tex]
[tex]6y-2x=8[/tex]
-------------------------
[tex]8y=112[/tex]
∴[tex]y=14[/tex]m
[tex]2y+2x=104 [/tex]
[tex]2(14)+2x=104[/tex]
[tex]28 +2x=104 [/tex]
[tex]2x=76[/tex]
∴[tex]x=38[/tex]m
What is the probability that a randomly selected person will have a birthday in march? assume that this person was not born in a leap year. express your answer as a simplified fraction or a decimal rounded to four decimal places?
Answers
there are 31 days in August
365 days in a year
31/365 = 0.0849 probability
Answer:
0.0849
Step-by-step explanation:
Let's consider the event: a birthday takes place in March. The probability (P) of such event is:
[tex]P = \frac{favorable\ cases }{possible\ cases}[/tex]
The favorable cases are 31 because March has 31 days.
The possible cases are 365 because there are 365 days in a year.
The probability of a birthday being in March is:
[tex]P=\frac{31}{365} =0.0849[/tex]
Elaine's jewelry box has a volume of 36 cubic centimeters. Which of the following could be the dimensions of Elaine's jewelry box? Choose all answers that apply: A 12 cm long, 12 cm wide, 12 cm high B. 4cm long, 4 cm wide, 2 cm high C 3 cm long, 4 cm wide, 3 cm high
Answers
A. 12*12*12=1728 cm^3
B. 4*4*2=32 cm^3
C. 3*4*3=36 cm^3
Only C is correct
Answer:
Option C is the correct answer.
Step-by-step explanation:
Volume of box = Length x width x height.
Volume of box = 36 cubic centimetres.
Option A:
Length = 12 cm
Width = 12 cm
Height = 12 cm
Volume = 12 x 12 x 12 = 1728 cubic centimetres.
Option A is wrong.
Option B:
Length = 4 cm
Width = 4 cm
Height = 2 cm
Volume = 4 x 4 x 2 = 32 cubic centimetres.
Option B is wrong.
Option C:
Length = 3 cm
Width = 4 cm
Height = 3 cm
Volume = 3 x 4 x 3 = 36 cubic centimetres.
Option C is correct.
Option C is the correct answer.
the sum of two numbers is 70. the larger number is five less than twice the smaller number . what are the two numbers
Answers
x=2y-5
x=larger number
y=smaller number
now plug in the value of x to only have one variable in the equation...
Way to solve for y:
(2y-5)+y=70
3y-5=70
add 5 on both sides...
3y=75
divide by 3 on both sides
y=25
1st way to solve for x:
now plug in y in the original equation...
x+25=70
subtract 25 on both sides...
x=45
2nd way to solve for x:
plug in y for the second equation...
x=25(2)-5
x=50-5
x=45
answer: the two numbers are 45 and 25.
Final answer:
To find the two numbers, we created a system of equations with the sum being 70 and the larger number as five less than twice the smaller number. Solving the equations, we found that the smaller number is 25 and the larger number is 45.
Explanation:
The question asks us to find two numbers where the sum is 70, and the larger number is five less than twice the smaller number. To solve this, we'll set up a system of equations with two variables, which we'll call x and y. The first equation would be x + y = 70, since their sum is 70. The second equation is derived from the statement that the larger number is five less than twice the smaller number, which can be expressed as y = 2x - 5. Now we substitute the second equation into the first to solve for x. We get x + (2x - 5) = 70. Simplifying, we have 3x - 5 = 70, thus 3x = 75, and therefore x = 25. With the smaller number found, we can determine the larger number using the second equation y = 2(25) - 5, resulting in y = 45. Hence, the two numbers are 25 and 45.
How do you work out a percentile rank of a score of 57
Answers
For example: Supposing that the mean of these scores were 52 and the standard deviation 3. You'd need to find the "z-score" of 57 in this case.
57 - 52
It is z = ------------ , or z = 5/3, or z = 1.67.
3
Find the area to the left of z = 1.67. Multiply that area by 100% to find the percentile rank of the score 57.
What is the quotient (3x4 – 4x2 + 8x – 1) ÷ (x – 2)?
Answers
Answer:
the quotient of this problem is 3x^3+6x^+8x+24+47/x-2
Let x and y represent two real numbers. Write an algebraic expression to denote the quotient obtained when the product of the two numbers is divided by their sum.
Answers
An algebraic expression to denote the quotient obtained when the product of the two numbers is divided by their sum is (xy)/(x+y).
What is an expression?The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.
Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.
Given that let x and y represent two real numbers. The expression to denote the quotient obtained when the product of the two numbers is divided by their sum is written as,
E = (xy)/(x+y).
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Use cylindrical coordinates. find the volume of the solid that lies between the paraboloid z = x2 + y2 and the sphere x2 + y2 + z2 = 2
Answers
Let [tex]R[/tex] be the region bounded by the two surfaces. Then the volume of the region is given by
[tex]\displaystyle\iiint_R\mathrm dV=\int_{\theta=0}^{\theta=2\pi}\int_{r=0}^{r=1}\int_{\zeta=r^2}^{\zeta=\sqrt{2-r^2}}r\,\mathrm d\zeta\,\mathrm dr\,\mathrm d\theta[/tex]
[tex]=\displaystyle2\pi\int_{r=0}^{r=1}r(\sqrt{2-r^2}-r^2)\,\mathrm dr[/tex]
[tex]=\displaystyle2\pi\int_0^1(r\sqrt{2-r^2}-r^3)\,\mathrm dr[/tex]
[tex]=\dfrac{(8\sqrt2-7)\pi}6[/tex]
To find the volume of the solid that lies between the paraboloid z = x^2 + y^2 and the sphere x^2 + y^2 + z^2 = 2 using cylindrical coordinates, set up the integral for the volume by rewriting the sphere equation in cylindrical coordinates, determining the limits for r and z, and evaluating the integral.
Explanation:To find the volume of the solid that lies between the paraboloid z = x^2 + y^2 and the sphere x^2 + y^2 + z^2 = 2
We can rewrite the sphere equation in cylindrical coordinates as r^2 + z^2 = 2. The limits for r are from 0 to √(2-z^2), and for z, they are from 0 to √(2-r^2).
The volume can be found by integrating the constant 1 over the limits of r and z: V = ∭1 dz dr dθ. Evaluate this integral to find the volume of the solid.
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Evaluate the given integral by making an appropriate change of variables, where r is the rectangle enclosed by the lines x - y = 0, x - y = 7, x + y = 0, and x + y = 6.
Answers
[tex]\mathbf J=\dfrac{\partial(u,v)}{\partial(x,y)}=\begin{bmatrix}\dfrac{\partial u}{\partial x}&\dfrac{\partial u}{\partial y}\\\\\dfrac{\partial v}{\partial x}&\dfrac{\partial v}{\partial y}\end{bmatrix}=\begin{bmatrix}1&-1\\1&1\end{bmatrix}[/tex]
[tex]\implies\det\mathbf J=2[/tex]
The area of the region is then given by
[tex]\displaystyle\iint_R\mathrm dA=\int_{u=0}^{u=7}\int_{v=0}^{v=6}2\,\mathrm dv\,\mathrm du=84[/tex]
A mixture of
12%
disinfectant solution is to be made from
10%
and
17%
disinfectant solution. How much of each solution should be used if
28
gallons of the
12%
solution are needed?
Answers
Gallon equation:
x+y = 28
Solution equation:
10x + 17y = (28)(12)
Solve with substitution:
y = 28-x (From 1st equation)
(Sub into 2nd equation)
10x + 17(28-x) = 336
10x + 476 - 17x = 336
-7x = -140
x = 20
y = 28-20 = 8
Therefore the solution is (20,8)
You need 20 gal of 10% solution mixed with 8 gal of 17% solution to obtain 28 gal of 12% solution.
Juan will rent a car for the weekend. He can choose one of two plans. The first plan has an initial fee of
$55
and costs an additional
$0.30
per mile driven. The second plan has no initial fee but costs
$0.80
per mile driven.
For what amount of driving do the two plans cost the same?
What is the cost when the two plans cost the same?
Answers
.09m + 57.98 = .14m + 49.98
9m + 5798 = 14m + 4998