The [tex]z[/tex]-score for a height of 16.4 feet is
[tex]z=\dfrac{16.4-14.2}{2.15}\approx1.023[/tex]
So
[tex]P(\text{height}>16.4\,\mathrm{ft})=P(Z>1.023)=1-F_Z(1.023)\approx0.153[/tex]
where [tex]F_Z(z)=P(Z\le z)[/tex] is the cumulative distribution function for the standard normal distribution.
PLEASE ANSWER 80 POINTS! PLEASE!
Riley’s mother, Ms. Cooper, owns Cooper’s Storage and Shipping Company. Ms. Cooper took Riley with her to work for the day to show Riley the different jobs the company does.
1. Riley noticed an aquarium in his mother’s office. The aquarium has the dimensions 16 in. by 8.5 in. by 10.5 in. The formula for volume is: V = l x w x h
(a) Riley noticed that the aquarium was three-fourths full of water. How many more cubic inches of water would be required to fill the tank? Show your work.
(b) Another aquarium in the building has dimensions that are each triple the dimensions of the aquarium in Riley’s mother’s office. Riley thought that the volume would also triple. Is Riley correct? How many times greater is the volume of the larger aquarium than the volume of the smaller one? Show your work and explain your reasoning.
(c) Riley also thought that the surface area would triple. Is Riley correct? How many times greater is the surface area of the larger aquarium than the surface area of the smaller one? Show your work and explain your reasoning.
Here is one more
Riley finds out that Cooper’s Storage and Shipping Company is working with a local business to package some office supplies. Some of the supplies are packed inside a cube-shaped box with side lengths of 4 1/2
in.
These boxes are then packed into a shipping box with dimensions of 18 in. 9 in. 4 1/2 in.
(a) How many boxes of office supplies can be packed into the larger box for shipping? Show your work.
(b) Sometimes the shipping boxes are protected with an outer covering because of weather. Draw a net of the shipping box. Use the net to find the surface area of the shipping box to help decide how much outer covering will be needed to protect one box. Show your work.
See the attached picture for the complete answers to both questions:
Question A asked how much more water was required to fill the tank, so you need to calculate the full volume, then subtract 3/4 of the volume.
Question B, volume is cubed, so you need to cube the scale factor of 3. 3^3 = 27, so the larger tank is 27 times more.
Question C) Area is squared so you need to square the scale factor. Surface area would be 9 times more. 3^2 = 9
For the 2nd question, calculate the volume of each box and divide to find the number of boxes.
Then see the picture for total surface area for part B.
A rectangle is inscribed in a right isosceles triangle, such that two of its vertices lie on the hypotenuse, and two other on the legs. What are the lengths of the sides of the rectangle, if their ratio is 5:2, and the length of the hypotenuse is 45 in? (Two cases)
CASE 1: (Blank) (blank) (Blank) (blank)
CASE 2: (Blank) (blank) (Blank) (blank)
Answer:
Part 1) The base is [tex]25\ in[/tex] and the height is [tex]10\ in[/tex]
Part 2) The base is [tex]7.5\ in[/tex] and the height is [tex]18.75\ in[/tex]
Step-by-step explanation:
case 1) Right isosceles triangle of the left
Let
x------> the base of the rectangle
y----> the height of the rectangle
Remember that
In a right isosceles triangle the lengths of the legs of the triangle is the same
[tex]y+x+y=45[/tex]
[tex]2y+x=45[/tex] ----> equation A
[tex]\frac{x}{y} =\frac{5}{2}[/tex]
[tex]x=2.5y[/tex] -----> equation B
substitute equation B in the equation A
[tex]2y+2.5y=45[/tex]
[tex]4.5y=45[/tex]
[tex]y=10\ in[/tex]
Find the value of x
[tex]x=2.5(10)=25\ in[/tex]
case 2) Right isosceles triangle of the right
Let
x------> the base of the rectangle
y----> the height of the rectangle
Remember that
In a right isosceles triangle the lengths of the legs of the triangle is the same
[tex]y+x+y=45[/tex]
[tex]2y+x=45[/tex] ----> equation A
[tex]\frac{y}{x} =\frac{5}{2}[/tex]
[tex]y=2.5x[/tex] -----> equation B
substitute equation B in the equation A
[tex]2(2.5x)+x=45[/tex]
[tex]6x=45[/tex]
[tex]x=7.5\ in[/tex]
Find the value of y
[tex]y=2.5(7.5)=18.75\ in[/tex]
In Case 1, the sides of the rectangle are 5x, 45, 45, and 2x. In Case 2, the sides of the rectangle are 5x, 45/sqrt(2), 45/sqrt(2), and 2x.
Explanation:Let's denote the length of one side of the rectangle as 5x and the length of the other side as 2x. Since the triangle is right isosceles, the two legs are of equal length. Let's denote the length of each leg as a. Using the Pythagorean theorem, we can write an equation: a^2 + a^2 = 45^2. Solving for a, we find that a = 45/sqrt(2).
Case 1: Since two vertices of the rectangle lie on the hypotenuse, the length of the rectangle's side that is parallel to the hypotenuse will be equal to the length of that hypotenuse, which is 45. Therefore, the sides of the rectangle will be 5x, 45, 45, and 2x.
Case 2: Since the two vertices of the rectangle lie on the legs of the triangle, the length of the rectangle's side that is parallel to the hypotenuse will be equal to the lengths of the legs, which is 45/sqrt(2). Therefore, the sides of the rectangle will be 5x, 45/sqrt(2), 45/sqrt(2), and 2x.
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Use the x-intercept method to find all real solutions of the equation x^3-6x^2+3x+10=0
Answer:
x ∈ {-1, 2, 5}
Step-by-step explanation:
The x-intercepts of the graph of the cubic are -1, 2, and 5. These are the values of x that are solutions to the equation.
Answer:
answer is D. -1, 2 , 5
Step-by-step explanation:
Which of the following expressions are equivalent to 4 + (14 - 2)
*First to answer gets the brainiest answer.
*Answer Fast Please!!!
14-2=12 +4= 16 is the answer to your question
Answer:
Step-by-step explanation:
there you go
Find the solution to Y=-x^2+3 for X=-3,0, and 3
Answer:
3-,3,0,3
Step-by-step explanation:
For this case we must evaluate the following quadratic equation, [tex]y = -x ^ 2 + 3,[/tex] for [tex]x = -3[/tex], [tex]x = 0[/tex]and [tex]x = 3[/tex]
For [tex]x = -3[/tex]:
[tex]y = - (- 3) ^ 2 + 3\\y = -9 + 3 =\\y = -6[/tex]
For [tex]x = 0[/tex]:
[tex]y = - (0) ^ 2 + 3\\y = 0 + 3\\y = 3[/tex]
For [tex]x = 3[/tex]:
[tex]y = - (3) ^ 2 + 3\\y = -9 + 3\\y = -6[/tex]
Answer:
[tex](-3, -6)\\(0,3)\\(3, -6)[/tex]
Find the missing terms of the geometric sequence 81,____, _____ 3
Subtract the two given numbers:
81 - 3 = 78
Divide by the number of missing numbers +1, so divide by 3:
78/3 = 26
This gives you the difference of each number in the sequence, so now subtract 26 from each one:
81 -26 = 55
55 - 26 = 29
29-26 = 3
The two missing numbers are 55 and 29.
To find the missing terms of a geometric sequence, divide each term by a common ratio.
Explanation:The given sequence is 81, ____, _____, 3.
To find the missing terms, we need to determine the common ratio of the geometric sequence.
Using the given sequence, we can see that each term is obtained by dividing the previous term by a constant factor. Therefore, the common ratio is 81 divided by the first missing term, which is then divided by the second missing term divided by the third missing term, which is then divided by 3.
To find the first missing term, we can divide 81 by the common ratio. Similarly, to find the second missing term, we can divide the first missing term by the common ratio. The missing terms of the geometric sequence are 27 and 9. Therefore, the complete sequence is 81, 27, 9, 3.
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Express 8.218.218, point, 21 as a mixed number.
Answer:
8 21/100
Step-by-step explanation:
8.21 is read "eight and twenty-one hundredths". The mixed number "eight and twenty-one hundredths" is written ...
8 21/100
Solve for x please I need help ASAP
Answer:
x > 2
Step-by-step explanation:
The inequality has constants and variables on both sides. It is convenient to find the variable term with the smallest (most negative) coefficient. Add the opposite of that term to both sides of the inequality.
2x -3 +5x > 11 -5x +5x . . . . added 5x
7x -3 > 11 . . . . . . . . . . . . . . . simplify
Now, find the constant on the side of the inequality that has the variable term. Add the opposite of that constant to both sides.
7x -3 +3 > 11 +3 . . . . . added 3
7x > 14 . . . . . . . . . . . . simplify
Finally, divide by the coefficient of x. It is positive, so we do not need to do anything to the relation symbol.
x > 14/7 . . . . . . . . . . divided by 7
x > 2 . . . . . . . . . . . . simplify . . . . This is your solution.
What is 27 3 over 8 minus
16 3 over 4
-6.625 or -6.63 rounded up
If △HLI ~ △JLK by the SSS similarity theorem, then is also equal to which ratio?
[tex]\rm \dfrac{HL}{JL}=\dfrac{IL}{KL}=\dfrac{HI}{JK}[/tex]
Step-by-step explanation:
Given :
[tex]\rm \bigtriangleup HLI \sim \bigtriangleup JLK[/tex] by the SSS similarity theorem.
[tex]\rm \dfrac {HL}{JL} = \dfrac{IL}{KL}[/tex]
Solution :
According to SSS postulate,
KL = IL ----- (1)
HL = JL ----- (2)
HI = JK ------ (3)
From equation (1), (2) and (3) we get,
[tex]\rm \dfrac{HL}{JL}=\dfrac{IL}{KL}=\dfrac{HI}{JK}[/tex]
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Triangle HLI has sides identical in measure to the three sides of triangle JLK, then the two triangles are congruent by SSS postulate. Therefore,
[tex]\rm \dfrac{HL}{JK}=\dfrac{IL}{KL}=\dfrac{HI}{JK}[/tex]
Given :
[tex]\rm \Delta HLI \sim \Delta JLK[/tex] by the SSS similarity.
According to SSS (Side Side Side) postulate:
If all three sides of a triangle are congruent to corresponding three sides of other triangle then the two triangles are congruent.
If triangle HLI has sides identical in measure to the three sides of triangle JLK, then the two triangles are congruent by SSS postulate.
Therefore, from triangle HLI and triangle JLK we get,
KL = IL ----- (1)
HL = JL ----- (2)
HI = JK ------ (3)
From equation (1) , (2) and (3) we get,
[tex]\rm \dfrac{HL}{JK}=\dfrac{IL}{KL}=\dfrac{HI}{JK}[/tex]
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What is 1/2x=18
can you please show steps. Thank you
Answer:
x = 36
Step-by-step explanation:
Multiply both sides of the equation by the inverse of the coefficient of x.
(2/1)·(1/2)x = (2/1)·18 . . . . . . . coefficient of x is 1/2, so we multiply by 2/1
x = 36
_____
This "multiplicative inverse" is also called the reciprocal. It has the property that when multiplied by the coefficient of x, the result is 1, the multiplicative identity element. So, your equation becomes ...
1·x = 36
The property of the multiplicative identity element is that anything multiplied by it is that thing. So, 1x = x, and your equation becomes
x = 36
This is the solution you're looking for.
The answer is 36 !!!!!
A delivery truck is transporting boxes of two sizes: large and small. The combined weight of a large box and a small box is 75 pounds. The truck is transporting 70
large boxes and 65 small boxes. If the truck is carrying a total of 5075
pounds in boxes, how much does each type of box weigh?
Answer:
large box weights 40 pounds & small box weights 35 pounds
Step-by-step explanation:
We can write 2 equations and solve simultaneous.
Let x be weight of large box and y be weight of small box
"The combined weight of a large box and a small box is 75 pounds.":
[tex]x+y=75[/tex]
"The truck is transporting 70 large boxes and 65 small boxes. If the truck is carrying a total of 5075 pounds":
[tex]70x+65y=5075[/tex]
Now we can solve for x in the first equation, which is x = 75 -y
We now substitute this into the 2nd equation and solve for y:
[tex]70x+65y=5075\\70(75-y)+65y=5075\\5250-70y+65y=5075\\-5y=5075-5250\\-5y=-175\\y=35[/tex]
Using y = 35 and plugging that into the 1st equation, we can solve for x:
[tex]x+y=75\\x+35= 75\\x= 75 - 35\\x=40[/tex]
Hence, large box weights 40 pounds & small box weights 35 pounds
Which choice correctly compares the two numbers?
A) 20.82 > 20.55
B) 21.02 < 20.55
C) 21.05 < 20.30
D) 21.22 > 21.30
Answer:
A) 20.82 > 20.55
Step-by-step explanation:
Hopefully, your issue is with the symbols (< vs >) rather than actually determining which number is larger or smaller.
The wide-open end of the symbol (the left side, in the case of >) indicates the larger (more positive) number.
So, the meanings of the symbols are ...
> — "is greater than"
< — "is less than"
The only true statement of those listed is ...
20.82 is greater than 20.55, or 20.82 > 20.55 . . . . selection A
_____
When writing number comparisons, I like to use the < symbol, because it puts the numbers in number-line order. That is, the smaller (or more negative) number is on the left, just as it is on a number line.
You trade the places of the numbers when changing the symbol. For example, answer choice A could be rewritten as ...
20.55 < 20.82
You know that 20.55 is to the left of 20.82 on the number line, so you know this statement is true.
A market sells strawberries for $5 per pint. Yesterday the market sold $46 worth of strawberries and today $52 worth. Which of the following is a good estimation of the total number of pints sold?
Answer:
yesterday 9 today 10
Step-by-step explanation:
46/5= 9.2
52/5= 10.4
A baker used 12 cups of batter to make muffins. It took 2/3 cup of batter to make 1 muffin. How many muffins did the baker make? (A) 6 muffins (B) 8 muffins (C) 18 muffins (D) 36 muffins
Dividing the total batter of 12 cups by the amount needed per muffin, which is 2/3 cup, reveals that the baker made 18 muffins. The answer is option C.
To find out how many muffins the baker made, we need to use the information given about the amount of batter used and how much batter is required for one muffin. According to the problem, the baker used 12 cups of batter in total and each muffin requires 2/3 cup of batter.
We start by setting up a simple division to find the answer:
(Total amount of batter) \/ (Amount of batter per muffin) = Number of muffins
12 cups \/ 2/3 cups per muffin = 18 muffins
This calculation reveals that the baker made 18 muffins. Therefore, the correct answer is (C) 18 muffins.
determine the down payment for this vehicle: $23, 400.00 sports car at 18% down
Answer:
$4212
Step-by-step explanation:
18% of $23,400 is ...
18/100 · $23,400 = $4212
Answer:
4212
Step-by-step explanation:
23400 * 0.18
As the angle θ increases to 90° the value of tan(θ)
A. decreases rapidly.
B. approaches +1.
C. increases rapidly.
D. approaches –1.
Answer:
C. increases rapidly.
Step-by-step explanation:
tan(θ) = sin(θ)/cos(θ)
Now, when sin 90 = 1
and cos 90 = 0
so, tan(90) = 1/0 = not defined.
(1/0 is infinity and its value is not defined)
So, when angle θ increases to 90°, then the value of tan(θ) increases rapidly, as shown in the figure below.
What is the radius of a circle whose equation is x2 + y2 + 8x – 6y + 21 = 0? units
Answer:
2
Step-by-step explanation:
x²+8x+y²-6y= -21;
(x+4)²+(y-3)²-25= -21;
(x+4)²+(y-3)²=2²;
it means r²=2²; ⇒ r=2.
Which of the following describes the given graph of the function over the interval [2, 6]? A. increasing B. constant C. decreasing D. decreasing to increasing
Final answer:
To determine the function's behavior over an interval, compare the y-values as the x-values increase. It could be increasing, decreasing, constant, or changing direction. Choose the description that matches the graph's movement over the given interval.
Explanation:
To determine how the function behaves over the interval [2, 6], we must look at the description of the graph presented. If during the interval the function is always moving upwards as the x-values increase, then it is increasing. If it moves downwards, it is decreasing. If the graph stays at the same level without moving up or down, then it would be constant. If the graph changes its direction, from either increasing to decreasing or vice versa, then it would be described as decreasing to increasing or increasing to decreasing, accordingly. Given these possibilities, the student should match the behavior of the graph with one of these descriptions.
Help me please! I need to fix this answer but dont know how to!! If you can help it would be greatly appreciated, just please remember to show your work. Btw ignore my attempt at the answer its obviously not correct.
Answer:
45.55 m
Step-by-step explanation:
The mnemonic SOH CAH TOA reminds you that the relationship between adjacent and opposite sides of an angle is ...
Tan = Opposite/Adjacent
Then the side opposite the angle of elevation will have length ...
tan(26°) = Opposite/(90 m)
Opposite = (90 m)·tan(26°) ≈ 43.90 m
This is the height of the top of the dam above Scarlett's height, so the total height to the top of the dam is ...
43.90 m + 1.65 m = 45.55 m
Two cyclists, Alan and Brian, are racing around oval track of length 450m on the same direction simultaneously from the same point. Alan races around the track in 45 seconds before Brian does and overtakes him every 9 minutes. What are their rates, in meters per minute?
Answer:
Alan: 200 m/min
Brian: 150 m/min
Step-by-step explanation:
Let "a" and "b" represent Alan and Brian's rates in meters per minute, respectively. The rate at which Alan is lapping Brian is the difference in their rates:
a - b = (450 m)/(9 min) = 50 m/min
This lets us write Brian's rate in terms of Alan's as ...
a -50 = b
The time to complete one oval differs by 3/4 minute (45 seconds), so we have ...
time = distance/speed
450/b - 450/a = 3/4
Multiplying by 4ab/3 gives ...
600(a -b) = ab
Substituting from above, we can rewrite this as ...
600·50 = a(a -50)
a^2 -50a -30000 = 0 . . . . . quadratic rearranged to standard form
(a -200)(a +150) = 0 . . . . . . .factored
a = 200 or -150 . . . . only the positive solution is useful here
Alan's rate is 200 m/min; Brian's rate is 150 m/min.
_____
Check
It takes Alan 450/200 min = 2.25 min to complete one oval. It takes Brian 450/150 = 3 min to complete one oval. That is 3-2.25 = 0.75 min = 45 seconds longer than Alan.
After 9 minutes, Alan will have gone (200 m/min)·(9 min) = 1800 m = 4 laps, while Brian will have gone (150 m/min)·(9 min) = 1350 m = 3 laps. Hence Alan will overtake Brian at the 9-minute mark.
The slope of a roof is called the pitch and is defined as follows:
pitch = rise of roof
½ span of roof
Find the pitch of a roof if the rise is 12 feet and the span is 30 feet.
Find the pitch of a roof if the rise is 4 feet and the span is 24 feet.
Answer:
a) 4/5
b) 1/3
Step-by-step explanation:
Put the numbers in the formula and evaluate.
a) pitch = (rise)/(1/2·span)
= (12 ft)/(1/2·30 ft) = 12/15 = 4/5
___
b) pitch = (4 ft)/(1/2·24 ft) = 4/12 = 1/3
The pitch of a roof is calculated by dividing the rise of the roof by half the span of the roof. The pitch of a roof with a 12 feet rise and 30 feet span is 0.8. The pitch of a roof with a 4 feet rise and 24 feet span is 0.333.
Explanation:The pitch of a roof can be calculated using the equation: pitch = rise of roof / (½ span of roof).
For the first part of your question, where the rise is 12 feet and the span is 30 feet, you substitute these values into the above equation: pitch = 12 / (½ * 30) = 0.8
For the second part, where the rise is 4 feet and the span is 24 feet, when these values substituted into the equation you get: pitch = 4 / (½ * 24) = 0.333
So, the pitch of the first roof is 0.8 and the pitch of the second roof is 0.333
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find the value of the greater root of x^2-6x+5=0
Answer:
5
Step-by-step explanation:
the roots are:
[tex]\left \{ {{x_1+x_2=6} \atop {x_1*x_2=5}} \right. \ => \ \left \{ {{x_1=1} \atop {x_2=5}} \right.[/tex]
Which of the following systems is equivalent to the given system?
2/3 x - y = 2 x + 1/2y = -3
a) 2x - 3y = 6 and 2x + y = 6
b)6x - 3y = 6 and 2x + y = 10
c) 2x - 3y = 6 and 2x + y = -6
Answer:
The correct answer is c.
Step-by-step explanation:
This is because it is the first equation being multiplied by 3 and the second being multiplied by 2.
None of the other examples are multiples of the originals.
Which of the following situations involve a permutation?
Select ALL the correct answers.
A) Determining how many different ways 7 runners can be assigned lanes on a track for a race
B) Determining how many 5-letter passwords can be made using the word "graph."
C) Determining how many different groups of 10 students can be chosen to go on a field trip from a group of 25 students
D) Determining how many different ways to choose 3 employees from a group of 9 employees.
E) Determining how many different seating charts can be made placing 6 people around a table
F) Determining how many different ways 4 cashiers can be chosen to work from a group of 6 cashiers.
Answer:
A, B, E
Step-by-step explanation:
Permutations are involved when order matters, as in lane assignment, passwords, and seating charts.
When the end result is a "group of 10 students", "3 employees", or "4 cashiers", clearly order does not matter. One student, employee, or cashier is as good as another in these cases.
Using it's definition, it is found that these following situations involve permutations:
A) Determining how many different ways 7 runners can be assigned lanes on a track for a race.
B) Determining how many 5-letter passwords can be made using the word "graph."
E) Determining how many different seating charts can be made placing 6 people around a table.
When are permutations used?Permutations are used when the order of the elements is important.The number of possible permutations of x elements from a set of n elements is given by:
[tex]P_{(n,x)} = \frac{n!}{(n-x)!}[/tex]
In this problem:
In items A, B and E, the order is important, for example, "rgaph" is a different word than "graph", hence they are permutations.In items C, D and F, the order is not important, hence they are not permutations, they are combinations.You can learn more about permutations at brainly.com/question/25247153
Explain why a rotation of 270∘ clockwise will result in the same transformation as a rotation of 90∘ counterclockwise
Because there is a maximum rotation of 360°, so if you rotate x° clockwise it's the same as if you rotate (360-x)° counterclockwise.
At least how many times must you flip a fair coin before there is at least a 50% probability that you will get at least three heads?
To get at least a 50% probability of getting at least three heads, you must flip a fair coin at least four times.
Explanation:To calculate the minimum number of times a fair coin must be flipped before there is at least a 50% probability of getting at least three heads, we can use the concept of probability. Let's assume that flipping a coin is a binary event with two possible outcomes: heads (H) or tails (T).
The probability of getting three or more heads in one flip is zero, since we can only get a maximum of two heads in one flip. However, we can calculate the probability of getting three or more heads in a series of flips.
The probability of getting at least three heads in one flip is zero, but we can calculate the probability of getting at least three heads in two flips, three flips, four flips, and so on. We continue calculating the probability until we find the number of flips that gives us a probability of at least 50%.
Using this method, we find that we need to flip the coin at least 4 times to have a probability of at least 50% of getting at least three heads. This means that after flipping the coin four times (or more), there is a better than 50% chance of getting at least three heads.
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I need help!!! Please explain how to do the question!
Answer:
Step-by-step explanation:
You do x+75=250 using the information given
Answer:
the answer is 175
Step-by-step explanation:
$250
-
$75
-------------
$175
Which function has an inverse that is not a function?
f(x) = x^2
f(x) = 2x
f(x) = x + 2
f(x) = √x
Answer:
your choice is correct
Step-by-step explanation:
f(x) = x^2 does not pass the horizontal line test (a horizontal line intersects its graph in two places), so its inverse does not pass the vertical line test. The inverse is not a function.
_____
Comment on the graph
The original function f(x)=x^2 is shown by the red curve. Its reflection across the orange dashed line y=x gives the inverse relation, in blue. The black horizontal and vertical lines show the multiple points of intersection with the curves, indicating the inverse relation is not a function.
Answer:
A) f(x)=x^2
Step-by-step explanation:
did it on i-ready
identify whether each equation has no solution, one solution, or infinitely many solutions.
1. 4x−x=2x+x
2. 2x+1=5
3. 4x+2=5x−x+4
4. 2(x+4)=4(x+2)
Answer:
infinitely manyone solutionno solutionone solutionStep-by-step explanation:
1. 4x−x=2x+x
Simplifies to 3x = 3x, which is true for all values of x. Hence there are infinitely many solutions.
___
2. 2x+1=5
True only for x=2; one solution.
___
3. 4x+2=5x−x+4
Simplifies to ...
4x +2 = 4x +4
2 = 4 . . . . . . . not true for any value of x; no solution.
___
4. 2(x+4)=4(x+2)
Simplifies to ...
2x +8 = 4x +8
0= 2x . . . . . . . . . subtract 2x+8
True only for x=0; one solution.