The degree of the product of two polynomials p(x) and q(x) is equal to the sum of the degrees of p(x) and q(x).
Explanation:The degree of the product of two polynomials p(x) and q(x) is related to the degrees of p(x) and q(x) by the following rule:
The degree of the product is equal to the sum of the degrees of the two polynomials.
For example, if p(x) is a polynomial of degree m and q(x) is a polynomial of degree n, then the product p(x)q(x) will be a polynomial of degree m+n.
A candy store sells walnuts for $8 per pound, and pecans for $2 per pound. If the store wants to create a bag of mixed nuts that sells for $6 per pound, how many pounds of pecans must be mixed with two pounds of walnuts?
the height y (in feet) of a ball thrown by a child is
[tex]y = \frac{1}{16} {x}^{2} + 4x + 3[/tex]
where x is the horizontal distance in feet from the point at which the ball is thrown
a).how high is the ball when it leaves the child's hand
b).what is the maximum high of the ball
c).how far from the child does the ball strike the ground
a. The height of the ball when it leaves the child's hand is 3 feet, as [tex]\(x = 0.[/tex]
b. The maximum height of the ball is 195 feet, calculated by finding the vertex of the parabolic function.
c. The distance from the child where the ball strikes the ground is approximately 47.24 feet, using the quadratic formula.
Given the equation representing the height y of a ball as a function of the horizontal distance x :
[tex]\[y = \frac{1}{16} x^2 + 4x + 3\][/tex]
Part (a): Height of the ball when it leaves the child's hand
The ball leaves the child's hand when x = 0.
Substituting x = 0 into the equation:
[tex]\[y = \frac{1}{16} \cdot 0^2 + 4 \cdot 0 + 3 = 3\][/tex]
Thus, the height of the ball when it leaves the child's hand is 3 feet.
Part (b): Maximum height of the ball
The given function represents a parabola, and since [tex]\(a = \frac{1}{16}[/tex] is positive, it opens upward.
To find the maximum height, we need the x-coordinate of the vertex, which can be found using:
[tex]\[x = -\frac{b}{2a} = -\frac{4}{2 \cdot \frac{1}{16}} = -\frac{4}{\frac{1}{8}} = -32\][/tex]
However, the correct derivation is:
[tex]\[x = -\frac{4}{2 \cdot \frac{1}{16}} = -\frac{4}{\frac{1}{8}} = -32 \cdot 8 = -64\][/tex]
But the correct value for x should be:
[tex]\[x = -\frac{4}{2 \times \frac{1}{16}} = -4 \times 8 = -32[/tex]
Considering this, let's correct the previous explanation and use the correct approach:
1. Find the x-coordinate of the vertex:
[tex]\[x = -\frac{4}{2 \times \frac{1}{16}} = -8 = 32\][/tex]
With the x-coordinate of the vertex, find the maximum height (the y-coordinate of the vertex):
[tex]\[y = \frac{1}{16} \cdot 32^2 + 4 \cdot 32 + 3 = \frac{1024}{16} + 128 + 3 = 64 + 128 + 3 = 195\][/tex]
Thus, the maximum height is 195 feet
Part (c): How far from the child does the ball strike the ground?
To find out when the ball strikes the ground, we need to set \(y = 0\) and solve for \(x\):
[tex]\[0 = \frac{1}{16} x^2 + 4x + 3\][/tex]
Using the quadratic formula, where[tex]\(a = \frac{1}{16}\), \(b = 4\), and \(c = 3\):[/tex]
[tex]\[x = \frac{-4 \pm \sqrt{4^2 - 4 \cdot \frac{1}{16} \cdot 3}}{2 \cdot \frac{1}{16}} = \frac{-4 \pm \sqrt{16 - \frac{12}{16}}}{\frac{1}{8}} \][/tex]
To find the x-coordinates where the ball strikes the ground, you can set (y = 0) and solve for (x):
[tex]\[0 = \frac{1}{16} x^2 + 4x + 3\][/tex]
Using the quadratic formula, where [tex]\(a = \frac{1}{16}\), \(b = 4\), and \(c = 3\) :[/tex]
[tex]\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\][/tex]
Substitute the given values of a ,b , and c into the quadratic formula:
Calculate [tex]\(b^2 - 4ac\) :[/tex]
- [tex]\(b^2 = 4^2 = 16\),[/tex]
- [tex]\(4 \cdot \frac{1}{16} \cdot 3 = \frac{12}{16} = \frac{3}{4}\),[/tex]
- So [tex]\(b^2 - 4ac = 16 - \frac{3}{4} = \frac{64}{4} - \frac{3}{4} = \frac{61}{4}\).[/tex]
Calculate the roots :
- The value for [tex]\(a\) is \(\frac{1}{16}\), so \(2a = \frac{1}{8}\),[/tex]
- Now we can find \(x\):
[tex]\[ x = \frac{-4 \pm \sqrt{16 - \frac{3}{4}}}{\frac{1}{8}} = \frac{-4 \pm \sqrt{61/4}}{1/8} = -32 \pm 2 \sqrt{61}. \][/tex]
Simplifying the roots :
- The roots for x are:
[tex]\[ x_1 = -32 + 2 \sqrt{61}, \, x_2 = -32 - 2 \sqrt{61}. \][/tex]
However, we consider only the positive root, as it represents the distance from the point where the ball is thrown to where it lands.
Thus, the value of [tex]\(x_1\)[/tex] is:
[tex]\[x = -32 + 2 \sqrt{61} \approx -32 + 15.6 = -16 + 16 \cdot \sqrt{61} \approx 47.24\][/tex]
Thus, the correct answer with [tex]\(x\approx 47.24\).[/tex]
The complete question is : The height y (in feet) of a ball thrown by a child is
[tex]y = (1)/(16) {x}^{(2)}+ 4x + 3[/tex]
where x is the horizontal distance in feet from the point at which the ball is thrown
a).how high is the ball when it leaves the child's hand ?
b).what is the maximum high of the ball ?
c).how far from the child does the ball strike the ground ?
The answer provides step-by-step explanations for finding the initial height, maximum height, and horizontal distance at which the ball strikes the ground, given the equation of the ball's height as a function of the horizontal distance.
To find the height of the ball when it leaves the child's hand (a), we need to determine the value of y when x is 0. Substituting x = 0 into the equation, we get:
y = -1/14(0)² + 2(0) + 3
y = 3 feet
Therefore, the ball is 3 feet high when it leaves the child's hand.
let's find the maximum height of the ball (b), we need to find the vertex of the quadratic equation. The vertex can be found using the formula x = -b/2a, where a, b, and c are the coefficients of the quadratic equation. In this case, a = -1/14, b = 2, and c = 3. Substituting these values into the formula, we get:
x = -2 / (2 * (-1/14))
x = 7 feet
Substituting x = 7 into the equation, we can find the maximum height:
y = -1/14(7)² + 2(7) + 3
y = 10.5 feet
Therefore, the maximum height of the ball is 10.5 feet.
let's find how far from the child the ball strikes the ground (c), we need to find the x-coordinate when y = 0. Setting y = 0 in the equation, we get:
0 = -1/14x² + 2x + 3
Solving this quadratic equation,
hence we find two possible values for x: x = -7 and x = 28. Since the ball is thrown in a positive direction, the ball strikes the ground 28 feet away from the child.
Suppose Tommy walks from his home at (0, 0) to the mall at (0, 8), and then walks to a movie theater at (5, 8). After leaving the theater Tommy walks to the store at (5, 0) before returning home. If each grid square represents one block, how many blocks does he walk?
The probability of winning a game is 25% . How many times should you expect to win if you play 36 times ?
1. 3 times
2. 7 times
3. 9 times
4. 11 times
A system of equations can be solved by elimination in table
ken and leah. are trying to solve a science homework question. They need to find out how much a rock that weights 4 pounds on earth would weight on Venus. They know they can multiply the amount the rock weights on earth by 0.91 to find its weight on Venus. Select the two partial products ken and leah would need to add to find the product of 4 and 0.91.
The Weight on Venus is 3.64 pounds
What is Unitary Method?The unitary technique involves first determining the value of a single unit, followed by the value of the necessary number of units.
For example, Let's say Ram spends 36 Rs. for a dozen (12) bananas.
12 bananas will set you back 36 Rs. 1 banana costs 36 x 12 = 3 Rupees.
As a result, one banana costs three rupees. Let's say we need to calculate the price of 15 bananas.
This may be done as follows: 15 bananas cost 3 rupees each; 15 units cost 45 rupees.
Given:
They need to find out how much a rock that weights 4 pounds on earth would weight on Venus.
So, the Weight on Venus
= 4 x 0.91
= 4x 0.9+ 4 x 0.01
= 3.6+ 0.04
= 3.64
Hence, the Weight on Venus is 3.64 pounds
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what is the distance between 1st and 3rd base?
The distance between the first base and third base in the given problem is 127.28 feet.
Distance is a measure of how far apart two objects are. It's a scalar quantity, which means that it has magnitude but no direction. The distance between two points can be gauged in various ways, including using a ruler, a tape measure, or a laser rangefinder. The primary difference between the distance and the displacement is that the latter has the direction but not the former.
As evident from the figure,
the running track is a square.
So, all the sides are equal to each other in length.
So, the distance between the first base and the second base is equal to the distance between that of the second base and the third base which is equal to 90 feet.
Let us name the square ABCD.
Join vertex A and vertex D.
The triangle ABC is the right-angled isosceles triangle.
So, by Pythagorean Theorem,
[tex]\sqrt{AC^2}[/tex] = [tex]\sqrt{AB^2+BC^2}[/tex]
= [tex]\sqrt{90^2 + 90^2}[/tex]
= [tex]\sqrt{8100+8100}[/tex]
= [tex]\sqrt{16200}[/tex]
= 127.28 (approximately)
So, the first base and third base are 127.28 feet apart.
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Help me with a Geometry problem! (10 points)
16,500 pounds
25,000 ounces
1,000 pounds 200 ounces
7 tons 200 pounds 100 ounces
From these from least to greatest Plz.
help please someone this is really important
A moving company had one sixth of a ton weight to move across town. If they wanted to split it equally amongst 4 trips, how much weight would they have on each trip
Answer:
And.Each Trip will have to take 41.667kilograms
Step-by-step explanation:
1/6×1000÷4=41.6666667
3. In a game, if you roll a 6 on a 6-sided number cube, you lose a turn.
(Part a) What is the probability that you roll a 6? Explain your reasoning.
(Part b) What is the probability that you either roll a 6 or do not roll a 6? Explain your reasoning.
(Part c) What is the probability that you don’t roll a 6? Explain your reasoning.
a dragonfly the fastest insect, can fly a distance of 50 feet at a speed of 25 feet per second. find the time in seconds. write the equation in form d=
Using the equation 't = d/v', where 'd' is distance and 'v' is velocity, the time 't' for a dragonfly to fly 50 feet at 25 feet per second is calculated to be 2 seconds.
To find the time it takes for a dragonfly to fly a distance of 50 feet at a speed of 25 feet per second, we can use the formula d = vt, where d is the distance, v is the velocity (speed), and t is the time. To solve for time (t), we rearrange the formula to t = d/v.
So, plugging in the values:
Distance (d) = 50 feetSpeed (v) = 25 feet/secondWe get:
t = d/v = 50 feet / 25 feet/second = 2 seconds
The dragonfly takes 2 seconds to fly 50 feet at a speed of 25 feet per second.
The coordinate of the vertices for square CDEF translated 3 units right and 4 units up if the vertices are C(-3,1), D(1,5), E(5,1), and F(1,-3) are. C’(0,5), D(4,6), E,(8,6) and F(4,-1). Is it true or false
Answer:
false
Step-by-step explanation:
fr ong
Garrett must inspect a bridge that goes over a river. He begins 18 feet above water level to inspect the lighting, descends 32 feet to inspect the base of the bridge underwater, then descends another 4 feet to check on the concrete anchoring the base. Which best represents Garrett’s position with respect to the water level?
–54 feet
–30 feet
–18 feet
–10 feet
Answer: –18 feet
Step-by-step explanation:
Given: Garrett begins 18 feet above water level to inspect the lighting, descends 32 feet to inspect the base of the bridge underwater, then descends another 4 feet to check on the concrete anchoring the base.
Then using integers, the expression which represents Garrett’s position with respect to the water level is given by :-
[tex]+18-32-4=-18[/tex]
Hence, Garrett’s position with respect to the water level =-18 feet.
Find the perimeter of right triangle WXY if the radius of the circle is 4 units and WY = 20 units.
The perimeter of the right triangle WXY is 60 units.
Explanation:To find the perimeter of a right triangle WXY, we need to know the lengths of the two legs. Given that the radius of the circle is 4 units and WY is 20 units, we can determine that the hypotenuse, XY, is equal to 20 units. So the perimeter of the triangle is the sum of the lengths of its three sides: 20 + 20 + 20 = 60 units.
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We calculate the perimeter of the right triangle by first finding the lengths of the legs using the Pythagorean theorem and the given radius, then summing these lengths with the hypotenuse.
Explanation:The subject of this question is Geometry, specifically involving the calculation of the perimeter of a right triangle. In a right triangle, the lengths of the three sides follow the Pythagorean theorem, written as [tex]a^2 + b^2 = c^2[/tex], where c is the hypotenuse and a and b are the lengths of the other two sides.
Here, the length of the hypotenuse WY is given as 20 units. In a right-angled triangle, where a circle is inscribed, the radius of the circle is equal to half the sum of the two legs of the triangle (radius = (a+b)/2). In this case, the radius is given as 4 units. Thus, the total of the two legs a and b is 8 units.
The Pythagorean theorem can now be used to solve for a and b, knowing that [tex]a^2 + b^2 = 20^2[/tex] and a + b = 8; then to find the perimeter, sum the hypotenuse and the lengths of the legs, i.e. Perimeter = a + b + c.
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Solve this system by graphing: y=3/2x-1 and x=-2
raj is visiting the united states and needs to convert 2000 rupees to us dollars
Answer:31.88
Step-by-step explanation:
Mary collects dolls. She has 4 dolls more then her friend Penney. Mary says if she tripled the number of dolls she has, she would have 27 dolls. How many dolls does Penney have?
what ia the length of the hypotenuse triangle below?
Answer: 18 - apex
Step-by-step explanation:
Solve the system of linear equations by substitution check your solution show work
Y-x=23
Y=3x+11
A) 3 ,20
B)6,29
C)24,47
D)4,27
2 x-y=30
2.5 +y=5
A)8,-22
B) 122.5, 92.5
C) 10 , -20
D)11, -22.5
true or false the area of a regular hexagon can be found by breaking the hexagon into 6 congruent triangles and then taking the sum of their areas
Joanna has a total of 7 rolls of coins worth $23. She has rolls of dimes worth $5 each, and rolls of nickel worth $2. How many rolls of each kind does she have?
There is 4 roll of coins of nickel and 3 rolls of coins of dimes.
Given that,
Joanna has a total of 7 rolls of coins worth $23.
She has rolls of dimes worth $5 each and rolls of nickel worth $2.
We have to determine,
How many rolls of each kind does she have?
According to the question,
Let the number of coins of dimes be x
And the number of coins of nickel be y.
Then,
The number of coins of dimes + Number of coins of nickel = Total number of coins.
[tex]\rm x+ y =7[/tex]
And the total price of the coins is equal to the price of each coin of dimes and nickel.
[tex]\rm 5x+2y=23[/tex]
Solving both the equation,
From equation 1,
[tex]\rm x+ y = 7\\\\x = 7-y[/tex]
Substitute the value of x in equation 2,
[tex]\rm 5x+2y = 23\\\\5(7-y) +2y = 23\\\\35-5y+2y=23\\\\-3y = 23-35\\\\-3y = -12\\\\y = \dfrac{-12}{-3}\\\\y = 4[/tex]
Substitute the value of y in equation 1,
[tex]\rm x+y = 7\\\\x+4 = 7\\\\x = 7-4\\\\x =3[/tex]
Hence, there is 4 roll of coins of nickel and 3 rolls of coins of dimes.
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Which of the following shows that polynomials are closed under subtraction when polynomial x + 5 is subtracted from 3x2 + 2x − 6?
3x2 + x − 11 may or may not be a polynomial
3x2 + x − 11 will be a polynomial
3x2 + x − 1 will be a polynomial
3x2 + x − 1 may or may not be a polynomial
A jar contains 5 blue marbles, 7 yellow marbles, and 8 green marbles. What is the probability of randomly choosing a blue marble, not replacing it, and then choosing a green marble?
The probability of randomly choosing a blue marble, not replacing it, and then choosing a green marble is,
⇒ 2 / 19
What is mean by Probability?The term probability refers to the likelihood of an event occurring. Probability means possibility. It is a branch of mathematics that deals with the occurrence of a random event. The value is expressed from zero to one.
Given that;
A jar contains 5 blue marbles, 7 yellow marbles, and 8 green marbles.
Hence, Number of marbles = 5 + 7 + 8 = 20
Thus, the probability of randomly choosing a blue marble is,
⇒ 5 / 20
⇒ 1 / 4
And, the probability of randomly choosing a green marble is,
⇒ 8 / 19
Thus, the probability of randomly choosing a blue marble, not replacing it, and then choosing a green marble is,
⇒ 1/4 × 8/19
⇒ 2 / 19
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Which of these four sets of side lengths will form a right triangle?
Set 1
4 cm, 5 cm, 6 cm
Set 2
8 in., 12 in., 20 in.
Set 3
10 mm, 20 mm, 30 mm
Set 4
12 ft, 16 ft, 20 ft
Set of sides that will make right triangle is
Set 4 : 12 ft, 16 ft, 20 ft
What is Pythagoras theorem?Pythagoras theorem states that in a right angled triangle square of the hypotenuse is equal to the sum of the square of perpendicular and square of base. The longest side is the hypotenuse.
Given measurements
Set 1:
4cm , 5cm , 6cm
By Pythagoras theorem
4² + 5² = 6²
16 + 25 = 36
41 ≠ 36
Pythagoras theorem is not satisfied,
this set of sides will not make right triangle
Set 2
8 in., 12 in., 20 in.
By Pythagoras theorem
8² + 12² = 20²
64 + 144 = 400
208 ≠ 400
Pythagoras theorem is not satisfied,
this set of sides will not make right triangle
Set 3
10 mm, 20 mm, 30 mm
By Pythagoras theorem
10² + 20² = 30²
100 + 400 = 900
500 ≠ 900
Pythagoras theorem is not satisfied
this set of sides will not make right triangle
Set 4
12 ft, 16 ft, 20 ft
By Pythagoras theorem
12² + 16² = 20²
144 + 256 = 400
400 = 400
Pythagoras theorem is satisfied
this set of sides will make right triangle.
Hence, Set 4 of sides 12 ft, 16 ft, 20 ft will form a right triangle.
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Please help me on number 8 and explain ?
Can people help me please to find the volume about question number 10 please and tell me the answer step by step please I need help
find two fractions with a difference of 1/2 but with neither denominator equal to 2
Two pairs of fractions that meet the criteria are 3/4 and 1/4, and 5/6 and 1/3, as the difference of each pair reduces to 1/2.
Explanation:
Finding two fractions with a difference of 1/2 where neither denominator is equal to 2 requires a bit of trial and error. An example of this could be the two fractions 3/4 and 1/4. If you subtract 1/4 from 3/4, you get 2/4, which reduces to 1/2. This meets the requirements of the question as neither denominator is equal to 2 and the difference of the two fractions is 1/2.
For another example, consider the fractions 5/6 and 1/3. Subtract 1/3 from 5/6, and your result is 3/6, which likewise reduces to 1/2. Again, this answer meets the criteria set by the question.
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A rectangular mat has a length of 12 in. and a width of 4 in. Drawn on the mat are three circles. Each circle has a radius of 2 in. Some friends play a game throwing darts onto the mat to see who can land a dart in one of the circles.
Assuming that a dart always lands on the mat, what is the probability that a dart thrown at random will land inside one of the circles?
Use 3.14 to estimate Pi.
Enter your answer, as a decimal rounded to the nearest hundredth, in the box.
P(landing in a circle) =
[tex] |\Omega|=4\cdot12=48\\
|A|=3\cdot3.14\cdot2^2=37.68\\\\
P(A)=\dfrac{37.68}{48}\approx0.79 [/tex]
Answer: 77.9
Step-by-step explanation: .