Final answer:
The area of grass that the goat can graze is calculated as a quarter-circle with a radius equal to the length of the rope, which is 6 metres, resulting in an area of approximately 28.27 square metres.
Explanation:
The area of grass a goat can graze when tethered by a 6 metre rope to the outside corner of a shed is a segment of the grassy field. To calculate this area, one must consider two scenarios based on the dimensions of the shed:
If the shed's dimensions of 4 metres by 5 metres do not allow the goat to reach the opposite corner, then the area grazed is a quarter-circle with the rope as the radius.
If the shed's dimensions allow the goat to reach around the corner, then additional area calculations are necessary for the extended grazing area beyond the corner.
In the stated problem, the rope's length does not allow the goat to graze beyond the opposite corner of the shed. Therefore, we calculate the area as a quarter-circle, which is a fraction (1/4) of the area of a full circle with radius 6 metres. The formula for the area of a circle is πr², where r is the radius.
To find the grazeable area, we use:
Area = (1/4) × π × radius² = (1/4) × π × 6² = (1/4) × π × 36
Area = 9π square metres (approximately 28.27 square metres)
The effective grazing area is 22.26 square meters.
The goat can graze in a quarter-circle sweep around the corner where the shed does not block its path.
Step-by-Step Calculation
The goat can graze around the corner up to a 6 metre radius, but the shed blocks its grazing in certain directions.
Calculate the total quarter-circle area the goat can cover:
Area = (1/4) * π * radius² = (1/4) * π * 6² = (1/4) * π * 36 = 9π square metres.
Subtract the area where the shed blocks grazing:
From one corner, the shed blocks a rectangular area measuring 4 meters by 6 meters.
Total area blocked by the shed: 6m * 4m = 24 square meters
Adjust for quarter-circle section, dividing the blockage by 4: 24 / 4 = 6 square meters
The area of grass the goat can graze = Total area - Blocked area
9π - 6 ≈ 28.26 - 6 ≈ 22.26 square metres
Thus, the total area grazed by the goat is 22.26 square meters.
round 0.7008 to the greatest nonzero place
The number 0.7008 rounded to the greatest non zero place is 1.
What is rounding of digit?Rounding is a process to estimate a particular number in a context.
To round a number look at the next digit in the right place, if the digit is less than 5, round down and if the digit is 5 or more than 5, roundup.
Now the given number is,
0.7008
To round the number to the greatest non zero place.
The greatest non zero digit in the number 0.7008 is 0.
The digit '0' in the given number is at ones place.
So, we will round the number 0.7008 to the nearest ones place.
Since, the digit to right the ones place that is tenths place is 7, which is greater than 5.
So, the digit will roundup to 1.
Thus, the number 0.7008 rounded to the greatest non zero place is 1.
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Find the highest common factor of 20 and 28
Answer:
4
Step-by-step explanation:
The factors of 20 are: 1, 2, 4, 5, 10, 20
The factors of 28 are: 1, 2, 4, 7, 14, 28
Then the greatest common factor is 4.
What mathematical relationship exists between human population and time
In his
garden Tim
plants the seed 4 and one fourth
in. below the ground. After one month the tomato plant has grown a total of 8 and one half
in. How many inches is the plant above the ground?
Express the area of this triangle as a monomial.
(picture below)
A. 28x^2
B. 1x
C. 11x^2
D. 14x^2
Iced tea, x, costs $4 per gallon and lemonade, y, costs $6 per gallon. You need to purchase at least 9 gallons of drinks for a neighborhood picnic, but have at most $55 to spend. Model the scenario with a system of inequalities. Which of the following options represents a possible solution to the system of inequalities?
Answer:
The system of inequalities is equal to
[tex]x+y\geq 9[/tex]
[tex]4x+6y\leq 55[/tex]
The graph in the attached figure
Step-by-step explanation:
Let
x------> the number of gallons of iced tea
y-----> the number of gallons of lemonade
we know that
[tex]x+y\geq 9[/tex] -----> inequality A
[tex]4x+6y\leq 55[/tex] -----> inequality B
using a graphing tool
the solution is the shaded area
see the graph in the attached figure
The answer is 10,1
The explanation is below. The graph is correct and the point 10,1 is in the shaded area that is covered by both equations.
Which of the following is equivalent to 4.5 m?
Select one:
a. 450 km
b. 450 cm
c. 0.45 km
d. 45 cm
Find parametric equations for the sphere centered at the origin and with radius 3. Use the parameters s and t in your answer.
Final answer:
Parametric equations for a sphere centered at the origin with radius 3 are x = 3 sin(t) cos(s), y = 3 sin(t) sin(s), and z = 3 cos(t), where t is the angle from the vertical and s is the angle from the x-axis on the xy-plane.
Explanation:
To find parametric equations for a sphere centered at the origin with radius 3 using parameters s and t, we make use of spherical coordinates. In spherical coordinates, a point on the sphere can be expressed as:
x = r sin(t) cos(s)
y = r sin(t) sin(s)
z = r cos(t)
Where r is the radius of the sphere, t is the angle with the vertical (ranging from 0 to π), and s is the angle on the xy-plane from the x-axis (ranging from 0 to 2π). Given that the radius r is 3, the parametric equations become:
x( s, t) = 3 sin(t) cos(s)
y( s, t) = 3 sin(t) sin(s)
z( s, t) = 3 cos(t)
Final answer:
To find the parametric equations for a sphere with radius 3 centered at the origin, we use spherical coordinates with parameters s and t to define the x, y, and z components in terms of trigonometric functions. The resulting parametric equations are x(s, t) = 3sin(t)cos(s), y(s, t) = 3sin(t)sin(s), and z(s, t) = 3cos(t).
Explanation:
Finding Parametric Equations for a Sphere
To find the parametric equations for a sphere centered at the origin with radius 3 using the parameters s and t, we use spherical coordinates. In spherical coordinates, you can represent any point on the surface of a sphere using two angles, commonly named θ (theta) and φ (phi), and the radius r. Since we're given a radius of 3, we can set r to be 3.
The standard spherical coordinates in terms of s and t can be expressed as:
x(s, t) = 3sin(t)cos(s)y(s, t) = 3sin(t)sin(s)z(s, t) = 3cos(t)Here, t represents the polar angle, measured from the positive z-axis, and s is the azimuthal angle in the xy-plane from the positive x-axis. The range of these parameters is usually 0 ≤ t ≤ π and 0 ≤ s < 2π to cover the entire surface of the sphere.
In our scenario, t and s would correspond to the angles θ and φ in spherical coordinates, respectively. However, we should specify the domain of the parameters to avoid ambiguity. This solution shows how to set up parametric equations for a sphere by using trigonometric functions to relate the spherical coordinates to Cartesian coordinates.
Write 4.4354 correct to 2 decimal places
The number 4.4354 correct to 2 decimal places is 4.44. The third digit, 5, indicates that the second decimal place is rounded up.
Explanation:The question is asking to write the number 4.4354 correct to 2 decimal places. When you're rounding a number to a certain number of decimal places, you look at the digit in the next spot decimal place. In this case, for two decimal places, we look at the third decimal. If this number is 5 or above, we 'round up' the second decimal place by one digit. If it is less than 5, we leave it as is. The third digit in this case is 5, so we 'round up' meaning the number 4.4354 written correct to 2 decimal places is 4.44.
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Given the function f(x) = 4 - 2x: If the domain is {-4, 0, 5}, find the range.
A) {-4, 4, -6}
B) {-4, 4, 14}
C) {12, 4, -6}
D) {12, 4, 14)
Which is bigger, 0.25 or 0.6?
Let f(x) = 8x3 - 28x + 61 and g(x) = 2x + 5. Find f(x) / g(x)
The result of the division is:
[tex]\[\boxed{4x^2 - 10x + 11 + \frac{6}{2x + 5}}\][/tex]
To find [tex]\( \frac{f(x)}{g(x)} \)[/tex] where [tex]\( f(x) = 8x^3 - 28x + 61 \)[/tex] and [tex]\( g(x) = 2x + 5 \)[/tex], we need to perform polynomial division.
Polynomial Division
1. Divide the leading term of the dividend by the leading term of the divisor:
[tex]\[ \frac{8x^3}{2x} = 4x^2 \][/tex]
2. Multiply the entire divisor by this result and subtract from the dividend:
[tex]\[ 8x^3 - 28x + 61 - (4x^2 \cdot (2x + 5)) = 8x^3 - 28x + 61 - (8x^3 + 20x^2) \][/tex]
Simplifying, we get:
[tex]\[ -20x^2 - 28x + 61 \][/tex]
3. Repeat the process with the new polynomial:
[tex]\[ \frac{-20x^2}{2x} = -10x \][/tex]
4. Multiply the entire divisor by this result and subtract:
[tex]\[ -20x^2 - 28x + 61 - (-10x \cdot (2x + 5)) = -20x^2 - 28x + 61 - (-20x^2 - 50x) \][/tex]
Simplifying, we get:
[tex]\[ 22x + 61 \][/tex]
5. Repeat the process again:
[tex]\[ \frac{22x}{2x} = 11 \][/tex]
6. Multiply the entire divisor by this result and subtract:
[tex]\[ 22x + 61 - (11 \cdot (2x + 5)) = 22x + 61 - (22x + 55) \][/tex]
Simplifying, we get:
[tex]\[ 6 \][/tex]
Putting it all together, we have:
[tex]\[\frac{f(x)}{g(x)} = 4x^2 - 10x + 11 + \frac{6}{2x+5}\][/tex]
So, the result of the division is:
[tex]\[\boxed{4x^2 - 10x + 11 + \frac{6}{2x + 5}}\][/tex]
The sum of two numbers is 28 . The first number, x, is three times the second number,y. Which system of equations can be used to find the two numbers
Answer:
d
Step-by-step explanation:
edge 2020
When mrs. myles gave a test, the scores were normally distributed with a mean of 72 and a standard deviation of 8. this means that 95% of her students scored between which two scores?
If you were solving a system of equations and you came to a statement like 1 = 3, what do you know about the solution to the system? (1 point)
The solution is (1, 3)
The solution is x = 1 and y = 3
There is no solution
There are infinitely many solutions
we know that
the statement
[tex]1=3[/tex] -----> is not true
so
The system has no solutions
For example
two parallel lines
[tex]y=5x+1[/tex] -----> equation A
[tex]y=5x+3[/tex] -----> equation B
The system has no solution. because the lines are parallel
If you equate equation A and equation B
[tex]5x+1=5x+3[/tex]
[tex]1=3[/tex] ------> is not true
therefore
the answer is the option
There is no solution
Answer:
There is no solution
Step-by-step explanation:
1 doesn't equal 3 so it can't have a solution
pleas help this problem
Answer: she needs to complete 8 more paintings
Step-by-step explanation:every month she completes 5 paintings
it has beeen 6 months
6*5=30 paintings
needs 38
38-30=8 paintings more
Ariel wants to write 8 entries for her blog. After 2 hours, she has 5 entries left. After 4 hours, she has 2 entries left. Which graph represents Ariel's remaining entries?
graph of line going through (0, 0) with a slope of 1.5
graph of line going through (0, 2) with a slope of 2
graph of line going through (0, 8) with a slope of 1.5
graph of line going through (0, 3) with a slope of 3
Let x be a time in hours and y be an amount of remaining entries. You have that:
At start Ariel wants to write 8 entries for her blog. This means that x=0 and y=8 (0 entries were written). The point with coordinates (0,8) belongs to needed graph.After 2 hours, she has 5 entries left. This gives you that at x=2, y=5 and graph passes through the point (2,5).After 4 hours, she has 2 entries left. This gives you that at x=4, y=2 and graph passes through the point (4,2).Find the slope of the line that is passing through the points [tex] (x_1,y_1)[/tex] and [tex] (x_2,y_2)[/tex] by formula
[tex] \dfrac{y_2-y_1}{x_2-x_1}.[/tex]
Thus, the slope is
[tex] \dfrac{5-2}{2-4}=-1.5.[/tex]
Answer: graph of line going through (0, 8) with a slope of -1.5
Joey had a summer job mowing lawns for 4 weeks. he made a total of $440. approximately how much did he make each week?
Answer: $110
Step-by-step explanation: $110 is the correct approximation. The easiest way is to divide the total by 4, since 25% represents 1 /4 .
Write a polynomial function of minimum degree with real coefficients whose zeros include those listed. Write the polynomial in standard form.
7, -11, and 2 + 6i
f(x) = x4 - 53x2 + 468x - 3080
f(x) = x4 - 9x3 - 42x2 + 234x - 3080
f(x) = x4 - 117x2 + 468x - 3080
f(x) = x4 - 9x3 + 42x2 - 234x + 3080
The polynomial function of minimum degree with real coefficients that has the zeros 7, -11, and 2 + 6i is obtained by also including the conjugate zero 2 - 6i and multiplying the corresponding factors. The proper polynomial is found after multiplying these factors and simplifying, but the given options do not match the correct polynomial. There may be an error in the given options.
Explanation:To write a polynomial function of minimum degree with real coefficients given the zeros 7, -11, and 2 + 6i, we must remember that complex zeros in polynomials with real coefficients always come in conjugate pairs. Therefore, the conjugate of 2 + 6i, which is 2 - 6i, must also be a zero of the polynomial.
First, we write a factor for each zero: (x - 7), (x + 11), (x - (2 + 6i)), and (x - (2 - 6i)). Multiplying these factors together will give us the required polynomial:
(x - 7)(x + 11)((x - 2) - 6i)((x - 2) + 6i)After multiplying and simplifying these factors, and combining like terms, the polynomial function in standard form with these zeros and with real coefficients is:
f(x) = x´ - 9x³ + 42x² - 154x + 3080
However, none of the given options match the correct polynomial derived from the provided zeros. It's possible there was a mistake in the multiplication or combining like terms, or there is an error in the options provided.
Which of these sentences includes an infinitive phrase??
A regular pentagon and a square share a mutual vertex B. The sides AB and BC are sides of a third polygon with vertex B. How many sides does this polygon have?
In a regular pentagon with equal sides and angles, the sum of the inside angles is 540 degrees. The third polygon P has 6 sides.
What is the Pentagon?In a regular pentagon with equal sides and angles, the sum of the inside angles is 540 degrees, and each angle is 108 degrees. Some actual items, including stars, soccer balls, and even street signs, contain pentagons.
Since P has a vertex at B, we can assume that its other sides emanate from B. Let's call the length of one of these sides x. Since the angle formed by AB and BC is a right angle, we can use the Pythagorean theorem to find the length of AC, which is the hypotenuse of the right triangle ABC:
AC² = AB^²+ BC²= s² + s² = 2s²
AC = √(2s²) = s x √(2)
Now, we can use the fact that AB and BC are sides of a regular pentagon and a square, respectively, to find the value of x. Since a regular pentagon has five sides of equal length, we know that AB is also a side of the Pentagon. Therefore:
AB = s
Similarly, since a square has all sides of equal length, we know that BC is also a side of the square. Therefore:
BC = s
Now, we can use the fact that the sum of the interior angles of a polygon with n sides is (n-2) x 180 degrees to find the value of n for polygon P. Since P has one right angle at B and two angles of 108 degrees (since a regular pentagon has interior angles of 108 degrees),
Set up the following equation:
90 + 108 + 108 + (n-3)180 = 180n
Simplifying, we get:
n = (540-90-108-108)/(180-180) + 3
n = 6
Therefore, the third polygon P has 6 sides.
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Talia took the bus from her home to the bank and then walked back to her home along the same route. The bus traveled at an average speed of 40 km/h and she walked at an average speed of 5 km/h. To determine the time, x, that it took Talia to walk home, she used the equation 40(0.9 – x) = 5x.
Talia took approximately 0.8 hours, or 48 minutes, to walk home.
Explanation:To determine the time it took Talia to walk home, we can solve the equation
40(0.9 – x) = 5x.
First, distribute the 40 to get 36 - 40x = 5x.
Combine like terms to get 36 = 45x.
Divide both sides by 45 to solve for x.
Therefore, Talia took approximately 0.8 hours, or 48 minutes, to walk home.
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To solve the equation 40(0.9 – x) = 5x, distribute 40, combine like terms, and solve for x. Talia took 0.8 hours or 48 minutes to walk home.
Explanation:The subject of this question is Mathematics and the grade level is High School.
To solve the equation 40(0.9 – x) = 5x, we can start by distributing 40 to the terms inside the parentheses: 36 – 40x = 5x.
Then, we combine like terms by adding 40x to both sides: 36 = 45x.
Finally, we divide both sides by 45 to find that x = 36/45 = 0.8.
Therefore, it took Talia 0.8 hours, or 48 minutes, to walk home.
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1. What is the slope of the line that passes through the points (-2, 5) and (1, 4)?
a. -3
b. -2
c. -1/3
d. 1/3
2. A line has a slope -5/3. Through which two points could this line pass?
a. (12,13) (17, 10)
b. (16, 15) (13, 10)
c. (0, 7) (3, 10)
d. (11, 13) (8, 18)
3. The pair of points (6,y) and (10, -1) lie on a line with slope 1/4. What is the value of y?
a. -5
b. -2
c. 2
d. 5
4. What is the slope of a vertical line?
a. -1
b. 0
c. 1
d. Undefined
5. The table below gives the best cost per person to rent a fishing charter boat. Find the rate of change given that it is a constant. Also explain what the rate of change means for this situation.
People l Cost ($)
2 l 110
3 l 165
4 l 220
5 l 275
a. 1/55
b. 110/1
c. 1/275
d. 55/1
The slope between points (-2, 5) and (1, 4) is -1/3. The pair of points (16, 15) and (13, 10) could lie on a line with a slope of -5/3. For a line with slope 1/4 passing through (6, y) and (10, -1), the value of y is 2. The slope of a vertical line is undefined. The rate of change in the charter boat cost is $55 per additional person.
Explanation:The slope of a line passing through two points can be found using the formula slope = (y2 - y1) / (x2 - x1). For the points (-2, 5) and (1, 4), the slope is (4 - 5) / (1 - (-2)) = -1 / 3, which is choice c, -1/3. For a line with a slope of -5/3, we can pick two points and use the slope formula to verify that the slope between them is -5/3. For example, using points (16, 15) and (13, 10), the slope would be (10 - 15) / (13 - 16) = -5 / -3, which is indeed 5/3, making choice b, (16, 15) (13, 10), correct.
For a pair of points (6, y) and (10, -1) lying on a line with slope 1/4, we can set up the equation 1/4 = (-1 - y) / (10 - 6) and solve for y to find that y = 2, so the answer is choice c, 2. The slope of a vertical line is undefined because the change in x is zero, hence choice d, Undefined, is the correct answer.
For the fishing charter boat cost problem, the rate of change or the cost per additional person is the difference in cost divided by the difference in the number of people. For example, for 3 and 2 people, the rate is (165 - 110) / (3 - 2) = 55, so the rate of change is 55 dollars per additional person, which is choice d, 55/1. This rate of change represents the additional cost for each extra person joining the fishing charter.
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20 POINT QUESTION
Solve the system of equations using the elimination method.
8x + 7y = 89
2x + 5y = 45
Explain each step and show how to verify your results.
PLEASE HURRY!!! i promise brainliest answer to the first CORRECT answer
8x + 7y =89
2x + 5y=45
You want to be able to subtract one away from the other one so you are just left with either x or y. to do this we could multiply the bottom equation by 4
so you would get
8x+7y=89
8x + 20y= 180
Now we can subtract equation 1 from equation 2 so you get
13y=91
this gives you y=7 so to find x we can simply substitute y into an equation;
2x + 5x7=45
2x + 35= 45
2x = 10
x =5
so the final answer is y = 7/ x = 5
An airplane flies 3/4 of a mile in 1/8 of a minute what is the airplane speed in miles per minute
Answer:
Step-by-step explanation:
Use proportions to figure this out. On the top you'll have miles and on the bottom you'll have minutes. I am going to change 3/4 to .75 and 1/8 to .125:
[tex]\frac{miles}{minute}:\frac{.75}{.125}[/tex]
We want to know how many miles per 1 minute that is. So we have the unknown on top in a new ratio, with 1 on the bottom:
[tex]\frac{miles}{minute}:\frac{.75}{.125}=\frac{x}{1}[/tex]
This second ratio is allowing us to calculate what the decimal ratio is in miles per 1 minute. Cross multiply to get
.125x = .75 so
x = 6 miles
That tells us that the decimal ratio is equivalent to 6 miles per minute
Which ordered pairs lie on the graph of the exponential function f(x)=128(0.5)x Question 1 options:
(0, 1)
(1, 64)
(3, 16)
(8, 0.5)
The ordered pairs lie on the graph of the exponential function is (3, 16).
What is Exponential Function?The formula f(x) = [tex]a^x[/tex]defines an exponential function, where x is the input variable as an exponent. The exponential curve is determined by the exponential function and the value of x.
We have,
f(x) = 180 (0.5[tex])^x[/tex]
Now, for x = 0,
So, f(x) = 128(0.5[tex])^0[/tex]
= 128(1)
= 128
Now, for x = 1,
f(x) = 128(0.5[tex])^1[/tex]
= 128(0.5)
= 256
Now, for x = 3,
f(x) = 128(0.5[tex])^3[/tex]
= 128(1/8)
= 16
So, the correct ordered pair is (3, 16).
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Right isosceles triangles are similar.
Always
Sometimes
Never
Answer:
always
I just did it on my website
Question Help
Factor the expression.
7y squared+10y+ 3
the length of lisa's rectangular dining room is 12 feet. if the area of the room is at least 96 square feet, what is the smallest width the room could have?
Expand (4x-3y)^4 using pascal's triangle ...?