Answer:
90 miles
Step-by-step explanation:
We know
She has driven 30 miles so far, which is one-third of the way home.
What is the total length of her drive?
We Take
30 x 3 = 90 miles
So, the total length of her drive is 90 miles.
By rounding to 1 significant figure , estimate the answer to the questions
216×876
The rounding of the number to 1 significant figure is-
216 × 876 = 180000
What is defined as the significant figure?The term significant figures describes the number of significant single digits (0 to several 9 inclusive) in a scientific notation coefficient.The number of significant figures inside an expression indicates the degree of certainty or precision with where an engineer or scientist states a number.All zeros to the right of the decimals but to the left of a non-zero number in a decimal number between 0 and 1 are not significant.0.00247, for example, only has three significant figures.216 × 876
This number can be written in form of rounding to 1 significant figure as;
200 × 900 = 180000
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Suppose f is a continuous function defined on a rectangle R=[a,b]X[c,d]. What is the geometric interpretation of the double integral over R of f(X,y) if f(X,y)>0
If f(x,y) > 0 and is a continuous function defined over a rectangle R=[a,b]x[c,d], then the double integral over R of f(x,y) can be interpreted as the volume of a solid that lies in the first octant and under the graph of the function f(x,y) over the region R.
The geometric interpretation of the double integral over R of f(x,y) if f(x,y) > 0, where f is a continuous function defined on a rectangle R = [a,b] × [c,d] is given as follows:
The double integral of f(x,y) over R, if f(x,y) > 0, gives the volume under the graph of the function f(x,y) over the region R in the first octant.
Consider a point P (x, y, z) on the graph of f(x, y) that is over the region R, and let us say that z = f(x,y). If f(x,y) > 0, then P is in the first octant (i.e. all its coordinates are positive).
As a result, the volume of the solid that lies under the graph of f(x,y) over the region R in the first octant can be found by integrating the function f(x,y) over the rectangle R in the xy-plane, which yields the double integral.
The following formula represents the double integral over R of f(x,y) if f(x,y) > 0:
∬Rf(x,y)dydx
The geometric interpretation of the double integral over R of f(x,y) if f(x,y) > 0 is given by the volume of the solid that lies under the graph of the function f(x,y) over the region R in the first octant.
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Line A has a y-intercept of 3 and is perpendicular to the line given by
y = 5x + 2.
What is the equation of line A?
Give your answer in the form y = mx + c, where m and c are integers or
fractions in their simplest forms.
Answer:
Step-by-step explanation:
The given line is y = 5x + 2. We know that any line perpendicular to this line will have a slope that is negative reciprocal of 5. The negative reciprocal of 5 is -1/5.
Line A is perpendicular to y = 5x + 2, so it has a slope of -1/5. We also know that the y-intercept of line A is 3. Therefore, the equation of line A can be written as:
y = (-1/5)x + 3
or in the form y = mx + c, where m = -1/5 and c = 3.
help me please........................................................
Answer:
To find the area of the given figure, we can divide it into two separate shapes, a rectangle and a triangle, and then add their areas together.
First, we can find the area of the rectangle by multiplying its length and width. From the diagram, we can see that the length of the rectangle is 12 cm and the width is 5 cm.
Area of rectangle = length x width
Area of rectangle = 12 cm x 5 cm
Area of rectangle = 60 cm^2
Next, we can find the area of the triangle by using the formula for the area of a triangle, which is:
Area of triangle = 1/2 x base x height
From the diagram, we can see that the base of the triangle is 5 cm and the height is 8 cm.
Area of triangle = 1/2 x base x height
Area of triangle = 1/2 x 5 cm x 8 cm
Area of triangle = 20 cm^2
Finally, we can find the total area of the figure by adding the area of the rectangle and the area of the triangle:
Total area = area of rectangle + area of triangle
Total area = 60 cm^2 + 20 cm^2
Total area = 80 cm^2
Therefore, the area of the given figure is 80 square centimeters.
Step-by-step explanation:
one gold nugget weighs 0.008 ounces. a second gold nugget weighs 0.8 ounces. how many times as much as the first nugget does the second nugget weigh? how many times as much as the second nugget does the first nugget weigh
Therefore , the solution of the given problem of unitary method comes out to be it weighs 0.01 times as much as the first nugget.
An unitary method is what?This common convenience, already-existing variables, or all important elements from the original Diocesan customizable survey that followed a particular event methodology can all be used to achieve the goal. If it does, there will be another chance to get in touch with the entity. If it doesn't, each of the crucial elements of a term proof outcome will surely be lost.
Here,
We can divide the weight of the second nugget by the weight of the first nugget to determine how many times as much the second nugget weights the first:
=> 0.8 oz / 0.008 oz = 100
The second piece is therefore 100 times heavier than the first.
We can divide the first nugget's weight by the second nugget's weight to determine how much the first nugget weights in relation to the second nugget:
=> 0.008 oz /0.8 oz = 0.01
In other terms, the second nugget weighs 100 times as much as the first nugget, or it weighs 0.01 times as much as the first nugget.
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linear equations in one variable
Answer:
x = [tex]\frac{3}{5}[/tex]
Step-by-step explanation:
[tex]\frac{x+1}{3}[/tex] - [tex]\frac{x-1}{2}[/tex] = [tex]\frac{1+2x}{3}[/tex]
multiply through by 6 ( the LCM of 2 and 3 ) to clear the fractions
2(x + 1) - 3(x - 1) = 2(1 + 2x) ← distribute parenthesis on both sides
2x + 2 - 3x + 3 = 2 + 4x
- x + 5 = 2 + 4x ( subtract 4x from both sides )
- 5x + 5 = 2 ( subtract 5 from both sides )
- 5x = - 3 ( divide both sides by - 5 )
x = [tex]\frac{-3}{-5}[/tex] = [tex]\frac{3}{5}[/tex]
a general principle in the field of tests and measurements is that longer tests tend to be more reliable than shorter ones. in your opinion, is that principle illustrated by the reliability coefficients shown in the table?
This principle is validated by the data shown in the table.
Tests and measurements is an essential aspect of the education process as it enables educators to gauge the level of knowledge and skills their students have acquired. The principle that longer tests tend to be more reliable than shorter ones has some merit because it allows educators to assess a broader range of skills and knowledge, which increases the validity of their assessments.In my opinion, the principle that longer tests tend to be more reliable than shorter ones is illustrated in the reliability coefficients shown in the table. This is because the data shows that the reliability coefficients for longer tests are consistently higher than those for shorter tests. Additionally, the results for the 10-item test indicate a higher reliability coefficient compared to the 5-item test, which supports the notion that longer tests are more reliable than shorter ones.The table displays that the longer tests have higher reliability coefficients compared to the shorter tests. For example, in the 5-item test, the reliability coefficient is .45, while the 10-item test's reliability coefficient is .73. This shows that the 10-item test is more reliable than the 5-item test, as the higher reliability coefficient indicates that the assessment is consistent in measuring the skill or knowledge it is intended to measure. As a result, this principle is validated by the data shown in the table.
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In a study of the effects of marijuana during pregnancy, measurements on babies of mother who used marijuana during pregnany were compared to measurements on babies of mothers who did not. A 95% confidence interval for the difference in mean head circumference (nonuse minus use) was .61 to 1.19 cm. What can be said from this statement about a p-value for the hypothesis that the mean difference is zero?
The 95% confidence interval of .61 to 1.19 cm provides strong evidence that there is a difference in mean head circumference between babies of mothers who used marijuana during pregnancy and those who did not. Furthermore, the small p-value suggests that the mean difference is statistically significant, and is not likely to be zero.
The 95% confidence interval for the difference in mean head circumference between babies of mothers who used marijuana during pregnancy and those who did not is .61 to 1.19 cm. This implies that the true mean difference is likely to be between .61 and 1.19 cm. The p-value is a measure of how likely it is that the difference in means is zero, and can be used to assess the statistical significance of the difference in means.
Therefore, the p-value for the hypothesis that the mean difference is zero is very small and can be considered statistically significant.
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The small p-value suggests that the mean difference is statistically significant, and is not likely to be zero.
A 95% confidence interval (CI) is a range of values that is expected to include the true population mean with a probability of 0.95. The CI for the difference in mean head circumference between non-users and users of marijuana during pregnancy is 0.61 to 1.19 cm.
This means that the sample mean difference (nonuse minus use) falls within this interval.
If the true population mean difference were zero, the sample mean difference would fall within the range of random sampling error, and the null hypothesis that there is no difference between the groups would be supported.
However, since the 95% CI does not include zero, we can conclude that the difference is statistically significant at the 0.05 level.
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In a distribution of 387 values with a mean of 72, at least 344 fall within the interval 64-80. Approximately what percentage of values should fall in the interval 56-88? Use Chebyshev’s theorem. Round your k and s values to one decimal place and final answer to two decimal places.
The required percentage of values that should fall in the interval 56-88 is approximately 74.37%.
Chebyshev’s Theorem:Chebyshev's Theorem states that, for any given data set, the proportion (or percentage) of data points that lie within k standard deviations of the mean must be at least (1 - 1/k2), where k is a positive constant greater than 1.Calculation:Given,Mean (μ) = 72N (Total number of values) = 387Interval (x) = 64-80 and 56-88Minimum values (n) = 344Minimum percentage (p) = (344 / 387) x 100 = 88.85%From the given data we have,1. Calculate the variance of the distribution,Variance = σ2 = [(n × s2 ) / (n-1)]σ2 = [(344 × 42) / 386]σ2 = 18.732. Calculate the standard deviation of the distribution,σ = √(18.73)σ = 4.33. Calculate k = (|x - μ|) / σ for the given interval 56-88,Here, x1 = 56, x2 = 88, k1 = |56-72| / 4.33 = 3.7, k2 = |88-72| / 4.33 = 3.7Thus, k = 3.74. Calculate the minimum percentage of values within the interval 56-88 using Chebyshev's Theorem,p = [1 - (1/k2)] x 100p = [1 - (1/3.7)2] x 100p = 74.37% (approximately)Therefore, the required percentage of values that should fall in the interval 56-88 is approximately 74.37%.
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A triangle is equal in area to a rectangle which measures 10cm by 9cm. If the base of the triangle is 12cm long, find its altitude
Answer:
h = 15 cm
Step-by-step explanation:
Area of triangle equals the area of rectangle. As the dimensions of the rectangle is given, we can first find the area of the rectangle.
[tex]\boxed{\bf Area \ of \ the \ rectangle = length * width}[/tex]
= 10 * 9
= 90 cm²
Area of triangle = area of rectangle
= 90 cm²
base of the triangle = b = 12 cm
[tex]\boxed{\bf Area \ of \ triangle = \dfrac{1}{2}bh}[/tex] where h is the altitude and b is the base.
[tex]\bf \dfrac{1}{2} b* h = 90 \\\\\dfrac{1}{2}*12* h = 90[/tex]
[tex]\bf h = \dfrac{90*2}{12}\\\\\boxed{\bf h = 15 \ cm}[/tex]
Write the equation of a line perpendicular to `y=3` that goes through the point (-5, 3).
Answer:
The equation of a line perpendicular to y=3 that goes through the point (-5, 3) is: x = -5.
Step-by-step explanation:
To find the equation of a line perpendicular to y=3 that goes through the point (-5, 3), we need to remember that the slope of a line perpendicular to another line is the negative reciprocal of the slope of the original line.
The equation y=3 is a horizontal line that goes through the point (0,3), and its slope is zero. The negative reciprocal of zero is undefined, which means that the line perpendicular to y=3 is a vertical line.
To find the equation of this vertical line that goes through the point (-5, 3), we can start with the point-slope form of a linear equation:
y - y1 = m(x - x1)
where m is the slope of the line and (x1, y1) is a point on the line. Since the line we want is vertical, its slope is undefined, so we can't use the point-slope form directly. However, we can still write the equation of the line using the point (x1, y1) that it passes through. In this case, (x1, y1) = (-5, 3).
The equation of the vertical line passing through the point (-5, 3) is:
x = -5
This equation tells us that the line is vertical (since it doesn't have any y term) and that it goes through the point (-5, 3) (since it has x=-5).
So, the equation of a line perpendicular to y=3 that goes through the point (-5, 3) is x = -5.
Answer:
x= -5
Step-by-step explanation:
The perpendicular line is anything with x= __.
x= -5 however, will go through the point (-5, 3) and that is our answer.
A 20ft ladder is leaning against the roof of a house that is 18ft high. How far away is the ladder from the house?
Therefore, the ladder is approximately 8.72 feet away from the house.
What is Pythagoras theorem?Pythagoras' theorem is a fundamental theorem in mathematics that states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In equation form, it can be written as:
c² = a² + b²
where c is the length of the hypotenuse, and a and b are the lengths of the other two sides of the right triangle.
Here,
We can use the Pythagorean theorem to solve this problem. Let x be the distance from the base of the ladder to the house.
In equation form, it can be written as:
c² = a² + b²
where c is the length of the hypotenuse, and a and b are the lengths of the other two sides of the right triangle.
Then we have:
x² + 18² = 20²
Simplifying and solving for x, we get:
x² = 20² - 18²
=400 - 324
= 76
x = √(76)
= 8.72 (rounded to two decimal places)
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find the length of the cord pt.3
According to the circle theorem, we can find the length of the cord, x = 4 units.
Define circle theorem?Geometrical assertions known as "circle theorems" set forward significant conclusions pertaining to circles. These theorems provide significant information regarding several aspects of a circle.
A circle's chord is a line segment that hits the circle twice on its edge, separating it into two equal pieces. The circle is divided into two equal pieces by the longest chord of the circle, which runs through its centre.
Here in the given circle,
As per the intersecting chords theorem,
AB × CB= BE × BD
⇒ 6 × 6 = 9× x
⇒ x = 36/9=4
Therefore, the length of the chord, x = 4 units.
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A group of 500 middle school students were randomly selected and asked about their preferred television genre. A circle graph was created from the data collected.
a circle graph titled preferred television genre, with five sections labeled drama 14 percent, sports 22 percent, documentaries, reality 20 percent, and sci-fi 20 percent
How many middle school students prefer the Documentaries television genre?
24
76
120
86.4
120 middle schοοl students prefer the dοcumentaries televisiοn genre.
What is percentage?A percentage in mathematics is a number οr ratiο that can be expressed as a fractiοn οf 100. If we need tο calculate a percentage οf a number, we shοuld divide it by 100 and multiply the result. Therefοre, the percentage refers tο a part per hundred. Per 100 is what the wοrd percentage means. The symbοl "%" is used tο represent it.
The tοtal is 100%. Subtract the οther parts οf the circle tο find the percent fοr spοrts.
100 - 14 -22-20 -20
24
Spοrts is 24%
Multiply the number οf students by the percentage οf students that prefer spοrts
500 *24%
500 *.24
120
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What is the difference in weight between the horse and combined weight of the dolphin and the ape?
Answer:
Step-by-step explanation:
44
The perimeter of a rectangular map of the world is 270 cm. It is 90 cm in height. How wide is it?
Answer:
The perimeter of a rectangle is given by:
P = 2(L + W)
where P is the perimeter, L is the length, and W is the width.
In this case, we know that P = 270 cm and L = 90 cm, so we can solve for W as follows:
270 = 2(90 + w)
Divide both sides by 2:
135 = 90 + w
Subtract 90 from both sides:
w = 45
Therefore, the width of the map is 45 cm.
Step-by-step explanation:
Can someone help me with this math problem pls! #Percents
Answer: $3.64
Step-by-step explanation:
At the store, you buy four toys for $1.5, which means you pay $1.5 * 4, or $6.
Then, you calculate the sales tax, which is 6%, which means you multiply $6 by (100% + 6%), or $6*(1.06) which is $6.36.
Finally, if you hand the cashier $10, and you spent $6.36, your change is $10 - $6.36, which is $3.64.
For which equation would x = 4 be a solution?
28 – 5.25 x = 2.75
4.25 x + 7 = 24
4.25 x ÷ 8 = 9
7 + 3.25 x = 29
Answer:
4.25 x + 7 = 24
Second choice
Step-by-step explanation:
Plug in x = 4 into each equation and see which one is consistent
The correct answer is 4.25x + 7 = 24
Left side = 4.25(4) + 7
= 17 + 7
=24
which matches the right side 24
Which statements are true? Select each correct answer.
A. 40m^6 -4=4(10m^6-1)
B. 32m^4 +12m^3 =4m^3(8m +3)
C. 15m^3-6m=3m(5m^2 -6m)
D. 6m^2+18m=6m^2(1+3m)
Answer:
C. 15m^3-6m=3m(5m^2 -6m) is true.
D. 6m^2+18m=6m^2(1+3m) is true.
A. 40m^6 -4=4(10m^6-1) is not true.
B. 32m^4 +12m^3 =4m^3(8m +3) is not true.
Answer:
C. 15m^3-6m=3m(5m^2 -6m) is true.
D. 6m^2+18m=6m^2(1+3m) is true.
Step-by-step explanation:
Gail averages 153 points per bowling game with a standard deviation of 14.5 points. Suppose Gail's points per bowling game are normally distributed. Let X= the number of points per bowling game. Then X∼N(153,14.5). If necessary, round to three decimal places.
Suppose Gail scores 108 points in the game on Thursday. The z-score when x = 108 is __
. The mean is __
Gail scores 153 points on average every bowling game, with a 14.5 point standard deviation. Assume Gail's bowling game points are evenly divided. With x = 108, the mean is 153, and the z-score is -3.103.
The z-score when Gail scores 108 points in a game is calculated as:
z = (x - μ) / σ
where x = 108 is the observed score, μ = 153 is the mean, and σ = 14.5 is the standard deviation.
Plugging in the values, we get:
z = (108 - 153) / 14.5 ≈ -3.103
Rounding to three decimal places, the z-score when Gail scores 108 points in a game is approximately -3.103.
The mean is μ = 153, which is given in the problem statement.
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A fair coin is flipped 3 times and a random variable X is defined to be 3 times the number of heads minus 2 times the number of tails. Find the probability mass function of X. (Write it in table format).
The probability mass function of X( -3, -1, 1 ,3) is P(X) 1/8 3/8 3/8 1/8.
A fair coin is flipped 3 times and the random variable X is defined as follows:
X = 3 times the number of heads - 2 times the number of tails
To find the probability mass function of X, we can list all the possible outcomes and calculate their probabilities.
The Possible outcomes are as shown:
3 heads (X = 3)
2 heads, 1 tail (X = 1)
1 head, 2 tails (X = -1)
3 tails (X = -3)
And the Probabilities are:
P(X = 3) = 1/8
P(X = 1) = 3/8
P(X = -1) = 3/8
P(X = -3) = 1/8
Therefore, the probability mass function of X is:
X( -3, -1, 1 ,3) is P(X) 1/8 3/8 3/8 1/8
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determine whether the series is convergent or divergent. 1/2 3/4 1/8 3/16 1/32 3/64..... convergent or divergent correct?. if it is convergent, find its sum. (if the quantity diverges, enter diverges.)
Answer: The geometric series is convergent and the value is 16.
Step-by-step explanation:
How to illustrate the information?
Recall that the sum of an infinite geometric series, S, given first term, t_1, and common ratio, r, is given by:
S = t_1/(1 - r)
Note that:
3 = (4)(3/4)
9/4 = (3)(3/4
27/16 = (9/4)(3/4)
So this a geometric series with t_1 = 4 and r = 3/4. Therefore:
4 + 3 + 9/4 + 27/16 + ... = 4/(1 - 3/4) = 4/(1/4) = 16
The correct option is 16.
Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum 4+3+ 9/4 +27/16 +???
choices are
1. 3/4
2. 12
3. 4
4. divergent
5. 16
In culinary class, you made fudge brownies and peanut butter brownies. Each batch of fudge brownies makes 1 pan. Each batch of peanut butter brownies makes 9 pans. The class made 5 batches and ended up with 29 pans. How many batches of each type of brownie were made?
Answer: 4 batches of fudge brownies and 1 batch of peanut butter brownies were made.
Step-by-step explanation:
Let x be the number of batches of fudge brownies made, and y be the number of batches of peanut butter brownies made.
From the problem, we can write two equations based on the information given:
Each batch of fudge brownies makes 1 pan: x = number of pans of fudge brownies.
Each batch of peanut butter brownies makes 9 pans: 9y = number of pans of peanut butter brownies.
We also know that the class made 5 batches in total, and ended up with 29 pans:
x + 9y = 29 (total number of pans)
We can now solve for x and y by using a system of two equations:
x + 9y = 29 (equation 1)
x + y = 5 (equation 2)
Solving for x in equation 2 and substituting into equation 1, we get:
(5 - y) + 9y = 29
Simplifying and solving for y:
8y = 24
y = 3
Substituting y = 3 into equation 2, we get:
x + 3 = 5
x = 2
Therefore, the class made 2 batches of fudge brownies (2 pans) and 1 batch of peanut butter brownies (9 pans), for a total of 29 pans. Alternatively, we can say that the class made 4 batches of fudge brownies (4 pans) and 1 batch of peanut butter brownies (9 pans) for a total of 29 pans.
Andre wrote the inequality 3x + 10 <= 30 to plan his time. Describe what x , 3x , 10 , and 30 represent in this inequality
Andre can make 6 small cranes. X is the number of small cranes, 3x is the minutes, 10 is the minute for large cranes and 30 is the total time.
3x + 10 is less than or equal to 30
3 is the minutes for the small cranes
X is the number of small cranes
10 is the minutes for the large crane
30 is the total time limit
first, subtract 10 from 30, ( 30-10) which gives you 20 so
3x is less than or equal to 20.
To figure this out, divide 20 by 3, which gives you 6 as a quotient with a remainder of two minutes.
Andre can make 6 small cranes.
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The Complete question is
Andre is making paper cranes to decorate for a party. He plans to make one large paper crane for a centrepiece and several smaller paper cranes to put around the table. It takes Andre 10 minutes to make the centrepiece and 3 minutes to make each small crane. He will only have 30 minutes to make the paper cranes once he gets home.
Andre wrote the inequality 3x + 10 ≤ 30 to plan his time. Describe what x, 3x, 10, and 30 represent in this inequality.
Solve Andre’s inequality and explain what the solution means.
PLEASE HELP MARKING BRAINLEIST JUST ANSWER ASAP
Answer:
3c + 19
Step-by-step explanation:
Perimeter: P = a + b + c
P = (c + 10) + (c + 6) + (c + 3) = 3c + 19
y= 2x + 1 and - 4x + y = 9, solve the system of equations without graphing
After solving the given equations we know that the value of x and y are -4 and -7 respectively.
What are equations?
A mathematical equation is a formula that uses the equals sign to represent the equality of two expressions.
Ax+By=C is the usual form for two-variable linear equations.
A standard form linear equation is, for instance, 2x+3y=5. When an equation is given in this format, finding both intercepts is rather simple (x and y).
This form is also highly useful when solving systems involving two linear equations.
So, we have the equations:
y=2x+1 ...(1)
-4x+y=9 ...(2)
Now, substitute y=2x+1 in equation (2):
-4x+y=9
-4x+2x+1=9
-2x=8
x=-4
Now, insert x=-4 in equation (1):
y=2x+1
y=2(-4)+1
y=-8+1
y=-7
Therefore, after solving the given equations we know that the value of x and y are -4 and -7 respectively.
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A square yellow rug has a orange square in the center. The side length of the orange square is x inches. The width of the yellow band that surrounds the orange square is 5 in. What is the area of the yellow band?
The yellow band has a surface area of 20x + 100 square inches.
The area of the yellow band is the difference between the area of the yellow square and the area of the orange square.
The yellow square has a side length of (x + 10) inches, where 10 inches is the sum of the widths of the two yellow bands that border the orange square.
So the area of the yellow square is:
A_ yellow = (x + 10)²
The orange square has a side length of x inches, so its area is:
A_ orange = x²
The area of the yellow band is the difference between these two areas:
A_ band = A_ yellow - A_ orange
= (x + 10)² - x²
= x² + 20x + 100 - x²
= 20x + 100
Therefore, the area of the yellow band is 20x + 100 square inches.
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The perimeter of a square is 8 root 2x units. The area of a square is 56 units square. Find the value of x
According to the perimeter, the value of x is 7/2.
The problem tells us that the perimeter of a square is 8√2x units. We can use this information to set up an equation. Since all four sides of a square are equal, we can let s represent the length of one side of the square. Then, we know that:
Perimeter of square = 4s = 8√2x
We can simplify this equation by dividing both sides by 4:
s = 2√2x
Now, we can use this expression for s to find the area of the square. The area of a square is simply the length of one side squared. So, we have:
Area of square = s² = (2√2x)² = 8x
The problem tells us that the area of the square is 56 units square, so we can set up another equation:
8x = 56
Solving for x, we get:
x = 7/2
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Jason and Scott plan on biking to the center of town to get ice cream at the convenience store. Since Scott
had to put air in his tires, Jason was able to get 1 mile ahead of Scott before Scott left the house. Both
bikers rode at a speed of 15 miles per hour.
Write an equation in y = mx + b form that represents Jason's trip. Jason =
a.
Write an equation in y = mx + b form that represents Scott's trip.
Will Jason and Scott meet before they both reach the store? Explain.
If you were to graph both lines on the same coordinate plane, predict what your graph would look
like.
Answer:
a. Jason's equation in y = mx + b form is y = 15x + 1.
b. Scott's equation in y = mx + b form is y = 15x.
Since both are moving at the same speed, they will meet at the point where their distances from the starting point are the same. Let d be the distance from Scott's starting point to the store. Then, the distance from Jason's starting point to the store is d + 1. Using the formula distance = rate × time, we can set up an equation:
15t = d
15t - 1 = d + 1
Solving for t in both equations, we get t = d/15 and t = (d+2)/15, respectively. Equating these expressions for t, we get d/15 = (d+2)/15, which simplifies to d = -2. This means that they will not meet before reaching the store, as Jason is already 1 mile ahead of Scott and will stay ahead throughout the trip.
If we were to graph both lines on the same coordinate plane, we would have two parallel lines with a slope of 15, where Jason's line would intersect the y-axis at 1.
Calculator may be used to determine the final numeric value, but show all steps in solving without a calculator up to the final calculation. The surface area A and volume V of a spherical balloon are related by the equation A’ = 364V? where A is in square inches and Vis in cubic inches. If a balloon is being inflated with gas at the rate of 18 cubic inches per second, find the rate at which the surface area of the balloon is increasing at the instant the area is 153.24 square inches and the volume is 178.37 cubic inches
In the equation A’ = 364V relating the surface area A and the volume V of a spherical balloon. We are also given that the volume is increasing at a rate of 18 cubic inches per second.so the rate at which the surface area of the balloon is increasing is 6552 square inches per second
We want to find the rate at which the surface area is increasing when A = 153.24 square inches and V = 178.37 cubic inches.
To find the rate of change of A with respect to time, we can use the chain rule of differentiation:
dA/dt = dA/dV × dV/dt
We know that dV/dt = 18 cubic inches per second, so we just need to find dA/dV and then we can find dA/dt.
To find dA/dV, we differentiate the equation A’ = 364V with respect to volume V:
dA/dV = 364
Now we can find dA/dt:
dA/dt = dA/dV × dV/dt ⇒ 364 × 18 ⇒ 6552 square inches per second
So the rate at which the surface area of the balloon is increasing is 6552 square inches per second when A = 153.24 square inches and V = 178.37 cubic inches.
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