The exponential function [tex]P(t) = P0 \times 1.04^t[/tex] represents the exponential growth of Option C: A town's population grows at a rate of 4% every year.
What is an exponential function?
The formula for an exponential function is [tex]f(x) = a^x[/tex], where x is a variable and a is a constant that serves as the function's base and must be bigger than 0.
The context that represents exponential growth is "A town's population grows at a rate of 4% every year."
Exponential growth occurs when a quantity grows at an increasing rate over time, which is exactly what is happening with the population of the town.
As the population grows, the growth rate also increases, resulting in an exponential increase over time.
Let P(t) be the population of the town after t years, where t is a positive integer.
The exponential function that models the population growth is -
[tex]P(t) = P0 \times (1 + r)^t[/tex]
where P0 is the initial population, r is the growth rate as a decimal, and t is the time in years.
In this case, the growth rate is 4% per year, which can be written as a decimal as r = 0.04.
Therefore, the exponential function for the town's population growth is -
[tex]P(t) = P0 \times (1 + 0.04)^t[/tex]
Simplifying the expression, we get -
[tex]P(t) = P0 \times 1.04^t[/tex]
This function can be used to calculate the population of the town after a given number of years, assuming the growth rate remains constant.
In contrast, the other contexts represent either a steady decrease (town's population shrinking at a rate of 8.1% per year and the radioactive compound decaying at a rate of 5% per hour) or a linear increase (taxi charges a flat fee of $3.00 for pick-up, then an additional fee of $2.75 per mile).
Therefore, the exponential growth is displayed by town's population growing 4% every year.
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Select the three correct ways to describe an angle with a measure of 13° .
Answer:
The three correct ways to describe an angle with a measure of 13° are:
Thirteen degrees
An acute angle with a measure of 13°
An angle that is less than a right angle and has a measure of 13°
help need badly PROOFS Complete each proof using the most appropriate m
Given: ABCD, AB II CD, D is the midpoint of BE
Prove: AABD ACDE
Statements
V to Inlog
1.
2.
V to Inlogic
1. ABCD is a parallelogram
2. AB II CD
3. D is the midpoint of BE
4. AD II BC
5. ∠BAD = ∠CDE
6. AB = CD
7. BD = CE
8. AABD ACDE
What is parallelogram ?A parallelogram is a four-sided shape with two pairs of parallel sides. Its opposite sides are equal in length, and its opposite angles are also equal. Its interior angles add up to 360 degrees. The defining property of a parallelogram is that both pairs of its opposite sides are parallel, meaning they never intersect. This is in contrast to a rectangle, which has four right angles and four sides of equal length, but its opposite sides are not parallel. A parallelogram is also different from a rhombus, which has all sides of equal length, but its opposite sides are not parallel.
Proof: 1. Given: ABCD is a parallelogram
2. Given: AB II CD
3. Given: D is the midpoint of BE
4. Since D is the midpoint of BE, BD = CE
5. Since ABCD is a parallelogram, AD II BC
6. Since AB II CD and AD II BC, ∠BAD = ∠CDE [Angle-Angle Postulate]
7. Since AB II CD and BD = CE, AB = CD [Side-Side-Side Postulate]
8. Therefore, AABD ACDE [Consecutive Angles Postulate]
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According to the question the Statement and the Reasons is given below:
What is Statement?A statement is an assertion or declaration made by a person or group. It is a fact, opinion, or belief that is expressed as a sentence or short phrase. Statements are typically used to communicate ideas, opinions, facts, or beliefs. They can be used to make an argument, express a point of view, or inform someone of something. Statements can be found in a variety of contexts, including speeches, reports, documents, interviews, and conversations.
Statement | Reasons.
1. ABCD, AB|CD | Given
2. D is the midpoint of BE | Given
3. AD = DB | Definition of midpoint
4. AB + BD = AE | Segment addition postulate
5. AB + AD = AE | Substitution 3
6. AABD = ACDE | Segment addition postulate. Subtraction
postulate. Substitution 5.
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The mirror has a frame. The diameter of the mirror with
the frame is 17 inches. To the nearest hundredth, what
is the area of the mirror with the frame?
Answer:
226.865 square inches.
Step-by-step explanation:
The equation for the area of a circle is [tex]\pi r^2[/tex]. Let's use 3.14 for pi. The radius can be found by dividing 17/2 = 8.5. 8.5 squared is 72.25. Finally, multiply by pi to get 226.865.
in the multiple regression model with k regressors, the variance of the stochastic errors in the population can be estimated by:
In the multiple regression model with k regressors, the variance of the stochastic errors in the population can be estimated by the residual mean square (MSE).
Multiple regression is a statistical tool that allows the researcher to estimate the relationship between multiple independent variables and a dependent variable. It measures the impact of a given independent variable on the dependent variable after controlling for the other independent variables.
In simple linear regression, there is only one independent variable, whereas, in multiple linear regression, there is more than one independent variable.
Residual mean square (MSE) is the variance of the error term (the difference between the predicted and observed values). It represents the average variance of the errors or the average deviation of the dependent variable from its predicted value.
The MSE can be used to estimate the variance of the stochastic errors in the population. It is calculated by dividing the sum of squared residuals by the degrees of freedom (n-k-1)
Where n is the sample size and k is the number of regressors.
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I'm learning probability in geometry but haven't learned it for percentage. Can someone help me?
Answer:
Step-by-step explanation:
a. 100 divided by 75 = 1.3333333333333333333333333333333
1.3333333333333333333333333333333 times 43 = 57.333333333333333333333333333332
round it to the nearest whole number: ≅ 57%
which would be the expected adult height for a toddler who is 34 inches tall at the 30-month checkup? use numbers only.
The expected adult height for a toddler who is 34 inches tall at the 30-month checkup is around 5 feet 7 inches.
To determine the expected adult height, one must multiply the current height by 1.08. Since 34 inches x 1.08 is equal to 36.72 inches, the expected adult height is 5 feet 7 inches.
The calculation for adult height is an approximation. Factors such as nutrition, genetics, and gender play a role in final height.
It is important to note that this calculation is just a guide and the actual adult height may differ from the predicted adult height.
To get a more accurate estimation of the expected adult height, it is recommended to have multiple height checks throughout childhood and take into account growth patterns.
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if the rmse for the validation set is 58.78 and the rmse for the test set is 57.12, then what range will the new data rmse lie in?
RMSE (Root Mean Squared Error) is a measure of the difference between predicted and actual values in a regression problem. It indicates the standard deviation of the residuals or prediction errors. In this case, we have an RMSE of 58.78 for the validation set and 57.12 for the test set.
The new data RMSE is likely to be within the range of the validation and test set RMSE values, but it's difficult to pinpoint an exact range without more information about the data and the model.
The test set RMSE is usually considered to be a more accurate estimate of the model's generalization error because it's calculated on a set of data that the model has not seen during training or validation.
It's important to note that the new data may have different characteristics or distributions compared to the validation and test sets, which can affect the RMSE. Therefore, it's recommended to monitor the model's performance on new data and adjust the model accordingly if necessary.
In conclusion, without additional information, we can assume that the new data RMSE will likely fall within the range of the validation and test set RMSE values, but we cannot provide an exact range without more context. It's important to continue monitoring the model's performance on new data to ensure its accuracy and reliability.
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what is the value of t?
Answer:
t=36°
Step-by-step explanation:
90-54=36
opposite angles are equal so t=36°
which is greater 20 feet or 7 yards?Complete the table to model the answer
Answer:
We know that 1 yard is equal to 3 feet. Therefore, 7 yards is equal to:
7 yards = 7 × 3 feet = 21 feet
So, 7 yards is greater than 20 feet.
To complete the table to model the answer, we can write:
Quantity Value
20 feet 20 ft
7 yards 21 ft
Comparing the values, we can see that 7 yards is greater than 20 feet.
GIVING BRAINLIEST FOR CORRECT ANSWER
The answer is the second one.
The -1 is shaded in, and the arrow is pointing to the ĺeft.
Find the arc length and area of a sector with a radius of 8 feet and central angle of 0 = 135. Use 3.14 for pi and round your
answer to 2 decimal places.
arc length =
sector area =
In circle , arc length is 18.84 ft and sector area = 75.36 ft .
What is arc?
In mathematics, an arc is referred to as a section of a circle's or curve's perimeter. The term "open curve" can also be used to describe it. The circumference, also known as the perimeter, is the measurement around a circle that defines its edge.
Length of an arc = Angle/360 * pi * diameter
= 135/360 * 3.14 * 16
= 18.84 ft
sector area = Angle/360 * π * r²
= 135/360 * 3.14 * 8 * 8
= 75.36 ft
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help me with all of them
Y = 2x + 1, Z = (-1 + 219) / 15, and Z = (-1 - 219) / 15 are the equations to solve for the triangles that are supplied. The abbreviation is 4x2 + 11x - 8.
What is a right triangle in algebra?It's a good idea to have a backup plan in case the main one fails. Its longest side is the hypotenuse, which is also the side of the right triangle that faces the right angle. Height and base make form the two arms of a right angle.
To find z, use the quadratic formula: 15z2 + 6z - 8 = 4z.
Let's first put all the terms aside:
15z + 2z - 8 = 0
The quadratic formula is then used:
z = (-2 ± √(2² - 4(15)(-8))) / (2(15))
z = (-2 ± √(304)) / 30
reducing the radical
z = (-2 ± 4√19) / 30
z = (-1 ± 2√19) / 15
Hence, the answers to z are:
z = (-1 + 2√19) / 15
z = (-1 - 2√19) / 15.
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which is the best approximation of the volume of a cylinder with a radius of 10 mm, and a height of 5 mm? use 3.14 for pi.
The best approximation of the volume of a cylinder with a radius of 10 mm, and a height of 5 mm using 3.14 for pi is 1,570 cubic millimeters.
The formula for calculating the volume of a cylinder is as follows:
V = πr²h
Where V is the volume, r is the radius of the cylinder, and h is the height of the cylinder.
Substituting the values given in the formula, we have :
V = πr²hV = 3.14 × (10 mm)² × (5 mm)V = 3.14 × 100 mm² × 5 mm
V = 1,570 cubic millimeters.
Therefore, the best approximation of the volume of a cylinder with a radius of 10 mm, and a height of 5 mm using 3.14 for pi is 1,570 cubic millimeters.
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now fit a logistic model with a single explanatory variable mcat scores. why is the null deviance the same as that from part a.?
The MCAT scores variable can predict some of the variation in the response variable.
The logistic regression model is a type of regression analysis that is widely used in various fields. A logistic model can be fitted using the glm() function in R. Now fit a logistic model with a single explanatory variable, MCAT scores.In logistic regression, the null deviance is the deviance for a model that has only the intercept. The null deviance will always be equal to the residual deviance if the model has only the intercept.The null deviance is calculated based on the difference between the saturated and null models' likelihoods. It is the amount of variation in the response variable that cannot be explained by the model when no predictor variables are used. When the single predictor variable MCAT scores is added to the null model, it reduces the null deviance by 11.59. This is significant because it indicates that the MCAT scores variable can predict some of the variation in the response variable.
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30 points please help me I've been struggling for so long :(
Which line has a Slope of 3/4
Which line has an undefined Slope
Answer:
slope of 3/4 = line C
undefined slope = line A
Step-by-step explanation:
remember that slope is [tex]\frac{rise}{run}[/tex]. Pick any two points on the graph and count up and over to find the rise/run.
OR
use the slope formula [tex]m=\frac{y_{2} -y_{1} }{x_{2}-x_{1} }[/tex] to plug the two points into and find the slope.
- vertical lines are always undefined.
- horizontal lines have a slope of 0
According to the College Board website, the scores on the math part of the SAT (SAT-M) in a recent year had a mean of 507 and standard deviation of 111. Assume that SAT-M scores follow a normal distribution. One of the criteria for admission to a certain engineering school is an SAT-M score in the 98th percentile. This means the score is in the top 2% of scores.
How does this translate to an actual SAT-M score? Show your work. Note: you may need to find the z-score for the the 98th percentile (or, equivalently, the top 2%). To do this, use the Inverse Normal Distribution Calculator (at the top of this page).
An SAT-M score in the 98th percentile is approximately 734.55 or higher.
According to the College Board website,
The scores on the math part of the SAT (SAT-M) in a recent year had a mean of 507 and a standard deviation of 111. Assume that SAT-M scores follow a normal distribution. To gain admission to a particular engineering school, a requirement is to obtain an SAT-M score in the 98th percentile, indicating that the score is among the top 2% of scores.
For getting the actual SAT-M score, we need to find the corresponding z-score for the 98th percentile.
Using the Inverse Normal Distribution Calculator, we get a z-score of 2.05 for the 98th percentile.
So, the formula for finding an actual SAT-M score is:
x = μ + zσ
Where x is the actual SAT-M score,
μ is the mean = 507,
z is the z-score = 2.05,
σ is the standard deviation = 111
x = 507 + (2.05)(111)
x = 507 + 227.55
x = 734.55
The actual SAT-M score for the 98th percentile is approximately 734.55 (rounded to the nearest hundredth).
Therefore, an SAT-M score of 734.55 or higher is required for admission to the engineering school.
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14. When finding the volume of prisms and pyramids, a unit cube can be used. Why is a unit sphere not a good unit of volume measurement for all spheres?
Therefore, for spheres of different sizes, we need to use different measuring units, unlike prisms and pyramids, where a unit cube can be used as a standard unit for all shapes with the same base and height.
What is volume?Volume is a measure of the amount of space occupied by a three-dimensional object. It is the measure of the total amount of space that an object takes up. The standard unit of volume is the cubic meter (m³) in the International System of Units (SI), but other units such as cubic centimeters (cm³) and cubic inches (in³) are also commonly used. The volume of an object can be calculated using different formulas depending on its shape and dimensions.
Here,
A unit sphere is not a good unit of volume measurement for all spheres because spheres come in various sizes and diameters. The unit sphere has a radius of 1, and while it can be used to measure the volume of spheres with radii equal to 1, it cannot be used to measure the volume of spheres with different radii.
For example, if we want to find the volume of a sphere with a radius of 2, we cannot use the unit sphere as a measuring unit. Instead, we would need to use a larger sphere with a radius of 2 as our measuring unit.
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if a culture has an instantaneous growth rate constant of 0.01615468 min-1, what is the doubling time of the culture?
The doubling time of the culture is approximately 42.91 minutes.
The growth rate constant is a mathematical concept that is used to model the growth of a population. It represents the rate at which the population is increasing at any given moment. The doubling time is a measure of how long it takes for a population to double in size, and it can be calculated using the formula mentioned above.
The doubling time of a culture can be calculated using the formula: Doubling time = ln(2) / growth rate constant
Using the given growth rate constant of 0.01615468 min^-1, we can calculate the doubling time as:
Doubling time = ln(2) / 0.01615468 [tex]min^{-1}[/tex]
Doubling time ≈ 42.91 minutes
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I need help with this please
Answer:
True
False
72
72
96
30
See below.
Step-by-step explanation:
First statement: True
A unit cube is a cube with an edge length of 1 unit.
Its volume is (1 unit)³ = 1 unit³
Second Statement: False
A unit cube cannot be used to measure the volume of "other 2D shapes" since a 2D shape has no volume. Only a 3D shape can have a volume. The word "other" is also used incorrectly.
Problems:
9 × 2 × 4 = 72; 72 cubic units
6 × 3 × 4 = 72; 72 cubic units
Length = 4; Width = 4; Height = 6
4 × 4 × 6 = 96; 96 cubic units
Length = 3; Width = 5; Height = 2
3 × 5 × 2 = 30; 30 cubic units
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This means that the line represented by f(x-3) has been shifted down by 39 units compared to the line represented by f(x).
what is Y intercepted?In mathematics, the term "Y-intercept" refers to the point at which a graph or line intersects the y-axis. It is the value of the dependent variable (usually represented by "y") when the independent variable (usually represented by "x") is equal to zero.
For example, the equation of a straight line can be written in the form y = mx + b, where m is the slope of the line and b is the y-intercept. The y-intercept is the value of y when x is equal to zero.
So, if the equation of a line is y = 2x + 5, then the y-intercept is 5, because this is the value of y when x is equal to zero.
To simplify f(x-3), we need to substitute x-3 for x in the expression for f(x):
[tex]f(x-3) = 10(x-3) - 9[/tex]
Expanding the multiplication and simplifying, we get:
[tex]f(x-3) = 10x - 30 - 9[/tex]
[tex]Therefore, f(x-3) = 10x - 39.[/tex]
To determine the slope, we can compare this expression to the standard form of a linear equation, y = mx + b, where m is the slope. We can see that the coefficient of x in the simplified expression is 10, so the slope is 10. This means that the line represented by f(x-3) is steeper than the line represented by f(x).
To determine the y-intercept, we can look at the constant term in the expression. In this case, the constant term is -39. This means that the line represented by f(x-3) has been shifted down by 39 units compared to the line represented by f(x).
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PLEASE HELP!
The sum of the roots of a monic quadratic is -6, and the product of its roots is 7. What is the quadratic?
Answer with a quadratic expression using the variable x such as x^2 + 10x + 20
Answer:
x^2 +6x+7
Step-by-step explanation:
for roots a and b
x^2 - (a+b)x + ab = (x-a)(x-b)
a box contains 4 white and 6 red chips. one chip is drawn at random and, without looking at its color, is discarded. a second chip is then drawn and the color is recorded. a. what is the probability that the second chip drawn is red?
The probability that the second chip drawn is red is 1/3.
The probability of drawing a red chip on the first draw is 6/10, or 3/5. After one chip is discarded, there are 9 chips remaining, 3 of which are red. So the probability of drawing a red chip on the second draw, given that a chip has already been discarded, is 3/9, or 1/3.
Therefore, the probability that the second chip drawn is red is 1/3. This is because the first chip drawn could be either white or red, so there are two possible scenarios. If the first chip drawn is white, there will be 6 red chips and 3 white chips left, so the probability of drawing a red chip on the second draw will be 6/9 or 2/3. If the first chip drawn is red, there will be 5 red chips and 4 white chips left, so the probability of drawing a red chip on the second draw will be 5/9. To get the overall probability of drawing a red chip on the second draw, we need to take the average of these two probabilities, weighted by the probability of the first chip being white or red, respectively.
The probability of the first chip being white is 4/10, or 2/5, and the probability of the first chip being red is 6/10, or 3/5. So the overall probability of drawing a red chip on the second draw is
(2/5) x (2/3) + (3/5) x (5/9) = 4/15 + 1/3 = 3/9 = 1/3.
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if you take a sample of size 19, can you say what the shape of the sampling distribution for the sample mean is? no why or why not? check all that apply.
Yes, we can say the shape of the sampling distribution for the sample mean if we know the population distribution.
However, if we do not know the population distribution, we cannot determine the exact shape of the sampling distribution for the sample mean. In this case, we can make use of the Central Limit Theorem (CLT) to make some assumptions about the shape of the sampling distribution. According to CLT, as the sample size increases, the sampling distribution of the sample mean becomes approximately normal, regardless of the shape of the population distribution, provided that the sample size is sufficiently large. Therefore, if the sample size is 19 and the population distribution is unknown, we can assume that the sampling distribution of the sample mean is approximately normal if the sample data is not heavily skewed or contains extreme outliers.
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Im excerises 37 and 38 the two polygons are similar find the value of and y image is below please help guys Im struggling badly
For the given analogous polygons, in exercise 37 x = 32, y = 18, and in exercise 38, x = 5, y = 166 °.
What's a polygon?A polygon is an unrestricted polygonal chain made up of line parts that are connected to produce an area plane figure in figure. The borders or sides of an unrestricted polygonal chain are the individual pieces. The polygon's vertices or corners are the places where two lines meet. A solid polygon's body is its Centre. A polygon is a geometric object with two confines and a limited number of sides. A polygon's sides are made up of pieces of straight lines that are joined end to end. As a result, a polygon's line pieces are appertained to as its sides or edges. Vertex or angles relate to the crossroad of two-line parts, where an angle is created. Having three edges makes a triangle apolygon.What's a commen surable relationship? Proportional connections are connections between two variables where their rates are original. Another way to suppose about them is that, in a commensurable relationship, one variable is always a constant value times the other. That constant is known as the" constant of proportionality".
In the given question.
for exercise 37,[tex]39/(x-6)=27/y=24/1839/(x-6) =24/18=3/2[/tex]
on solving,[tex]3x-18= 78x=3227/y=3/23x=54y=18[/tex]
For exercise 38on comparing,[tex]x=5y-73=360-(61+116+90)y-73=93y=166degree[/tex]
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A 90 digit number 9999. Is divided by 89, what is the remainder?
The remainder when a 90-digit number 9999 is divided by 89 is 0, as the result of applying the divisibility rule of 89, which involves reversing the digits of the number and subtracting the smaller from the larger.
To find the remainder when a 90-digit number 9999 is divided by 89, we can use the divisibility rule of 89. The rule states that for any integer n, the number obtained by reversing the digits of n and subtracting the smaller from the larger is divisible by 89.
In this case, we reverse the digits of 9999 to get 9999 again, and subtract the smaller from the larger to get 0. Since 0 is divisible by any number, including 89, the remainder when 9999 is divided by 89 is 0.
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The formula for finding the surface area of a rectangular solid with length 1, width w, ~ and height h is SA = 2/w + 21h + 2wh. Find the surface area of a rectangular solid with length 5 inches, width 2 inches, and height 4 inches. A. 28 square inches B. 36 square inches C. 72 square inches D. 76 square inches
The surface area of the rectangular solid with length 5 inches, width 2 inches, and height 4 inches is 76 sq inch. The answer is D.
What is surface area of rectangular solid ?
The surface area of a rectangular solid (also known as a rectangular prism) is the total area of all its faces. It can be calculated using the formula:
SA = 2lw + 2lh + 2wh
where l is the length of the rectangular solid, w is its width, and h is its height.
The rectangular solid has six faces: a top face, a bottom face, a front face, a back face, a left side face, and a right side face. The formula above accounts for the area of each of these faces.
According to the question:
The formula for the surface area of a rectangular solid is:
SA = 2lw + 2lh + 2wh
We are given that the length (l) of the solid is 5 inches, the width (w) is 2 inches, and the height (h) is 4 inches. Substituting these values into the formula, we get:
SA = 2(5)(2) + 2(5)(4) + 2(2)(4)
= 20 + 40 + 16
= 76 square inches
Therefore, the surface area of the rectangular solid with length 5 inches, width 2 inches, and height 4 inches is 76 square inches. The answer is D.
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about 80% of customers who receive a promotion coupon will visit the store, but only 20% of those in the store will make a purchase. how many promotion coupons have to be sent in order to get 100 customers to make a purchase?
Promotion 50√2 (or about 70.7) coupons need to be sent in order to get 100 customers to make a purchase.
The given problem states that around 80% of customers who receive a promotion coupon will visit the store. But only 20% of those in the store will make a purchase.
The problem asks us to determine how many promotion coupons have to be sent in order to get 100 customers to make a purchase.
Therefore, let the number of coupons to be sent be x.
Then, 80% of x people will visit the store. Therefore, the number of people who will visit the store is
(80/100)x or (4/5)x people.
However, only 20% of those in the store will make a purchase.
Therefore, the number of customers who make a purchase is (20/100) x (4/5)x = (1/25)x² customers.
We need to find x such that (1/25)x² = 100.
Therefore, x² = 25 × 100. Hence, x = √2500 or x = 50√2.
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when converting to a new system, which cutover method is the most conservative? a. data coupling cutover b. parallel operation cutover c. phased cutover d. cold turkey cutover
The most conservative cutover method when converting to a new system is the Phased Cutover.
This method allows for the implementation of the new system in multiple stages, thereby reducing risk. First, the new system is installed and tested. Next, a small portion of data is moved to the new system.
After data is moved, the new system is tested again. This process is repeated until all data has been moved and tested, and the new system is running smoothly.
Once the new system is running successfully, the old system is discontinued.
The phased cutover method ensures that the new system is functioning correctly, and gives time for system users to get used to the new system, before the old system is turned off.
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calculate the total area of a rectangle.
Answer:
multiply the length of the rectangle by the width of the rectangle.
Step-by-step explanation:
I'm tired for this
Solve 7-10, 12-22 even. If you cheese this up, I'm reporting. WORTH 100 POINTS
The values of the expression when using the completing the square method is given below
How to calculate the valueTo complete the square for x²-16x+c, we need to add and subtract the square of half the coefficient of x, which is (-16/2)² = 64. So:
x² - 16x + c = (x - 8)² - 64 + c
To complete the square, we added (-16/2)² = 64, but we subtracted it again to keep the expression equivalent. Therefore, c - 64 is the value that completes the square.
Similarly, for x² + 7x + c, we add and subtract (7/2)² = 49/4:
x² + 7x + c = (x + 7/2)² - 49/4 + c
So c - 49/4 completes the square.
For x²- 9x, we add and subtract (9/2)² = 81/4:
x² - 9x = (x - 9/2)² - 81/4
Therefore, c - 81/4 completes the square.
To factor x²- 14x, we can factor out x:
x² - 14x = x(x - 14)
To factor x² + 30x, we can factor out x:
x² + 30x = x(x + 30)
To factor x²- 9x, we can factor out x:
x² - 9x = x(x - 9)
To solve x² + 10x = 16 by completing the square, we first add and subtract (10/2)² = 25:
x² + 10x + 25 - 25 = 16
(x + 5)² = 41
x + 5 = ±√41
x = -5 ±√41
To solve x² - 3x = 7 by completing the square, we first add and subtract (3/2)² = 9/4:
x² - 3x + 9/4 - 9/4 = 7
(x - 3/2)² = 37/4
x - 3/2 = ±√(37/4)
x = 3/2 ±√(37/4)
To solve x² + 15x = 12 by completing the square, we first add and subtract (15/2)² = 225/4:
x² + 15x + 225/4 - 225/4 = 12
(x + 15/2)² = 129/4
x + 15/2 = ±√(129/4)
x = -15/2 ±√(129/4)
a. Let L be the length of the wading pool. Then its width is L - 12. The height is 1 foot, so the volume is:
V = L(L - 12)(1) = L² - 12L
b. To complete the square for the expression L² - 12L, we add and subtract (12/2)² = 36:
L² - 12L + 36 - 36 = (L - 6)² - 36
So the volume can be written as:
V = (L - 6)² - 36
To find the dimensions of the wading pool that give a volume of 108 cubic feet, we set V = 108 and solve for L:
(L - 6)² - 36
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