Answer:
[tex]\text{Slope} \; m = \dfrac{11}{6}[/tex]
Step-by-step explanation:
[tex]m = \dfrac{rise}{run} = \dfrac{\Delta y}{\Delta x}[/tex]
[tex]m = \dfrac{y_2 - y_1}{x_2 - x_1}[/tex]
[tex]m = \dfrac{-1/4 - 5/2}{-1/2 - 1}[/tex]
[tex]m = \dfrac{\dfrac{-11}{4}}{\dfrac{-3}{2}}[/tex]
[tex]m = \dfrac{-11}{4} \times \dfrac{2}{-3}[/tex]
[tex]m = \dfrac{-22}{-12}[/tex]
[tex]m = \dfrac{11}{6}[/tex]
View the photo and solve the probability
Therefore, the probability that at least one of the next six births is a girl is 1 - 0.033 = 0.967 (rounded to three decimal places).
What is Probability?Probability is a measure of the likelihood that an event will occur. It is a number between 0 and 1, with 0 indicating that the event is impossible and 1 indicating that the event is certain.
To calculate the probability of an event, you divide the number of ways that event can occur by the total number of possible outcomes. For example, if you flip a fair coin, there are two possible outcomes - heads or tails - and each has an equal probability of 0.5 (or 50%) of occurring.
Given by the question.
To find the probability that at least one of the next six births is a girl, we can find the probability that all six of them are boys and subtract it from 1.
The probability that one birth is a girl is 1 - 0.513 = 0.487.
The probability that all six births are boys is. [tex]0.513^{6}[/tex] = 0.033.
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Tim and Al are brick makers. T can make an outdoor grill in ll day. If Al helps they can do it in 3 days. How long would it take Al to build the grill alone?
Answer:
It would take Al 7.5 days to build the grill alone.
Step-by-step explanation: Since Tim and Al are bricklayers, and Tim can construct an outdoor grill in 5 days, and if Al helps Tim, they can build it in only 3 days, to determine how long would it take Al to build the grill alone should be done the following calculation:
1/5 + X = 1/3
0.20 + X = 0.333
X = 0.333 - 0.20
X = 0.1333333
X = 1 / 7.5
Therefore it would take Al 7.5 days to build the grill alone.
Carlos reads 1/2 hour every night. how many hours will he read in 11 nights?
Answer: 5 1/2 hours
Step-by-step explanation:
for every 2 nights he reads he will gain an hour of time so the easiest way to solve is to divide 11 by 2 for an answer of 5.5 or 5 1/2 hours.
If the unit rate is 1/2 hours per night,
Carlos will read for 5.5 hours in 11 nights.
We have,
If Carlos reads for 1/2 hour every night,
This is the unit rate.
Now,
In 11 nights he will read for:
(1/2) x 11 = 11/2
= 5.5 hours
Therefore,
Carlos will read for 5.5 hours in 11 nights.
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which is more 7,000 millimeters or 7 liters?
please answer this question: the length of a rectangle is 6 centimeters less than its width. what are the dimensions of the rectangle if it's area is 160 square centimeters?
SOMEONE PLEASE HELP ME!!
Measure of the arc or angle indicated is 150.9 degrees.
Describe Arc?In geometry, an arc is a portion of the circumference of a circle or any other curved shape. It is defined by two endpoints and all the points on the curve that lie between them. An arc is usually named by its two endpoints, with a small arc symbol above them to indicate that it is an arc.
The length of an arc can be calculated using the formula:
Arc length = (central angle/360) x 2πr
where r is the radius of the circle, and the central angle is the angle subtended by the arc at the center of the circle, measured in degrees.
The measure of an arc is the degree measure of the central angle subtended by the arc. A semicircle is an arc that subtends a central angle of 180 degrees, and a full circle is an arc that subtends a central angle of 360 degrees.
Since DE and PE are chords of the circle, and they intersect at point P, we can use the intersecting chords theorem to find the length of DP. Let x be the length of DP. Then:
DP * PE = DE * PC
x * (x + PE) = ([tex]\frac{CE}{2}[/tex]) * ([tex]\frac{CE}{2}[/tex])
x² + x(PE) - [tex]\frac{CE}{2}^{\frac{2}{4} }[/tex] = 0
Since angle DPE is 60 degrees, we can use the law of cosines to find PE. Let y be the length of PE. Then:
y² = DE² + DP² - 2 * DE * DP * cos(60)
y² = ([tex]\frac{CE}{2}[/tex])² + x^2 - ([tex]\frac{CE}{2}[/tex]) * x
Substitute this expression for y^2 into the equation for x and simplify:
[tex]x^{2} +x((\frac{CE}{2})^{2} + x^{2} -\frac{CE}{2} *x)^{0.5} -(\frac{CE}{2} ^{\frac{2}{4} } )=0[/tex]
Solve for x:
x = [tex]\frac{CE^{2} -4*(CE^{2} -3*\frac{CE}{2} ^{2} )^{0.5} }{4}[/tex]
x = [tex]\frac{3}{4} *\frac{CE^{2} }{CE^{2} -12}[/tex]
Now we can find the measure of the CD arc by using the formula for the central angle:
CD arc = [tex]2*arctan(\frac{DP}{CE})[/tex]
CD arc = [tex]2*arctan(\frac{x}{\frac{CE}{2} } )[/tex]
CD arc = [tex]2* arctan(CE^{2} -4*(CE^{2} -3*\frac{(\frac{CE}{2} ^{2})^{0.5}}{CE^{2}-12 } ))[/tex]
Simplifying this expression, we get:
[tex]CD arc=2*arctan(2*3^{\frac{1}{2} } -1)[/tex]
CD arc ≈ 150.9 degrees.
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Measure of the arc or angle indicated in the given figure is 150.9 degrees.
Describe Arc?In geometry, an arc is a portion of the circumference of a circle or other curved shape. It is defined by his two endpoints and all midpoints of the curve. Arcs are usually named after their two endpoints, with a small arc symbol above them to indicate that they are arcs.
Arc Length = (Center Angle/360) x 2πr
where r is the radius of the circle and the central angle is the angle the arc makes at the center of the circle, measured in degrees.
The arc measurement is the angle of the central angle defined by the arc. A half circle is an arc that spans a central angle of 180 degrees, and a full circle is an arc that spans a central angle of 360 degrees.
Since DE and PE are chords of the circle and intersect at P, we can use the chord rule to find the length of DP. Let x be the length of DP. Then:
DP × PE = DE × PC
x × (x + PE) = (CE/2) × (CE/2)
x² + x(PE) - [tex]\frac{CE}{2} ^{\frac{2}{4} }[/tex] = 0
Since angle DPE is 60 degrees, we can use the law of cosines to find PE. Let y be the length of PE. Then:
y² = DE² + DP² - 2 × DE × DP × cos(60)
y² = ([tex]\frac{CE}{2}[/tex])² + x² - ([tex]\frac{CE}{2}[/tex]) × x
Substitute this expression for y² into the equation for x and simplify:
[tex]x^{2} +x((\frac{CE}{2} ^{2}) + x^{2} - \frac{CE}{2}*x)^{0.5} -[/tex] [tex]\frac{CE}{2} ^{\frac{2}{4} }[/tex] = 0
Solve for x:
x = [tex]\frac{CE^{2} - 4*(CE^{2}-3*\frac{CE}{2} ^{2})^{0.5} }{4}[/tex]
x = (3/4) × (CE²/CE²-12)
Now we can find the measure of the CD arc by using the formula for the central angle:
CD arc = 2arc tan(DP/CE)
CD arc = 2arc tan [x/(CE/2)]
CD arc = 2arc tan (CE² - 4 × (CE² - 3 × [tex]\frac{(\frac{CE}{2} ^{2}) ^{0.5} }{CE^{2}-12 }[/tex] )
Simplifying this expression, we get:
CD arc ≈ 150.9 degrees.
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Glados has a savings account with $700 in it that earns 7.9% simple interest per year. How much money, to the nearest penny, will Glados have in 5 years?
Answer:
To calculate the amount of money Glados will have in 5 years with simple interest, we can use the formula:
A = P(1 + rt)
where A is the final amount, P is the principal (initial amount), r is the annual interest rate as a decimal, and t is the time in years.
In this case, P = $700, r = 0.079 (since 7.9% is 0.079 as a decimal), and t = 5 years. Substituting these values into the formula, we get:
A = 700(1 + 0.079 x 5)
A = 700(1 + 0.395)
A = 700(1.395)
A = $976.50
Therefore, Glados will have $976.50 in 5 years, to the nearest penny.
How can you make your topic relevant (culturally, socially, personally) for the audience? Consequently, what can the audience learn from your presentation? 100 word count please
1. Making a topic relevant to an audience can involve finding ways to connect the subject matter to their cultural, social, or personal experiences.
2. The audience can learn a great deal from a well-crafted presentation that effectively conveys important information and insights.
How can we make a topic relevant for the audience?For example, if the topic is about climate change, one might discuss how it affects their local community or how it relates to their personal lifestyle choices. Similarly, if the topic is about a historical event, one might highlight its relevance to current social issues or draw connections to cultural traditions and values.
What can the audience learn from your presentation?For instance, they may gain a deeper understanding of a particular issue or topic, or learn about new ideas and perspectives. A good presentation can also inspire the audience to take action or make changes in their own lives.
In order to achieve these outcomes, it is important for the presenter to communicate clearly and engagingly, and to connect the topic to the interests and concerns of the audience. By doing so, the presentation can be a valuable learning experience for everyone involved.
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Geneva wants to save $12,000 to buy a new car. She just received an $8,000 bonus and plans to invest it in an account earning 7% Annual simple interest. How long will she need to leave her money in the account to accumulate the $12,000 she needs?
Answer:
7.14 years
Step-by-step explanation:
We can use the formula for simple interest to solve this problem:
Simple Interest = Principal x Rate x Time
where the principal is the initial investment, the rate is the annual interest rate, and the time is the number of years.
We know that Geneva wants to save $12,000, and she already has $8,000 from her bonus, so she needs to earn an additional $4,000 in interest. We can use this amount as the principal in the formula.
Simple Interest = $4,000
Principal = $8,000
Rate = 0.07 (7% expressed as a decimal)
Plugging in the values, we get:
$4,000 = $8,000 x 0.07 x Time
Simplifying, we get:
Time = $4,000 / ($8,000 x 0.07)
Time = 7.14 years (rounded to the nearest hundredth)
Therefore, Geneva will need to leave her money in the account for approximately 7.14 years to accumulate the $12,000 she needs
Problem 13 and problem 20 ?
The answers to both questions are as:
(a) the value of the investment after 5 years is $14,917.95, after 10 years is $23,673.58, and after 15 years is $37,337.35,
b) the amount of the final payment is $688.32.
What is compound interest?
Compound interest is when you earn interest on both the money you've saved and the interest you earn.
We can use the formula for compound interest to solve this problem:
[tex]A = P(1 + r/n)^{(nt)}[/tex]
where A is the final amount, P is the principal (initial investment), r is the annual interest rate (as a decimal), n is the number of times the interest is compounded per year, and t is the time (in years).
(a) For 5 years with semiannual compounding:
n = 2 (compounded semiannually)
r = 0.08 (8% annual rate)
t = 5
[tex]A = 9900(1 + 0.08/2)^{(2*5)}[/tex]
A = $14,917.95
(b) For 10 years with semiannual compounding:
n = 2 (compounded semiannually)
r = 0.08 (8% annual rate)
t = 10
[tex]A = 9900(1 + 0.08/2)^{(2*10)}[/tex]
A = $23,673.58
(c) For 15 years with semiannual compounding:
n = 2 (compounded semiannually)
r = 0.08 (8% annual rate)
t = 15
[tex]A = 9900(1 + 0.08/2)^{(2*15)}[/tex]
A = $37,337.35
Therefore, the value of the investment after 5 years is $14,917.95, after 10 years is $23,673.58, and after 15 years is $37,337.35, assuming the interest is compounded semiannually.
To calculate the final payment, we can first find the balance of the loan at the end of the fifth year, and then use this as the principal to calculate the balance at the end of the tenth year.
The interest rate is 5% per year, compounded quarterly. This means that the quarterly interest rate is:
r = 5% / 4 = 0.0125
Let B be the balance of the loan after 5 years. Then we have:
B = 1000*(1 + r)²⁰- 200*(1 + r)⁴
where the first term is the future value of the initial loan after 5 years, and the second term is the present value of the first payment of $200.
Plugging in the values, we get:
B = 1000*(1 + 0.0125)²⁰ - 200*(1 + 0.0125)⁴
B = 1000*(1.0125)²⁰ - 200*(1.0125)⁴
B = 1346.49
So the balance of the loan after 5 years is $1346.49. We can use this as the principal to calculate the balance at the end of the tenth year, which is the final payment we are looking for.
Let P be the final payment. Then we have:
P = 1346.49*(1 + r)⁴⁰ - 800*(1 + r)²⁰
where the first term is the future value of the balance after 10 years, and the second term is the present value of the second payment of $800.
Plugging in the values, we get:
P = 1346.49*(1 + 0.0125)⁴⁰ - 800*(1 + 0.0125)²⁰
P = 1346.49*(1.0125)⁴⁰ - 800*(1.0125)²⁰
P = 688.32
So the amount of the final payment is $688.32.
hence, the answers to both questions are as:
(a) the value of the investment after 5 years is $14,917.95, after 10 years is $23,673.58, and after 15 years is $37,337.35,
b) the amount of the final payment is $688.32.
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A home has a rectangular kitchen. If listed as ordered pairs, the corners of the kitchen are (11, 5), (−6, 5), (11, −2), and (−6, −2). What is the area of the kitchen in square feet?
119 ft2
49 ft2
48 ft2
15 ft2
Answer:
First, we need to find the length and width of the rectangle. The length is the distance between the points (11, 5) and (-6, 5), which is 11 - (-6) = 17 feet. The width is the distance between the points (11, 5) and (11, -2), which is 5 - (-2) = 7 feet.
The area of the rectangle is the product of the length and width, so A = 17 * 7 = 119 square feet.
Therefore, the answer is 119 ft2.
Please help i need the answers to this statistic questions, i am very lost.. #10
a-f
There is not enough evidence to support the analyst's claim that the stocks lost value from one hour to the next on that business day.
How to explain the hypothesisThe null hypothesis is that there is no significant difference between the two sets of prices, while the alternative hypothesis is that there is a significant difference.
Using a t-test calculator, we can find the t-value and p-value:
t-value = -0.322
p-value = 0.757
At a significance level of 0.01, the critical t-value with 6 degrees of freedom is ±3.707. Since the calculated t-value of -0.322 is within this range, we fail to reject the null hypothesis.
Therefore, there is not enough evidence to support the analyst's claim that the stocks lost value from one hour to the next on that business day.
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1/4 x5/3 as a fraction
The product of the fraction 1/4 x5/3 is 4/12
What are fractions?Fractions are defined as the part of a whole number, variable or element.
The different types of fractions in mathematics are;
Simple fractions.Proper fractions.Improper fractions.Complex fractions.Mixed fractions.Examples of simple fraction; 1/4, 2/3
Examples of improper fractions; 3/2. 4/3
Examples of proper fractions; 1/2, 2/3
From the information given, we have that;
1/4 x5/3
To determine the product, we need to multiply the numerator, then multiply the denominator
5/12
We can no longer divide the values as there is no common divisor.
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Solve for x. Round to the nearest tenth.
x =
(35 points)
The value of x rounded to the nearest tenth is equal to 24.3 units.
How to determine the value of x?In order to determine the value of x, we would apply basic trigonometry. From the information provided about this right angled triangle, we can logically deduce the following parameters:
Adjacent side (Adj) = xHypotenuse (Hyp) = 26.Angle = 21 degrees.Therefore, we would use the cosine trigonometry to determine the value of x as follows:
Cosθ = Adj/Hyp
Cos21 = x/26
x = 26cos21
x = 26(0.9336)
x = 24.3 units.
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Complete the ratio table to convert the units of time from hours to weeks or weeks to hours.
Hours:
168 1 week
1,008. ____week
_____. 5 weeks
Answer:
6 weeks and 840 hours
Step-by-step explanation:
There are 168 hours in one week.
24 hrs/day * 7 days = 168 hours
1008 hours ÷ 24 hours(1 day) = 42 days ÷ 7 days in a week = 6 weeks
168 hours/week * 5 weeks = 840 hours
I need help with my homework
To find the length of a line segment in a circle, use the formula [tex]d = 2r[/tex] [tex]sin(t/2)[/tex] , where r is the radius of the circle and t is the angle between the radii. The length of segment DE is [tex]5[/tex] units.
What is the formula for circle segment length?We can use the similar triangles property to find the missing length of segment DE in the given figure. Because triangles ABD and CBE are similar, we can use a proportion to find the length of DE:
[tex]CB/BE = AB/BD[/tex]
With the given values, we get:
[tex]3/6 = 5/(5 + DE)[/tex]
When we simplify and solve for DE, we get:
[tex]3(5 + DE) = 6 * 5 \s15 + 3DE = 30[/tex]
[tex]3DE = 15 \sDE = 5[/tex]
Therefore, segment DE has a length of 5 units.
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Which of the following sets of numbers could represent the three sides of a
triangle?
{7, 15, 20}
{7, 16, 23}
O {11, 22, 35}
O {5, 17, 22}
The following sets of numbers could represent the three sides of a
triangle is {7, 15, 20}.
How to determine the set that represent the three sides of a triangleThe set of numbers {7, 15, 20} could represent the three sides of a triangle, but the other sets of numbers do not satisfy the triangle inequality theorem, which states that the sum of any two sides of a triangle must be greater than the third side. Checking the sets of numbers:
For {7, 15, 20}, 7 + 15 > 20, 7 + 20 > 15, and 15 + 20 > 7, so they satisfy the triangle inequality theorem and could represent the sides of a triangle.
For {7, 16, 23}, 7 + 16 < 23, so they do not satisfy the triangle inequality theorem and cannot represent the sides of a triangle.
For {11, 22, 35}, 11 + 22 < 35, so they do not satisfy the triangle inequality theorem and cannot represent the sides of a triangle.
For {5, 17, 22}, 5 + 17 < 22, so they do not satisfy the triangle inequality theorem and cannot represent the sides of a triangle.
Therefore, the answer is {7, 15, 20}.
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What is 7% as a decimal ?
Answer: 0.07%
Step-by-step explanation:
Answer:
0.07
Step-by-step explanation:
x^2+y^2-10x-18y+5 write in standard form
The expressiοn in standard fοrm is: [tex](x - 5)^2 + (y - 9)^2 = 101[/tex]
What is an Equatiοns?Equatiοns are mathematical statements with twο algebraic expressiοns οn either side οf an equals (=) sign. It illustrates the equality between the expressiοns written οn the left and right sides.
Tο determine the value οf a variable representing an unknοwn quantity, equatiοns can be sοlved. A statement is nοt an equatiοn if there is nο "equal tο" symbοl in it. It will be regarded as an expressiοn.
Starting with the x terms:
[tex]x^2 - 10x[/tex]
Tο cοmplete the square, we need tο add and subtract [tex](10/2)^2 = 25:x^2 - 10x + 25 - 25[/tex]
Again, we need tο add and subtract [tex](18/2)^2 = 81:[/tex]
[tex]y^2 - 18y + 81 - 81[/tex]
This is nοw a perfect square:
[tex](y - 9)^2 - 81[/tex]
Therefοre, the expressiοn in standard fοrm is: [tex](x - 5)^2 + (y - 9)^2 = 101[/tex]
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Find the annual percentage rate for an account earning compound interest at a rate of 3.425%
Complete the table to find the APR when compounded semiannually and quarterly.
The APR when compounded semiannually and quarterly would be :
Compounded semi - annually - (1.034543) ^ t - 3.4543%Compounded quarterly - (1.034692) ^ t - 3.4692%How to find the APR ?To find the APR when compounded semi - annually, we first need to find the periodic rate to be :
= Annual rate / 2 semi annual periods
= 3. 425 / 2
= 1.7125%
Then use the Effective Annual Rate (EAR) to find the APR to be:
= ( 1 + periodic rate ) ^ number of periods - 1
= ( 1 + 1. 7125 % ) ² - 1
= 3.4543%
For the APR when compounding quarterly, you can variate the EAR formula to the original version of:
= ( 1 + annual rate / number of periods ) ^ number of periods - 1
= ( 1 + 3. 425 / 4 ) ⁴ - 1
= 3.4692%
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Write the polynomial in standard form. Then classify the polynomial by degree and by number of terms.
7x² +9x² - 6x²
Write the polynomial in standard form.
(Simplify your answer.)
Answer:
Step-by-step explanation:
When we combine like terms, we get:
7x² +9x² - 6x² = (7+9-6)x² = 10x²
So the polynomial in standard form is 10x².
The degree of the polynomial is 2 (since the highest power of the variable x is 2) and the number of terms is 1 (since there is only one term). Therefore, we classify this polynomial as a quadratic monomial.
Easy (7th-grade math)
Answer:
384 cm
Step-by-step explanation:
hope this helps ! good luck on ur assignment! <3
What is the mode of Region A?
The mode for the region A is 2.6.
What exactly is mode?
In statistics, the mode is the value that has highest frequency in a data set. It is one of the three measures of central tendency, along with the mean and median.
To find the mode of a data set, you simply identify the value that appears most often. If no value is repeated, the data set has no mode.
The mode is particularly useful when dealing with categorical data or data that can be easily grouped into categories, such as colors, types of fruit, or letter grades. In such cases, the mode can provide insight into which category is most common or prevalent.
Now,
As given Values for Region A are
2.3 2.5 2.6 2.6 2.6 2.7 2.7 2.8
Here 2.6 comes most times or frequency of 2.6 is highest of all.
Hence,
The mode for the region A is 2.6.
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why is ÷ by five a step, there did this 5 come from if its not in the inequality?
Answer:
You divide by five because it's the common factor of 5 and 15 in the 4th line of work and the goal is to solve for x. The only way to get x alone in the inequality 5x>15 is to divide each side by 5. This would give you x and 3, hence the answer: x > 3
[tex]\frac{5x}{5} =x\\\\\frac{15}{5} =3[/tex]
Polly's sister-in-law is going to have a baby! For the baby shower, Polly decided to sew pillow to give as a gift. She is using a flower-printed rectangular piece of fabric that is 26 inches long and 22 inches wide.
Answer:
The answer is 96
Step-by-step explanation:
2*(26+22)
2*48
96
part e Theresa’s computer at home has 1 terabyte of storage. How many more bits of storage does her computer have than the combined bits available on her family’s phones, tablets, and cloud storage? Type your answer in scientific notation.
Answer:
Step-by-step explanation:
We need to convert the 1 terabyte into bits, which is:
1 terabyte = 1,000 gigabytes (GB) x 1,000 megabytes (MB) x 1,000 kilobytes (KB) x 1,000 bytes (B) x 8 bits (b)
1 terabyte = 1,000,000,000,000 bytes x 8 bits
1 terabyte = 8,000,000,000,000 bits
Let's assume the total storage of all the family's phones, tablets, and cloud storage is x bits. Then, the number of more bits of storage Theresa's computer has is:
8,000,000,000,000 bits - x bits
We are not given the value of x, so we cannot calculate the exact difference.
However, we can still express the answer in scientific notation by noting that:
8,000,000,000,000 = 8 x 10^12
So, the difference in bits of storage can be written as:
(8 x 10^12) - x bits
there are 4 types of ice cream, 3 different cones and 3 choices of toppings. how many different ways can an ice cream cone be ordered
Answer:
25
Step-by-step explanation:
A,B and B,C form a right angle at point B. If A = (-3,-1) and B = (4,4), what is the equation of B,C?
Answer:
the equation of line BC is y = (-7/5)x + (48/5).
Step-by-step explanation:
To find the equation of the line that passes through points B and C, we first need to determine the coordinates of point C. Since the angle at B is a right angle, we can use the slope of line AB to find the slope of line BC.
The slope of line AB is:
mAB = (yB - yA) / (xB - xA)
= (4 - (-1)) / (4 - (-3))
= 5/7
Since lines AB and BC are perpendicular, the slope of line BC is the negative reciprocal of the slope of line AB:
mBC = -1 / mAB
= -7/5
Now we can use the point-slope form of the equation of a line to find the equation of line BC. We can use point B as the known point, since we already know its coordinates:
y - yB = mBC(x - xB)
Substituting the values we have:
y - 4 = (-7/5)(x - 4)
Expanding and simplifying:
y - 4 = (-7/5)x + (28/5)
y = (-7/5)x + (48/5)
Can someone help please
let's bear in mind that complex roots never come alone, their conjugate sister is always with her, so if we have the complex root of "i" or namely "0 + i", her conjugate is also coming along, or "0 - i", so we really have four roots, so
[tex]\begin{cases} x = 0+i &\implies x -i=0\\ x = 0-i &\implies x +i=0\\ x = \sqrt{2} &\implies x -\sqrt{2}=0\\ x = 3 &\implies x -3=0\\ \end{cases} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{original~polynomial}{a ( x -i )( x +i )( x -\sqrt{2} )( x -3 ) = \stackrel{0}{y}}\hspace{5em}\stackrel{\textit{we are assuming that}}{a=1} \\\\\\ 1( x -i )( x +i )( x -\sqrt{2} )( x -3 ) = y\implies ( x -i )( x +i )( x -\sqrt{2} )( x -3 ) = y \\\\[-0.35em] ~\dotfill[/tex]
[tex]\stackrel{ \textit{difference of squares} }{( x -i )( x +i )}\implies x^2 - i^2\implies x^2-(-1)\implies x^2+1 \\\\[-0.35em] ~\dotfill\\\\ (x^2+1)( x -\sqrt{2} )( x -3 )\implies (x^2+1)(x^2-3x-x\sqrt{2}+3\sqrt{2}) \\\\\\ (x^2+1)[x^2-x(3+\sqrt{2})+3\sqrt{2}] \\\\\\ x^4-x^3(3+\sqrt{2})+3x^2\sqrt{2}+x^2-x(3+\sqrt{2})+3\sqrt{2} \\\\\\ \boxed{x^4-x^3(3+\sqrt{2})+x^2(3\sqrt{2}+1)-x(3+\sqrt{2})+3\sqrt{2}~~ = ~~y}[/tex]
The Yellowstone National Park sits at an elevation of 8,000 to 11,000 feet above sea level. During the month of June, the temperature at 8,000 feet above sea level was recorded as 650 Fahrenheit, and, at a distance of 10,500 feet feet above sea level, the temperature was recorded as 15⁰ Fahrenheit. For every additional 100 feet above sea level, the temperature decreases by tº Fahrenheit, where t is a constant. What is the value of t?
Answer:
Step-by-step explanation:
To find the value of t, we can use the information given about the temperature at two different elevations and use the formula for a linear equation.
Let's assume that the temperature decreases by tº Fahrenheit for every additional 100 feet above sea level, where t is the constant we want to find.
Then, using the two points given:
- (8000, 65) where 65 is the temperature in Fahrenheit at 8000 feet above sea level
- (10,500, 15) where 15 is the temperature in Fahrenheit at 10,500 feet above sea level.
We can calculate the slope of the line connecting these two points:
slope = (15 - 65) / (10,500 - 8,000) = -25
The slope of the line represents the amount that the temperature decreases per unit increase in elevation. In this case, we've found that the temperature decreases by 25 Fahrenheit for each increase in elevation of 1000 feet.
To find the value of t, we need to express the slope in terms of the decrease in temperature per 100 feet of elevation. We can do this by dividing the slope by 10:
t = slope / 10 = -2.5
Therefore, the value of t is -2.5.