Using scales, the distance of the two cities on the map would be of:
distance on the map = actual distance/40
What is the scale of a map?A scale on the map represents the ratio between the actual length of a segment and the length of drawn segment, hence:
Scale = actual length/drawn length
In this problem, the scale is of 1 inch to 40 miles, meaning that:
Each inch drawn on the map represents 40 miles.
Then the distance of the two cities on the map, in inches, would be given as follows:
distance on the map = actual distance/40.
If the distance was of 200 miles, for example, the distance on the map would be of 5 inches.
The problem is incomplete, hence the answer was given in terms of the actual distance of the two cities. You just have to replace the actual distance into the equation to find the distance on the map.
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Decide whether or not the number 46/45 Could represent the probability
Solution
- A probability, by definition, is a fraction that is less than or equal to 1.
- The number 46/45 is a number greater than 1.
- Thus, the number 46/45 CANNOT be a probability
5. Kara earns a 3.5% commission on all sales made by recommendations to the hair salon. If the total amount of sales from referrals by Karo was $3,670, how much did Kara make?
Let's begin by listing out the information given to us:
Commission (C) = 3.5% = 0.035
Total amount of sales (T) = $3,670
To determine how much Kara made, we will find the product of the commission & total amount of sales:
[tex]\begin{gathered} Kara(K)=Commission(C)\cdot TotalAmountOfSales(T) \\ K=C\cdot T=0.035\cdot3670=128.45 \\ K=128.45 \end{gathered}[/tex]We therefore, see that Kara made $128.45 from referrals
A storage box has a volume of 56 cubic inches. The base of the box is 4 inches by 4 inches. Lin’s teacher uses the box to store her set of cubes with an edge length of 1/2 inch.If the box is completely full how many cubes are in the set?
To answer this question, we have to find the volume of one of the cubes from Lin's set. To do it use the formula to find the volume of a cube, which is the edgelength raised to 3:
[tex]\begin{gathered} V=ed^{3^{}} \\ V=(\frac{1}{2})^3 \\ V=\frac{1}{8} \end{gathered}[/tex]In this case, each cube has a volume of 1/8 cubic inches. To find the number of cubes in the set, perform the division of the volume of the box by the volume of one cube:
[tex]n=\frac{V_{box}}{V_{cube}}=\frac{56}{\frac{1}{8}}=8\cdot56=448[/tex]The set has 448 cubes.
If the 10 letters are {aa,aa,aa,aa,bb,bb,cc,cc RR,RR} are available and all 10 of them are to be selected without replacement,what is the number of different permutations?
In order to calculate the number of permutations, first we start with the factorial of the number of letters.
There are 10 letters, so we start with the factorial of 10.
Then, we need to check the number of repetitions. Each repetition will be a factorial in the denominator:
[tex]x=\frac{10!}{a!\cdot b!\operatorname{\cdot}...}[/tex]We have four repetitions of aa, two repetitions of bb, two repetitions of cc and two repetitions of RR, therefore the final expression for the number of permutations is:
[tex]x=\frac{10!}{4!2!2!2!}[/tex]Calculating this expression, we have:
[tex]x=\frac{10\operatorname{\cdot}9\operatorname{\cdot}8\operatorname{\cdot}7\operatorname{\cdot}6\operatorname{\cdot}5\operatorname{\cdot}4!}{4!\operatorname{\cdot}2\operatorname{\cdot}2\operatorname{\cdot}2}=\frac{10\operatorname{\cdot}9\operatorname{\cdot}8\operatorname{\cdot}7\operatorname{\cdot}6\operatorname{\cdot}5}{8}=18900[/tex]Therefore there are 18900 permutations.
The number of microbes in a tissue sample is given by the functionN (t) = 34.8 + In(1 + 1.2t)where N(t) is the number of microbes (in thousands) in the sample after thours.a.) How many microbes are present initially?b.) How fast are the microbes increasing after 10 hours?
Explanation
[tex]N(t)=34.8+\ln (1+1.2t)[/tex]we have a function where the number of microbes ( N) depends on the time(t)
hence
Step 1
a.) How many microbes are present initially?
to know this, we need replace time I= t = zero, because it was "initially"
so
when t=0
replace.
[tex]\begin{gathered} N(t)=34.8+\ln (1+1.2t) \\ N(0)=34.8+\ln (1+1.2\cdot0) \\ N(0)=34.8+\ln (1) \\ N(0)=34.8+0 \\ N(0)=34.8 \end{gathered}[/tex]so, initially there were 34.8 microbes
Step 2
b)How fast are the microbes increasing after 10 hours?
to know this, let t=10
so
[tex]\begin{gathered} N(t)=34.8+\ln (1+1.2t) \\ N(10)=34.8+\ln (1+1.2\cdot10) \\ N(10)=34.8+\ln (1+12) \\ N(10)=34.8+\ln (13) \\ N(10)=34.8+2.56 \\ N(10)=37.36 \end{gathered}[/tex]therefore , after 10 hours the number of microbes is 37.36
I hope this helps you
Which lines are parallel?
M: y + 1 = -3 (x-1)
K: y =3(x+2)
P: y + 4 = 3x
The most appropriate choice for equation of line in slope intercept form will be given by-
line K is parallel to line P
line M is not parallel to both line K and line P.
What is equation of line in slope intercept form?
The most general form of equation of line in slope intercept form is given by y = mx +c
Where m is the slope of the line and c is the y intercept of the line.
Slope of a line is the tangent of the angle that the line makes with the positive direction of x axis.
If [tex]\theta[/tex] is the angle that the line makes with the positive direction of x axis, then slope (m) is given by
m = [tex]tan\theta[/tex]
The distance from the origin to the point where the line cuts the x axis is the x intercept of the line
The distance from the origin to the point where the line cuts the y axis is the y intercept of the line
For M
y + 1 = -3(x - 1)
y + 1 = -3x + 3
y = -3x + 3-1
y = -3x + 2
Slope of M = -3
For K
y = 3(x+2)
y = 3x + 6
Slope of K = 3
For P
y + 4 = 3x
y = 3x - 4
slope of P = 3
Since Slope of K = Slope of P,
line K is parallel to line P
Since slope of M is different from slope of both K and P,
line M is not parallel to both line K and line P.
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plssss help!!!! Right triangle RST is drawn below. A square is drawn on to each leg of the triangle, and a square is drawn onto the hypotenuse of the triangle. The area of square A is 25 cm² . The area of square B is 144 cm². Determine the length in centimeters of x, the hypotenuse of the right triangle.O 13 cmO 17 cm O 169 cmO 119 cm
the hypotenuse of the right angled triiangle = x = 13cm (option A)
Explanation:
Area of B = 144cm²
Area of a square = length²
length = √Area of a square
shape of B is a square, hence the length of B:
the length of one of the side of B = √144 = 12cm
Area of A = 25cm²
shape of A is a square
the length of one of the side of A = √25 = 5cm
Triangle is a right angled-triangle
From the diagram, the length of the side of B = opposite = 12cm
The length of the side of A = adjacent = 5cm
The length of the third square marked x = hypotenuse
Using pythagoras' theorem:
Hypotenuse² = opposite² + adjacent²
x² = 12² + 5²
x² = 144 + 25
x² = 169
x = √169
x = 13cm
Hence, the hypotenuse of the right angled triiangle = x = 13cm (option A)
What does the y-intercept mean? What does the x-intercept mean? Explain what each intercept means and then Identify the x-intercept and y-intercept from each equation.A. y=7/2x -2B. x=-3
x intercept = where the function crosses the x axis (x,0)
y intercept = where the function crosses the y-axis (0,y)
A. y=7/2x-2
x intercept , replace y by 0 and solve for x:
0 =7/2x-2
2= 7/2 x
2 / (7/2) = x
x= 4/7
y-intercept, replace x by 0 and solve for y
y= 7/2x-2
y= 7/2 (0) -2
y=-2
B.
x-intercept:
x=-3
y-intercept
0=-3
It doesn't have a y-intercept.
Scientists are conducting an experiment with a gas in a sealed container. The mass of the gas is measured and the scientists realize that the gas is leaking over time in a linear way. Eight minutes since the experiment started the gas had a mass of 302.4 grams. Seventeen minutes since the experiment started the gas had a mass of 226.8 gramsLet x be the number of minutes that have passed since the experiment started and let y be the mass of the gas in grams at that moment. Use a linear equation to model the weight of the gas over time.a) This lines slope-intercept equation is [ ] b) 39 minutes after the experiment started, there would be [ ] grams of gas left. c) if a linear model continues to be accurate, [ ] minutes since the experiment started all gas in the container will be gone.
Here, we want to model an experiment linearlly
From the question, we have it that;
The coordinates are written as;
(number of minutes, mass of gas)
So, what we have to do know is to set up the two given points
These are the points;
(8,302.4) and (17,226.8)
Now, using these two points, we can model the equation
We start by getting the slope of the line that passes through these two points
To do this, we shall use the slope equation
We have this as;
[tex]\begin{gathered} \text{slope m = }\frac{y_2-y_1}{x_2-x_1} \\ \\ (x_1,y_1)\text{ = (8,302.4)} \\ (x_2,y_2)\text{ = (17,226.8)} \\ \text{substituting these values;} \\ m\text{ = }\frac{226.8-302.4}{17-8}\text{ = }\frac{-75.6}{9}\text{ = -8.4} \end{gathered}[/tex]The general equation representing a linear model is ;
[tex]\begin{gathered} y\text{ = mx + b} \\ m\text{ is slope} \\ b\text{ is y-intercept} \\ y\text{ = -8.4x + b} \end{gathered}[/tex]To get the y-intercept so as to write the complete equation, we use any of the two points and substitute its coordinates
Let us substitute the coordinates of the first point
[tex]\begin{gathered} 302.4\text{ = -8.4(8) + b} \\ 302.4\text{ = -67.2 + b} \\ b\text{ = 67.2 + 302.4} \\ b\text{ = 369.6 } \end{gathered}[/tex]a) Thus, we have the complete linear model as;
[tex]y\text{ = -8.4x + 369.6}[/tex]b) To get this, we simply substitute the value of x given into the linear model
[tex]\begin{gathered} y\text{ = -8.4(39) + 369.6} \\ y\text{ = -327.6 + 369.6} \\ y\text{ = 42} \end{gathered}[/tex]39 minutes after the experiment started, there would be 42 grams
c) If all the gas is gone, then the value of y will br zero at this point
To get the corresponding x-value which is the time, we have it that;
[tex]\begin{gathered} 0\text{ = -8.4x +369.6} \\ 8.4x=\text{ 369.6} \\ x\text{ = }\frac{369.6}{8.4} \\ x\text{ = 44} \end{gathered}[/tex]In 44 minutes, all the gas in the container will be gone
a) rewrite each equation using function notation f(x) b) find f(3)
Hello there. To solve this question, we'll simply have to isolate y and plug in the value for x.
Given the equation:
4x - 3y = 8
To rewrite it using the notation y = f(x), subtract 4x on both sides and divide the equation by -3
-3y = 8 - 4x
y = - 8/3 + 4x/3
Now, plug in x = 3 in order to find f(3)
f(3) = -8/3 + 4 * 3/3 = -8/3 + 12/3 = 4/3
4. Driving on the highway, you can safely drive 65 miles per hour. How far can you drive in ‘h’ hours? What is the domain of the function which defines this situation?A) 65B) the number of hours you driveC) the distance you driveD)the amount of gas you use
Answer:
[tex]\text{ The number of hours you drive.}[/tex]Step-by-step explanation:
For distance, we can apply the following equation:
[tex]\begin{gathered} d=s*h \\ where, \\ s=\text{ speed} \\ h=\text{ hours} \end{gathered}[/tex]Since we know that we can safely drive 65 miles per hour, the domain will be defined by the number of hours you drive:
[tex]d=65h[/tex]Each coordinate grid shows the graph of a system of two equations. Which graph represents a system of equations with no solution? Select all that apply.
System of Linear Equations with No Solutions
A system has no solutions if two equations are parallel.
Therefore, The answer would be option:
As an incoming college freshman, Tina received a 10-year $15,100 Federal Direct
Unsubsidized Loan with an interest rate of 4.29%. She knows that she can begin making
loan payments 6 months after graduation but interest will accrue from the moment the
funds are credited to his account. How much interest will accrue while she is still in
school and over the 6-month grace period for this freshman year loan?
O $2,813.28
O $2,915.06
O $3,001.32
O $3,102.38
The interest that will accrue while the college freshman is still in school and over the 6-month grace period for this freshman year loan is, approximately, B. $2,915.06.
How is the interest determined?The interest for federal college loans is based on the simple interest formula instead of compounding.
By compounding, we mean that interest is computed on the principal and accumulated interest.
Federal Direct Unsubsidized Loan = $15,100
Number of years for college = 4 years
The number of years before repayment = 4.5 years (4 years + 6 months)
Simple interest for 4.5 years = $2,915.06 ($15,100 x 4.29% x 4.5 years)
On the other hand, one can compute the compounded interest using an online finance calculator.
Compounded Interest:N (# of periods) = 4.5 years
I/Y (Interest per year) = 4.29%
PV (Present Value) = $15,100
PMT (periodic payment) during the 4.5 years = $0
Results:
FV = $18,241.85
Total Interest = $3,141.85
Thus, the interest is Option B.
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tom has a rectangular prism - shaped suitcase that measures 9 inches by 9 inches by 24 inches. he needs a second suitcase that has the same volume but smaller surface than his current suitcase. which suitcase size would fit Toms needs
ANSWER:
18 inches by 9 inches by 12 inches
EXPLANATION:
The volume of Tom's rectangular prism-shaped suitcase which measures 9 inches by 9 inches by 24 inches is;
[tex]\begin{gathered} Volume=l*w*h \\ \\ =9*9*24 \\ \\ =1944\text{ }square\text{ }inches \end{gathered}[/tex]So the volume of Tom's suitcase is 1944 cubic inches
The surface area will be;
[tex]\begin{gathered} SA=2(lw+wh+hl) \\ \\ =2(9*9+9*24+24*9) \\ \\ =2(81+216+216) \\ \\ =2(513) \\ \\ =1026\text{ }square\text{ }inches \end{gathered}[/tex]So the volume of the suitcase is 1026 square inches
*Let's go ahead and determine the volume and surface area of a suitcase that measures 18 inches by 18 inches by 6 inches;
[tex]\begin{gathered} Volume=l*w*h \\ \\ =18*18*6 \\ \\ =1944\text{ cubic inches} \end{gathered}[/tex][tex]\begin{gathered} Surface\text{ }Area=2(18*18+18*6+6*18) \\ \\ =2(324+108+108) \\ \\ =2(540) \\ \\ =1080\text{ square inches} \end{gathered}[/tex]We can see that the suitcase that measures 18 inches by 18 inches by 6 inches has the same volume as the first one but a higher surface area which doesn't fit Tom's needs
*Let's go ahead and determine the volume of a suitcase that measures 12 inches by 10 inches by 9 inches;
[tex]\begin{gathered} Volume=12*10*9 \\ \\ =1080\text{ cubic inches} \end{gathered}[/tex]We can see that the suitcase that measures 12 inches by 10 inches by 9 inches has a different volume from the first one which doesn't fit Tom's needs.
Let's go ahead and determine the volume of a suitcase that measures 16 inches by 5 inches by 9 inches;
[tex]\begin{gathered} Volume=16*5*9 \\ \\ =720\text{ cubic inches} \end{gathered}[/tex]We can see that the suitcase that measures 16 inches by 5 inches by 9 inches has a different volume from the first one which doesn't fit Tom's needs.
*Let's go ahead and determine the volume and surface area of a suitcase that measures 18 inches by 9 inches by 12 inches;
[tex]\begin{gathered} Volume=l*w*h \\ \\ =18*9*12 \\ \\ =1944\text{ cubic inches} \end{gathered}[/tex][tex]\begin{gathered} Surface\text{ }Area=2(18*9+9*12+12*18) \\ \\ =2(162+108+216) \\ \\ =2(486) \\ \\ =972\text{ square inches} \end{gathered}[/tex]We can see that the suitcase that measures 18 inches by 9 inches by 12 inches has the same volume as the first one and s smaller surface area which fits Tom's needs
Suppose that the probability that you will win a contest is 0.0002, what is theprobability that you will not win the contest? Leave your answer as a decimal and donot round or estimate your answer.
Answer:
0.9998
Explanation:
The probability that you will not win the contest can be calculated as 1 less the probability that you will win a contest, so
1 - 0.0002 = 0.9998
Therefore, the answer is 0.9998
find the size of each interior angle of a regular hexagon
Answer:
Each interior angle = 180° -60° = 120°
Step-by-step explanation: We know that the three angles in a triangle, add up to 180°, and all the three angles are 60° in an equilateral triangle. The total number of angles of an enclosed space is 180° (n-2) where in is the number of sides.
A hexagon has six sides, so: s= 180° (6-2)
s= 180° x 4
s= 720°
now since in a regular shape, each interior angle is equal. We just divide the total interior angle with a number of sides
6.
720° divided by 6 is equal to 120°
Solve the inequality a < 5 and write the solution using: Inequality Notation:
Answer:
Step-by-step explanation:
In the triangle below, if B = 69°, A = 32°, c = 5.7, use the Law of Sines to find a. Round your answer to the nearest hundredth.
We know that the interior angles have to add to 180°, then we have that:
[tex]\begin{gathered} C=180-69-32 \\ C=79 \end{gathered}[/tex]Hence angle C=79°.
Now that we know the angle C we can use the law of sines to find a; the law of sines states that:
[tex]\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}[/tex]From this we have the equation:
[tex]\frac{\sin A}{a}=\frac{\sin C}{c}[/tex]Plugging the values given and solving for a we have:
[tex]\begin{gathered} \frac{\sin32}{a}=\frac{\sin 79}{5.7} \\ a=(5.7)\frac{\sin 32}{\sin 79} \\ a=3.08 \end{gathered}[/tex]Therefore a=3.08
Does the least-squares fit line always go through at least one point in the plot?
Not necessarily. The least-squares line is the best fit for all the points in the scatterplot. if it so happens that in order to get close to some point on the plot the line has to go a little further away from some other point, the line will be adjusted to accommodate that.
Hence, the least square line does not always pass through at least one point on the line.
Introduction to Probability
The probability of birthday being in July is 1/12.
What do you mean by probability?
The area of mathematics known as probability deals with numerical representations of the likelihood that an event will occur or that a statement is true. An event's probability is a numeric value and 1, where, broadly speaking, 0 denotes the event's impossibility and 1 denotes certainty. The likelihood that an event will occur increases with its probability. A straightforward illustration is flipping a fair (impartial) coin. The chance of either "heads" or "tails" is half because there are only two possible outcomes (heads or tails), and because the coin is fair, both outcomes (heads and tails) are equally likely.
Probability of birthday being in July = 1/12 (12 months in a year)
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find the missing values in the figure below ( I need help as soon as possible only have 5 minutes available)
You can see in the figure attached that there are two Right triangles.
By definition, Right triangles are those triangles that have an angle that measures 90 degrees.
The larger triangle is the triangle ABC, but you only know the lenght of the side BC, which is:
[tex]BC=15m+2.5m=17.5m[/tex]And for the smaller triangle you only know the side whose lenght is 2.5 meters.
Therefore, since the exercise does not provide any other lenght and it does not provide another angle, you can conclude that the missing values cannot be determine with the given information.
So, the answer is OPTION D.
when a graph is a smooth curve it means that there is not a definite law connecting the two quantities which are plotted true or false
Question:
When a graph is a smooth curve it means that there is not a definite law connecting the two quantities which are plotted.
Solution:
a smooth curve is by definition a function, so by definition of a function, we have that there is a definite law connecting the two variables (quantities).
Answer: false.
Chords WP and KZ intersect at point L in the circle shown.Wz*3x - 22KIL5РWhat is the length of KZ?7.5910O 12
For the cicle with intersecting chords the relation between the length of segments of chords is,
[tex]KL\cdot ZL=WL\cdot PL[/tex]Substitute the values in the equation and solve for x.
[tex]\begin{gathered} 2\cdot(3x-2)=x\cdot5 \\ 6x-4=5x \\ 6x-5x=4 \\ x=4 \end{gathered}[/tex]So value of x is 4.
The equation for the length of chord KZ is,
[tex]\begin{gathered} KZ=KL+LZ \\ =2+3x-2 \\ =3x \end{gathered}[/tex]Substitute the value of x in equation to determine the length of KZ.
[tex]\begin{gathered} KZ=3\cdot4 \\ =12 \end{gathered}[/tex]So answer is 12.
find the probability of tossing 5 tails, them 5 heads. on the first 10 tosses of a fair coin
When a coin is tossed, the probability iof getting a head or a tail is 1/2.
The probability of tossing 5 tails = (1/2)^5
The probability of tossing 5 heads = (1/2)^5
The probability of tossing 5 talis and 5 heads = (1/2)^5 X (1/2)^5
= (1/2)^10
Describe the correlation in the scatter plot below.----------------The scatter plot shows (positive linear, positive linear with one outlier, negative linear, negative linear with one outlier, nonlinear, or no) correlation because as the plotted values of x increase, the values of y generally (decrease, increase, show no pattern or follow a nonlinear pattern).
A scatter plot shows a positive correlation when the values of y tend to increase as the values of x increases.
From this scatter plot, we can see that as the values of x increase, the values of y also increase. Therefore, this scatter plot shows a positive linear correlation.
An outlier is the the dot which doesn't fit with other dots or is far away from the rest of the dots. Here, we have one outlier.
Therefore, we can say the scatter plot shows positive linear with one outlier correlation because as the plotted values of x increase, the values of y generally increase.
ANSWER:
The scatter plot shows positive linear with one outlier correlation because as the plotted values of x increase, the values of y generally increase.
cabrinha run 3/10 mile each day for 6 days how many miles did she run in off
3/10 mile per day for 6 days.
To find how many miles did she run multiply the miles per day by 6days:
[tex]\frac{3\text{mile}}{10\text{day}}\cdot6\text{days}=\frac{18}{10}\text{mile}=\frac{9}{5}\text{mile}[/tex]Then, in 6 days she run 9/5 mileA manufacturing process has a 70% yield, meaning that 70% of the products
Answer:
are acceptable and 30% are defective.
Step-by-step explanation:
I need some kind of tutor really smart on math
To solve this problem we will need a system of equations.
Step 1. Find the first equation.
Using the statement "Emma rented a bike for 4 hours and paid £18", we will call the cost per hour h, and the flat fee f. Thus, the first equation is:
[tex]4h+f=18[/tex]This is because Emma rented the bike for 4 hours but she had to pay a flat fee f, and the total was £18.
Step 2. Find the second equation.
We do the same but now with the statement "Louise rented a bike for 7 hours and paid £25.5". Remember that for our equation, h represents the cost per hour and f the flat fee. The second equation is:
[tex]7h+f=25.5[/tex]Step 3. In summary, our system of equations is:
[tex]\begin{gathered} 7h+f=25.5 \\ 4h+f=18 \end{gathered}[/tex]Step 4. To solve part a. we have to find the cost per hour "h".
To find it, we use the elimination method in our system of equations, which consists of adding or subtracting the equations in order to eliminate one variable.
Since we are interested in finding "h", we can subtract the second equation from the first one, and we get the following:
Applying the subtraction:
And we start subtracting 7h-4h, which results in 3h:
The next subtraction is f-f, which results in 0.
And then, subtract 25.5-18:
The equation we have as a result is:
[tex]3h=7.5[/tex]Which is an equation we can use to solve for the cost per hour h.
Dividing both sides by 3:
[tex]\begin{gathered} h=\frac{7.5}{3} \\ h=2.5 \end{gathered}[/tex]The cost per hour is £2.5
Step 5. To find part b we need to find the rental feed, in our case, this means to find "h".
Using the first equation of the system:
[tex]7h+f=25.5[/tex]And substituting the previous result:
[tex]h=2.5[/tex]We get:
[tex]7(2.5)+f=25.5[/tex]Solving the operations:
[tex]17.5+f=25.5[/tex]And solving for f:
[tex]\begin{gathered} f=25.5-17.5 \\ f=8 \end{gathered}[/tex]the flat fee is £8.
Step 6. To find part c, we consider the cost per hour and the flat fee.
Michael rented the bike for 2 hours.
Since the cost per hour is £2.5, and the flat fee is £8, he will pay:
[tex]2(2.5)+8[/tex]Solving these operations:
[tex]5+8=13[/tex]It will cost £13.
Answer:
a. £2.5
b. £8
c. £13
Find the Standard form of the line that has a slope of -6 and y intercept of 10.
We have to write the standard form, Ax+By=C, of the line that has a slope m = -6 and y-intercept b = 10.
To do that we use that information to write the slope-intercept form of the equation and then rearrange it into standard form.
The slope-intercept for the line is:
[tex]\begin{gathered} y=mx+b \\ y=-6x+10 \end{gathered}[/tex]We can then rearrange it as:
[tex]\begin{gathered} y=-6x+10 \\ 6x+y=10 \end{gathered}[/tex]Answer: 6x + y = 10
Yea I think and her dad is doing great so
Given the following function:
[tex]tan\text{ }\theta=\frac{10}{y}[/tex]Both θ and y are functions of the time (t)
We will find the derivatives of θ and y with respect of the time (t) as follows:
[tex]sec^2θ*\frac{dθ}{dt}=-\frac{10}{y^2}*\frac{dy}{dt}[/tex]Now, we will find dy/dt when θ = π/6 and dθ/dt = π/12
First, we need to find the value of y when θ = π/6
[tex]\begin{gathered} tan(\frac{\pi}{6})=\frac{10}{y} \\ \frac{1}{\sqrt{3}}=\frac{10}{y} \\ \\ y=10\sqrt{3} \end{gathered}[/tex]so, we will substitute the values to find dy/dt as follows:
[tex]\begin{gathered} sec^2(\frac{\pi}{6})*\frac{\pi}{12}=-\frac{10}{(10\sqrt{3})^2}*\frac{dy}{dt} \\ \\ so,\frac{dy}{dt}=-\frac{(10\sqrt{3})^2}{10}*sec^2(\frac{\pi}{6})*\frac{\pi}{12}=-10.4719755 \end{gathered}[/tex]Rounding to 2 decimal places
So, the answer will be:
[tex]\frac{dy}{dt}=-10.47\text{ feet/hour}[/tex]