P (8, 4) is a point on a circle, centre O.
The tangent at P intersects the axes at points A and B.
a) Find the gradient, m, of the tangent.
Your final line should say, m =...
(2)
b) Find the equation of the tange the
form y = mx + c where m and c are integers. (1)
y =
X +
c) Find the coordinates of point A.
d) Work out the length of AB,
giving your answer in the form a√5 where a is an integer.
(1)
B
P (8,4)
A
5 (2)
Total marks: 6
Answers
The equation of tangent is y = -2x + 20. The slope of the tangent line is negative 2. And coordinate of point A is (10, 0). And the length of AB is 22.36 units.
What is the equation tangent line to circle?The equation of the circle with center at origin and radius r is given as,
x² + y² = r²
The slope of the tangent line to the circle at points (8, 4) is given as,
2x + 2y (dy/dx) = 0
dy / dx = - 8 / 4
m = - 2
Then the equation of the line is given as,
y = -2x + c
The equation is passing through (8, 4), then we have
4 = -2(8) + c
c = 4 + 16
c = 20
The equation is y = -2x + 20. Then the x-intercept is given as,
y = 0
-2x + 20 = 0
2x = 20
x = 10
Then the length of the line is given as,
AB² = (10 - 0)² + (0 - 20)²
AB² = 100 + 400
AB = √(500)
AB = 22.36 units
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Related Questions
Evaluate the finite geometric series. 3 + 4.5 + 6.75 + … + 34.171875
Answers
First determine the common ratio between consecutive terms.
4.5/3 = 9/6 = 3/2
6.75/4.5 = 27/18 = 3/2
and so on. The [tex]n[/tex]-th term in the sequence is given by
[tex]a_n = 3 \cdot \left(\dfrac32\right)^{n-1}[/tex]
Solve for [tex]n[/tex] that gives the last term to find how many terms are in the sum.
[tex]n = 1 + \log_{3/2}\left(\dfrac{a_n}3\right)[/tex]
Plug in the last term.
[tex]n = 1 + \log_{3/2} \left(\dfrac{34.171875}3\right) = 1 + 6 = 7[/tex]
Then the sum of the first 7 terms of this sequence is
[tex]\displaystyle \sum_{n=1}^7 3 \cdot \left(\frac32\right)^{n-1} = 3\cdot\frac{1 - \left(\frac32\right)^7}{1-\frac32} = \boxed{\frac{6177}{44}} = 96.515625[/tex]
where we use the formula
[tex]\displaystyle \sum_{n=1}^k r^{n-1} = \frac{1-r^n}{1-r}[/tex]
Crossfire Company segments its business into two regions—East and West. The company prepared a contribution format segmented income statement as shown below:
Total Company East West
Sales $ 1,120,000 $ 770,000 $ 350,000
Variable expenses 840,000 616,000 224,000
Contribution margin 280,000 154,000 126,000
Traceable fixed expenses 155,000 65,000 90,000
Segment margin 125,000 $ 89,000 $ 36,000
Common fixed expenses 70,000
Net operating income $ 55,000
Required:
1. Compute the companywide break-even point in dollar sales.
2. Compute the break-even point in dollar sales for the East region.
3. Compute the break-even point in dollar sales for the West region.
4. Prepare a new segmented income statement based on the break-even dollar sales that you computed in requirements 2 and 3. What is Crossfire’s net operating income (loss) in your new segmented income statement?
5. Do you think that Crossfire should allocate its common fixed expenses to the East and West regions when computing the break-even points for each region?
Answers
PLEASE HELP!!!
In ΔJKL, m∠J = 78° and m∠L = 90°. Determine the measure of the exterior angle to ∠K.
168°
102°
57°
12°
Answers
The sum of all interior angles of a triangle is 180° thus the measure of the exterior angle to ∠K is 168° so option (A) is correct.
What is a triangle?A triangle is a 3-sided shape that is occasionally referred to as a triangle. There are 3 sides and three angles in every triangle, some of which may be the same.
It is known that the sum of all three angles inside a triangle will be 180°.
So, m∠J + m∠L + m∠K = 180°
m∠K = 180° - 78° - 90°
m∠K = 12°.
The exterior angle of k = 180- 12 = 168°.
Hence "The sum of all interior angles of a triangle is 180° thus the measure of the exterior angle to ∠K is 168°".
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2. The area of the base of the cone is 64π cm2 and the altitude is 6 cm. Calculate the cone: (a) the length of the constituent;(b) the area of the lateral area of the cone; c) the volume.
Answers
Answer: We have to calculate the (i) volume and (ii) the surface area of the cone:
[tex]\begin{gathered} A=\pi r^2+\pi rl\rightarrow(1) \\ \\ V=\pi r^2\frac{h}{3}\rightarrow(2) \end{gathered}[/tex]The formula (1) is for the lateral surface of the cone and the formula (2) is for the volume of the cone, the unknowns are determined as follows:
[tex]\begin{gathered} r=\sqrt{\frac{64\pi}{\pi}}=8cm \\ \\ r=8cm \\ \\ l=\sqrt{r^2+h^2}=\sqrt{(8cm)^2+(6cm)^2} \\ \\ l=\sqrt{100}=10 \\ \\ l=10 \end{gathered}[/tex]Therefore the volume and the lateral surface area is calculated as follows:
[tex]\begin{gathered} \begin{equation*} A=\pi r^2+\pi rl \end{equation*} \\ \\ A=\pi(8)^2+\pi(8)(10) \\ \\ A=144\pi cm^2\Rightarrow(x) \\ \\ \begin{equation*} V=\pi r^2\frac{h}{3} \end{equation*} \\ \\ V=\pi(8cm)^2\frac{6cm}{3} \\ \\ V=128\pi cm^3\Rightarrow(y) \end{gathered}[/tex]Therefore x and y are the answers.
A coin is flipped and a spinner is spun.What is the sample space of the experiment?
Answers
The sample space is the set of all different outcomes that can show up from the experiment.
Flipping a coin has two possible outcomes: heads or tails.
Spinning the spinner has three possible outcomes: A, B or C.
Draw a tree diagram to represent the sample space:
If we represent results using ordered pairs, the sample space is:
[tex]\mleft\lbrace(H,A\mright),(H,B),(H,C),(T,A),(T,B),(T,C)\}[/tex]Where H stands for heads and T stands for tails.
. La cuarta parte de un número, disminuida en 17,
es igual a 4. Calcula la mitad de dicho número.
Answers
Resolviendo una ecuación lineal veremos que el número es 76, y la mitad de ese número es 38.
¿Como calcular la mitad del número?Definamos la variable "x" como el número de la ecuación.
Tenemos la oración matematica:
"La cuarta parte de un número, disminuida en 17, es igual a 4"
Esto se puede escribir como la ecuacion:
(1/4)*x - 17 = 4
Esto es una ecuación lineal que puede ser resuelta para x.
(1/4)*x - 17 = 4
(1/4)*x = 4 + 17 = 19
x = 4*19 = 76
Entonces la mitad del número es:
x/2 = 76/2 = 38
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The function C(x) = 150 + 3.3x models the cost for a company to produce x units of a product. The function R(x) = 15x models the revenue the company earns if they sell x units of the product. Which function, P(x), models the profit the company earns if they sell x units of the product? (Profit = Revenue - Cost) O P(x) = 18.3x - 150 O P(x) = 11.7x - 150 OPlx) = 150 - 11.7x O P(x) = 18.3x + 150
Answers
Answer: We need to find the expression p(x) for profit.
[tex]\begin{gathered} \text{profit = revenue -cost} \\ \therefore\rightarrow \\ P(x)=R(x)-C(x) \\ P(x)=15x-(150+3.3x)=15x-150-3.3x=18.3x-150 \\ \therefore\rightarrow \\ P\mleft(x\mright)=18.3x-150 \\ \end{gathered}[/tex]Therefore the first option represents the profit of the company.
What is the measure of the base of the rectangle if the area of the triangle is 28 ft?
Answers
Answer:
14 ft
Step-by-step explanation:
The base of the rectangle and triangle is same.
Triangle area formula:
A = bh/2GivenA = 28 ft²,h = 4 ft.Find bSubstitute the values:
28 = 4b/228 = 2bb = 28/2b = 14 ftAnswer is 14 ft.
Explanation.Triangle Area Use Formula.
A = bh/2A = 28 ft2
H = 4 ft
B Value.
28 = 4b/2
28 = (4/2)b
28 = 2b
b = 28/2
b = 14
_____________________
Class: Middle School
Lesson: Triangle
[tex]\boxed{ \colorbox{lightblue}{ \sf{ \color{blue}{ Answer By\:Cyberpresents}}}}[/tex]
Eric can choose Plan A or Plan B for his long distance charges. For each plan, cost (in dollars) depends on minutes used (per month) as shown below.
Answers
(a) Plan B, 8
At 150 minutes, Plan B costs $20 and Plan A costs $12.(b) 250, Plan A
The costs are the same when the graphs intersect.Before 250 minutes, the graph for Plan A is lower than the graph of Plan B, meaning the graph of Plan A will cost less.The amount of cereal in a box is normally distributed with a mean of 24 oz. and a standarddeviation of 0.62 oz. If 7,500 boxes of cereal are packaged in one day, approximately how many boxeswould contain less than 22.9 oz.?
Answers
Given data:
The given mean is m=24.
The given standard devaition is s=0.62.
The given vallue is x=22.9.
The expression for the value of z is,
[tex]\begin{gathered} P(x<22.9)=P(z<\frac{x-m}{s}) \\ =P(z<\frac{22.9-24}{0.62}) \\ =P(z<-1.774) \end{gathered}[/tex]Refer the standard z-table.
[tex]P(x<22.9)=0.03836[/tex]The number of box required are,
[tex]\begin{gathered} N=7,500\times0.03836 \\ =287.7 \\ \approx288 \end{gathered}[/tex]Thus, the number of boxes are 288.
how do you know that the slope of every horizontal line is zero
Answers
Let:
[tex]\begin{gathered} P1=(x1,y1) \\ P2=(x2,y2) \end{gathered}[/tex]If the line is horizontal, then:
[tex]y1=y2[/tex]The slope is given by:
[tex]\begin{gathered} m=\frac{y2-y1}{x2-x1} \\ since\colon \\ y1=y2 \\ m=\frac{y1-y1}{x2-x1}=\frac{0}{x2-x1} \end{gathered}[/tex]Zero over any quantity is always zero, so:
[tex]m=\frac{0}{x2-x1}=0[/tex]Evaluate the expression 8÷2+54÷9×2×2
Answers
We are asked to evaluate the mathematical expression below;
[tex]8\div2+54\div9\times2\times2[/tex]To do this we would use "PEMDAS". PEMDAS is an acronym for the words parenthesis, exponents, multiplication, division, addition, subtraction.
Given two or more operations in a single expression, the order of the letters in PEMDAS tells you what to calculate first, second, third, and so on, until the calculation is complete.
We will then proceed to solve the expression according to PEMDAS.
[tex]\begin{gathered} 8\div2+54\div9\times2\times2 \\ =\frac{8}{2}+\frac{54}{9}\times2\times2 \\ =4+6\times4 \\ =4+24 \\ =28 \\ \end{gathered}[/tex]Answer: 28
This is 2.Dotty and Lionel are making a graph comparing a car's age with its gasmileage. Dotty says the graph should show discrete points because the car'sage is stated in whole numbers of years. Lionel says they should make acontinuous line because age increases gradually, like time. Which student doyou agree with and why?
Answers
Lionel should be the correct answer.
What is an expression?
Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.
Given that;
Dotty and Lionel are making a graph comparing a car's age with its gas mileage.
Dotty says the graph should show discrete points because the car sage is started in whole numbers of years.
Lionel says they should make a continuous line because age increases gradually, like time.
Now,
Clearly, The Lionel is correct because when we talk about the age, that means time and Time is continuous , never stopping or slowing down for anyone.
Thus, Lionel should be the correct answer.
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What is 2/3−3 5/6
Help please.
Answers
Answer:
Step-by-step explanation:
First you want to make the denominator the first multiple of both numbers so: 2/3 -3 5/6 = 2/6 - 3 5/6
Next you want to multiply the numerator by the amount of times you had to multiply the denominator so: 4/6 - 3 5/6
Then subtract the answer: 3 1/6
Could you teach me how to solve this problem without using a graphing calculator? (My algebra teacher doesn’t allow graphing calculators on the test.)
Answers
The type of function we are going to work with here is an exponential function. These functions are given by equations like:
[tex]f(x)=a\cdot b^x+c[/tex]Where x is the variable and a, b and c fixed parameters. In this case c=0, a=1 and b=13/4. This number b, i.e. the one affected by the exponent is called the base so I'm going to refer to it with that word.
All exponential functions have similar graphs, they pass from being very close, almost pararel to the x-axis and then they begin to rapidly increase toward infinite y values. In the following picture you can see the graph of two exponential functions:
The red graph tends to zero for positive x values, this means that its equation has a negative sign in the exponent:
[tex]\text{red(x)}=b^{-x}[/tex]The blue graph on the other hand tends to zero for negative x values which means that there's no negative sign in the exponent. This is the case of our function r(x) so we can discard the upper left option.
Another thing to take into account is the offset of the function, that number that I labeled as "c" in the first equation. This offset translates the graph upwards or downwards depending on its sign.
For example, in this image the graphs decrease for negative x values which means their exponents don't have a negative sign but they all tend to different values which means their offsets are different. Blue's offset is 0 since it tends to 0 while gray's offset is 1 because it tends to 1. The same way you can deduce that the orange graph has an offset of -2. In the case of our problem, function r(x) doesn't have an offset which means that in one direction it has to tend to 0. This means we can discard the lower rigth option.
We have already discarded two options, if we discard one more we have the answer.
Now let's take r(x) and evaluate it in x=0:
[tex]r(0)=(\frac{13}{4})^0=1[/tex]You don't need a calculator for this because any number raised to 0 lays 1. This means that at x=0 ,i.e. the y-intercept of the graph, the function has y=1.
Lower left graph intercepts the y-axis in 1 while the upper right does it closer to 0 so we can discard this last option.
In summary, we have only one possible option left, the one in the lower left. This is the correct option.
Hello, I need help with this precalculus homework question, please?HW Q9
Answers
Solution
- The question would like us to determine the difference expression that corresponds to the following logarithm
[tex]\log_a(\frac{M}{N})[/tex]- In order to solve this question, we should apply the Law of Logarithm which states that:
[tex]\log A-\log B=\log(\frac{A}{B})[/tex]- Applying this law, we have:
[tex]\begin{gathered} \text{ Let A from the formula be M and Let B from the formula be N from the question} \\ Note\text{ that }a\text{ in the question is represented by 10 from the formula. The base 10 in logarithm is usually not } \\ \text{ indicated} \\ \\ \log_a(\frac{M}{N})=\log_aM-\log_aN \end{gathered}[/tex]Final Answer
The answer to the question is
[tex]\operatorname{\log}_{a}(\frac{M}{N})=\operatorname{\log}_{a}M-\operatorname{\log}_{a}N[/tex]Write the set in set-builder notation.
1) The set of all calculus books
Answers
The set-builder form is { x | x is a calculus book}.
A set is a collection of quantities based on a specific criteria. A set can be expressed in two forms.
We usually use the form {a, b, c,....} in which we include each element of the set separated with a comma in between. This form of set notation is called the roster form.
Another set notation is the set-builder form. This form is more concise and precise and easily understandable also. In this form, all the elements are included in the set which has a same property. That is, here we don't write each and every element separately in the set but just emphasize the property of the elements which are included in the set.
We are asked to write the set of all calculus books in set-builder form.
The set-builder form for 'The set of all calculus books' is { x | x is a calculus book}.
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Solve for z.
z² = 0.64
Answers
Answer:
Step-by-step explanation:
z 3 = 0.64 Take the specified root of both sides of the equation to eliminate the exponent on the left side. z = 3 √ 0.64 The result can be shown in multiple forms. Exact Form: z = 3 √ 0.64 Decimal Form: z = 0.86177387
use the figures below what shows that rectangles are not similar PS.THIS IS JUST HOMEWORK THAT I NEED HELP WITH
Answers
Answer is option A
Because the other side (AB) is 15 and the other rectangle (JM) is 10, so the realation between them is 15/10 = 3/2
But, since DC and ML relation is NOT 3/2, then the rectanlges are not similar
What strategy would be the best to solve this problem?To get into shape, Miles ran for 1 minute on Monday, 3 minutes on Tuesday, 6 minutes on Wednesday, and 10 minutes on Thursday. If he continues, how many minutes will Miles run on Sunday?A. guess, check and reviseB. work backwardsC. use a formulaD. look for patterns
Answers
As we can see from the given data, that, it gives some informtaion about, an on going process, that is almost in rythm.
[tex]\begin{gathered} 1\text{ minute:Monday} \\ 3\text{minute: Tuesday} \\ 6\text{minute: Wednesday} \\ 10\text{ minutes: Thursday} \end{gathered}[/tex]So, from that we can see that, Mile had increased the running time in a particular pattern, at first he had increased the time by 2, then by 3, then by 4, and so on.
So to find the running time for sunday, we can solve this problem by analysing this pattern.
Therefore the answer is (D): looking for patterns.
f(x) = x2What is f(x) + f(x) + f(x)?DONE
Answers
If we have the function
[tex]f(x)=x^2[/tex]Then
[tex]\begin{gathered} f(x)+f(x)+f(x)=x^2+x^2+x^2 \\ =3x^2 \end{gathered}[/tex]This also can be written in the form
[tex]f(x)+f(x)+f(x)=3f(x)=3x^2[/tex]We see that both answer are the same, like it should be.
Divide. Round your answer to the nearest tenth.
160 divided by 0.749 =
Answers
rounded to the nearest tenth is 213.6
A custormer spends $240 at Kal's on a shopping trip The customer hands the cashier a 25% off coupon. How much should the customor pay for their purchase? Enter your answer in the field below. Thero is a sciontific calculator avalable at tho top of the screen, if you need it.
Answers
Let's begin by listing out the information given to us:
Amount spent = $240
discount = 25%
[tex]\begin{gathered} Pay=Amountspent-(AmountSpent\cdot discount)_{} \\ P=240-(240\cdot\frac{25}{100})=240-60 \\ P=180 \end{gathered}[/tex]Therefore, the customer paid $180 for his purchase
in the figure below m
Answers
The value of x is 15
Here, we want to get value of x
From the question, we have that AC = AD
This means that what we have is an isosceles triangle
Mathematically, the bases of an isosceles triangle have same angle
For triangle ACD; the measure of angle D and the measure of angle C is same
That means the value of x is 15 too
Jonah picked 287 cherry tomatoes from his garden in May and 339 in June.
How many cherry tomatoes did Jonah pick in all?
The estimate by rounding to the nearest hundred. Then solve.
Answers
Answer:
600
Step-by-step explanation:
the total is 626 but round it to the nearest hundred is 600 because the rule 5 or more go up one more 4 or less stay at rest
Put the rational numbers in order from least to greatest. -4.2, -4 1/2, -4, 0, 4
Answers
Let's express the mixed fraction in decimal.
[tex]-4\frac{1}{2}=-4.5[/tex]Then, from least to greatest, the numbers are
[tex]-4.5,-4.2,-4,0,4[/tex]Given a normal distribution with m = 100 and s = 10, what is the probability that a. X >75 b. X < 70 c. X < 80 or X > 110
Answers
Given a normal distribution with m = 100 and s = 10, the probabilities are;
a) X > 75 = 0.0062
b) X < 70 = 0.00135
c) X < 80 or X > 110 = 0.053954
What is the Probability from z-score?
We are given;
Population mean; m = 100
standard deviation; s = 10
The z-score is gotten from the formula;
z = (X - m)/s
Thus;
1) Probability that X > 75 is;
z = (75 - 100)/10
z = -2.5
From online p-value from z-score calculator, we have;
probability = 0.0062
2) Probability that X < 70 is;
z = (70 - 100)/10
z = -3
From online p-value from z-score calculator, we have;
probability = 0.00135
3) Probability that X < 80 is;
z = (80 - 100)/10
z = -2
From online p-value from z-score calculator, we have;
probability = 0.02275.
Probability that X > 110 is;
z = (110 - 100)/10
z = -1
From online p-value from z-score calculator, we have;
probability = 0.158655
Thus, P(0.02275 < x < 0.158655) = 0.053954
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If the length is 100 yards and the width is 55 yards, how many yards is the perimeter?
Answers
Answer:
perimeter = 310 yards
Step-by-step explanation:
If the length is 100 yards and the width is 55 yards, how many yards is the perimeter?
p = 2l + 2w
perimeter = 2(100) + 2(55)
perimeter = 200 + 110
perimeter = 310 yards
Length of the land= 100 yards
Width of the land = 55 yards
Perimeter of the land = ? Let it be P
We know that , Perimeter is given by the formula, P = 2(l+w)
Now , Perimeter of the land = 2(100+55) yards
=> P = 2(155)yards
=> P = 310 yards
Paul sold 162 cups of lemonade in 5 hours. About how many cups of lemonade did she sell each hour?
Answers
The number of cups of lemonade sold in each hour is 32.4.
The number of cups of lemonade in each hour is the rate of selling. The rate of selling will be calculated by the formula : number of cups ÷ time.
Keep the values in formula to find the number of cups of lemonade sold each hour
Number of cups of lemonade sold each hour = 162 ÷ 5
Performing division to find the number of cups sold each hour
Number of cups of lemonade sold each hour = 32.4
Thus, each hour, 32.4 cups of lemonade are sold.
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Find the slope of each line.
Answers
Given any line, to find the slope of the line, follow the procedure below:
First, pick any two points (x1,y1) and (x2,y2) on the line.
Then, substitute the points into the formula for slope below:
[tex]\begin{gathered} \text{Slope},m=\frac{Change\text{ in y-axis}}{Change\text{ in x-axis}} \\ m=\frac{y_2-y_1}{x_2-x_1} \end{gathered}[/tex]You can use this method to find the slope of any given line.
let x = 12; then y = 7/2(12) = 2 = 42 + 2 = 44
slope = m.... m = (y1 - y2)/(x1 - x2) then plug in the x & y values...
m = (30 - 44)/(8 -12) = -14/-4 = 7/2
thus, slope of line 'y' = 7/2