Line g passes through the points (-2.6,1) and (-1.4.2.5), as shown. Find theequation of the line that passes through (0,-b) and (c,0).
Answers
The blue line passes through the points
(-2.6, 1) and (-1.4, 2.5)
I will label the coordinates as follows for reference:
[tex]x_1=-2.6,y_1=1,x_2=-1.4,y_2=2.5[/tex]Step 1: Find the slope of the blue line
The slope between two points is calculated with the formula:
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]We substitute the values and we get that the slope of the blue line is:
[tex]m=\frac{2.5-1}{-1.4-(-2.6)}=\frac{1.5}{1.2}=1.25[/tex]The slope m of the blue line is 1.25.
step 2: With that slope, calculate b (the intercept of the blue line with the y axis).
For this we use the point - slope equation:
[tex]y=m(x-x_1)+y_1[/tex]Where we will use the sane x1 and x2 as in the previous step, so we get
[tex]\begin{gathered} y=1.25(x-(-2.6))+1 \\ y=1.25(x+2.6)+1 \\ y=1.25x+3.25+1 \\ y=1.25x+4.25 \end{gathered}[/tex]We compare this with the slope-intercept equation
[tex]y=mx+b[/tex]And we can see that the incercept b is 4.25
[tex]b=4.25[/tex]step 3: Find the value of c.
to find the value of c, we need to know at which point the blue line crosses the x axis.
Since we already have the equation of the blue line y=1.25x+4.25, and the line crosses the x axis at y=0, we substitute this to find the x value that is equal to c:
[tex]\begin{gathered} 0=1.25x+4.25 \\ -4.25=1.25x \\ \frac{-4.25}{1.25}=x \\ -3.4=x \end{gathered}[/tex]The blue line crosses the x axis at (-3.4,0), thus we can conclude that
[tex]c=-3.4[/tex]Step 4: Define the two point where the orange line passes through.
We know from the picture that the orange line passes through (c,0) and (0,-b)
Since we have the values of c = -3.4 and b=4.25, we can say that the orange line passes through (-3.4, 0) and (0, -4.25)
Step 5: Calculate the slope of the orange line.
the orange line passes through (-3.4, 0) and (0, -4.25), so we define:
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Use the remainder theorem to find P (1) for P(x) = 2x - 3x' + 3x -3.Specifically, give the quotient and the remainder for the associated division and the value of P (1).미미2Quotient = 0Х$2Remainder =0P(1) =
Answers
Using the remainder theorem, we must find P(1) for:
[tex]P(x)=2x^4-3x^3+3x-3[/tex]1) Because we want to evaluate P(x) for x = 1, we must compute
[tex]\frac{2x^4-3x^3+3x-3}{x-1}[/tex]2) Now we make the synthetic division by putting a 1 in the division box:
The remainder from the division is:
[tex]R=-1[/tex]The quotient of the division is:
[tex]2x^3-x^2+2x+2[/tex]3) From the synthetic division we get a remainder R = -1, applying the Remainder Theorem we get that:
[tex]P(1)=R=-1[/tex]Summary
The answers are:
1)
[tex]Quotient=2x^3-x^2+2x+2[/tex]2)
[tex]Remainder=-1[/tex]3)
[tex]P(1)=-1[/tex]Khalid is investigating two linear functions. The first linear function is defined by the equation 2x + 3y = 12. The second linear functionpasses through the points (3,-2) and (-2, k).For the case where the two linear functions have the same y-intercept, what must be the value of k?k=
Answers
According to the given data we have the following:
first linear function is defined by the equation 2x + 3y = 12
second linear function passes through the points (3,-2) and (-2, k)
the two linear functions have the same y-intercept
k?
To calculate k first we have to do the following:
we would have to use the formula y=mx +b
the two linear functions have the same y-intercept, therefore, b=12.
So, y=mx +12
As second linear function passes through the points (3,-2) we are going to substitue the x and y with 3 and -2.
So, -2=m*3+12
-2-12=m*3
-14=m*3
m=-14/3
m=-4
Finally we would calculate k by writiing the equation of the line that passes through each pair of points as follows:
y2-y1/x2-x1=m
So
[tex]\frac{k\text{ -(-2)}}{\text{-2 - 3}}\text{ }=\text{ -4}[/tex]So, k +2/-5=-4
k+2=20
k=20-2
k=18
is this right triangle shown a right triangle? 50 cm2 40mc2 20cm2 Explain your reasoning.
Answers
Solution:
Note that :
[tex]2500=50^2\ne\text{ }40^2+20^2=2000[/tex]and If the square of the length of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle. In this case, this statement is not true. We can conclude that it is not a right triangle.
Write the equation of a quadratic in general form, given its solutions.x=4 ; x=-1
Answers
The given solutions of the quadratic equation:
x = 4 and x = -1
First we make them factors of the equation:
x= 4 becomes:
[tex]x-4\text{ = 0}[/tex]and x = -1 becomes:
[tex]x\text{ + 1 = 0}[/tex]So (x-4) and (x+1) are the factors.
To get the general quadratic equation, we would expand the factors
[tex]\begin{gathered} (x-4)(x+1)\text{ = 0 } \\ x(x+1)\text{ -4(x+1) = 0} \\ x^2+x\text{ -4x-4 = 0} \end{gathered}[/tex][tex]\begin{gathered} Adding\text{ like terms} \\ x^2-3x-4\text{ = 0} \end{gathered}[/tex]The general quadratic equation:
[tex]x^2\text{ - 3x - 4 = 0}[/tex]look at the screenshots
Answers
Answer:
c for the first one and a for the second
Drag each tile to the correct box.The figures in the graph below can be shown to be similar by a sequence of transformations.Choose the correct sequence of transformations that take figure A to figure B.
Answers
Answer
Rotate 270 degrees clockwise about the origin → Translate 3 units right and 3 units up → Dilate by a scale factor of 3
Step-by-step explanation
Rotation 270 degrees clockwise about the origin transforms the point (x, y) into (-y, x). Applying this rule to the vertices of figure A, we get:
(-5, 5) → (-5, -5)
(-4, 4) → (-4, -4)
(-5, 1) → (-1, -5)
(-4, 1) → (-1, -4)
Translation 3 units right and 3 units up transform the point (x, y) into (x+3, y+3). Applying this rule to the previous points, we get:
(-5, -5) → (-5+3, -5+3) → (-2, -2)
(-4, -4) → (-4+3, -4+3) → (-1, -1)
(-1, -5) → (-1+3, -5+3) → (2, -2)
(-1, -4) → (-1+3, -4+3) → (2, -1)
Dilation by a factor of 3 transforms the point (x, y) into (3x, 3y). Applying this rule to the previous points, we get:
(-2, -2) → (3x-2, 3x-2) → (-6, -6)
(-1, -1) → (3x-1, 3x-1) → (-3, -3)
(2, -2) → (3x2, 3x-2) → (6, -6)
(2, -1) → (3x2, 3x-1) → (6, -3)
These vertices coincide with the vertices in figure B
A commercial citrus farm has decided to mechanise the planting operation. A tractor was
purchased that can plant and water seedlings automatically with pneumatic tubes. In order to
ensure the saplings receive the correct amount of water, a maximum variance of 55mm2
is
tolerated when watering. A sample of 31 planting lines were measured and the variance was
found to be 68mm2
. Test at 1% level of significance if the tractor is not operating correctly.
Answers
From the checks and calculation the tractor is not operating correctly.
What is standard deviation?Standard deviation refers to by how how much the data varies from the mean
How to determine if the tractor is not operating correctlyGiven data form the question
1% level of significance
variance was found to be 68mm2
A sample of 31 planting lines
a maximum variance of 55mm2
Definition of variables
1% level of significance is equivalent to 99% confidence interval
mean sample, μ = ?
standard deviation, SD = √variance = √68 = 8.246
Z score, Z = 2.576
from z table z score of 99%confidence interval = 2.576sample size, n = 31
maximum variance, X = 55mm2
The formula in term s of Z is
Z = ( X - μ ) / SD
2.576 = (55 - μ) / 8.246
(55 - μ) = 2.576 * 8.246
55 - μ = 21.242
μ = 55 - 21.242
μ = 33.758 mm²
For the tractor to be working correctly the difference between the mean and 2 * SD should not be more than the maximum variance which is 55mm²
55mm² ≥ mean ± 2 * SD
55mm² ≥ 33.758 mm² ± 2 * 8.246
55mm² ≥ 50.25
Since 50.25 is less than the maximum variance the tractor is operating correctly
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Note: Enter your answer and show all the steps that you use to solve this problem in the space provided
Answers
From the given picture we can see
ACB is a right triangle at C
AC = b
CB = a
AB = c
Since mSince a = 5 ft
Then to find b and c we will use the trigonometry ratios
[tex]\begin{gathered} \sin A=\frac{a}{c} \\ \sin 60=\frac{5}{c} \end{gathered}[/tex]Substitute the value of sin 60
[tex]\begin{gathered} \sin 60=\frac{\sqrt[]{3}}{2} \\ \frac{\sqrt[]{3}}{2}=\frac{5}{c} \end{gathered}[/tex]By using the cross multiplication
[tex]\begin{gathered} \sqrt[]{3}\times c=2\times5 \\ \sqrt[]{3}c=10 \end{gathered}[/tex]Divide both sides by root 3
[tex]\begin{gathered} \frac{\sqrt[]{3}c}{\sqrt[]{3}}=\frac{10}{\sqrt[]{3}} \\ c=\frac{10}{\sqrt[]{3}} \end{gathered}[/tex]To find b we will use the tan ratio
[tex]\begin{gathered} \tan 60=\frac{a}{b} \\ \tan 60=\frac{5}{b} \end{gathered}[/tex]Substitute the value of tan 60
[tex]\begin{gathered} \tan 60=\sqrt[]{3} \\ \sqrt[]{3}=\frac{5}{b} \end{gathered}[/tex]Switch b and root 3
[tex]b=\frac{5}{\sqrt[]{3}}[/tex]The exact values of b and c are
[tex]\begin{gathered} b=\frac{5}{\sqrt[]{3}} \\ c=\frac{10}{\sqrt[]{3}} \end{gathered}[/tex]The length of the diagonal of a Rectangle is 14cm,and it forms a 30 degree angle in one corner of the rectangle.What is the area of the rectangle.(A=LxW)
Answers
We can find W and L using the sine and the cosine functions:
[tex]\begin{gathered} \sin (30)=\frac{W}{14} \\ so\colon \\ W=14\cdot\sin (30) \\ --------- \\ \cos (30)=\frac{L}{14} \\ so\colon \\ L=14\cdot\cos (30) \end{gathered}[/tex]Therefore:
[tex]\begin{gathered} A=L\cdot W \\ A=14\cdot\cos (30)\cdot14\cdot\sin (30) \\ A=49\sqrt[]{3} \\ A\approx84.87cm^2 \end{gathered}[/tex]Answer:
84.87 cm²
A bourse named northern dancer won the Kentucky derby by running 1 1/4 miles in exactly 2 minutes. At this constant rate, how long does it take northern dancer to run the 1 1/2 mile Belmont stakes? Use unit rate
Answers
It is given that there are
[tex]1\frac{1}{4}=\frac{5}{4}\text{miles}[/tex]run in 2 minutes.
So, we have to determine time required to run
[tex]1\frac{1}{2}=\frac{3}{2}\text{miles}[/tex]Apply the unitary method,
For 5/4 miles required 2 minutes.
So , for 1 miles, time required
[tex]\frac{2}{\frac{5}{4}}=\frac{2\times4}{5}=\frac{8}{5}\min [/tex]Therefore,for 3/2 miles , time required is
[tex]\frac{3}{2}\times\frac{8}{5}=\frac{12}{5}\text{min}=2.4\min [/tex]Hence the time required is 2.4 minutes.
Higher Order ThinkingIn 448,244, how is the relationship between the first pair of 4s the same as the relationship between the second pair of 4s?4 grade studentLesson place value relationship
Answers
In the number 44,244, we can see two pairs of 4's.
The first pair (to the left) has a higher value than the second pair to the right, but the 4's have something in common: The leftmost 4 is ten times as high as the rightmost 4.
For this reason, we start the number as forty-four thousand and end up with forty-four.
What is the product of 11/12 and its reciprocal?
Answers
Answer:
The product of 11/12 is 0.916.This as as a fraction would be 0.916/1.00. This means it's reciprocal is 1.
Step-by-step explanation:
The reciprocal is basically the bottom part, denominator, of the fraction, being siwtched to on top of the fraction (numerator, and vice versa. The fraction would be 0.916/ 1.00 because the closest whole number to 0.916 is 1, meaning the fraction would be 0.916 out of 1. DO NOT MISTAKE THIS FOR 100. When you do the final step, finding the recirprocal of 0.916/1.00, we siwtch the numerator and denominators position, making out answer:
The product of 11/12 is 0.916 which is 0.916/1.00 in fraction form. The reciprocal of this is 1.00/0.916. Hope this helped!
Please ANSWER this
The table shows the parts of gelatin and water used to make a dessert.
Boxes of Gelatin Powder (oz) Water (cups)
3 9 6
7
At this rate, how much gelatin and water will Jeff use to make 7 boxes?
Jeff will use 14 oz of powder and 21 cups of water to make 7 boxes of gelatin.
Jeff will use 13 oz of powder and10 cups of water to make 7 boxes of gelatin.
Jeff will use 27 oz of powder and 18 cups of water to make 7 boxes of gelatin.
Jeff will use 21 oz of powder and 14 cups of water to make 7 boxes of gelatin.
Answers
Jeff needs 21 oz of gelatin and 14 cups of water to make 7 boxes
How to determine the amount of gelatin and water needed to make 7 boxes?The table of values is given as
Boxes Gelatin Powder (oz) Water (cups)
3 9 6
From the above table, we can see that
Gelatin Powder = 3 * Boxes
Water = 2 * Boxes
When there are 7 boxes, the equations become
Gelatin Powder = 3 * 7
Water = 2 * 7
Evaluate the products in the above equation
So, we have
Gelatin Powder = 21
Water = 14
Hence, the amount of gelatin and water needed to make 7 boxes are 21 oz and 14 cups respectively
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Answer: D: jeff will use 21 oz of powder and 14 cups of water
Step-by-step explanation:
the chart says there are
3 boxes of gelatin ) 9 oz of powder ) 6 cups of water
that equals the same as
1 box of gelatin ) 3 oz of power ) 2 cups of water
so for every box of gelatin, there is 3oz of powder and 2 cups of water
if he wants to make 7 boxes.....
7x3oz=21 oz
7x2cups=14 cups
so the Answer is D
the equation of line u is y=2x+8/9. line v includes the point (7,9) and is parallel to line u. what is the equation of.line v
Answers
The linear equation parallel to line u that passes through (7,9) is y = 2x - 5
How to find the equation of line V?
Two lines are parallel if have the same slope, we know that line V is parallel to:
y = 2x + 8/9
Then line V will be of the form:
y = 2x + c
To find the value of c, we use the fact that the line passes through (7, 9), replacing these values we get:
9 = 2*7 + c
9 = 14 + c
9 - 14 = c
-5 = c
The linear equation is y = 2x - 5
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I need help me answering #10 I'm really bad with ratios. (also sorry for the bad lighting in the picture)
Answers
The ratio of Perch to total fish is:
3 ( Number of perch that he caught)
------------------------------------------------------
9 ( The result of the addition of all values)
Simplifying, we have:
1
---
3
So, the answer would be 1 : 3
NO LINKS!! Please help me with this probability question 2a
Answers
====================================================
Work Shown:
A = he eats pizza
B = he drinks cola
P(A) = 0.40
P(B) = 0.60
P(A and B) = 0.30
Then apply the conditional probability formula.
P(A given B) = P(A and B)/P(B)
P(A given B) = 0.30/0.60
P(A given B) = 0.5 exactly
P(A given B) = 50%
If we know for certain he drinks cola, then there's a 50% chance of him eating pizza.
An emperor penguin has
76,634 feathers. The penguin has about 27 times as many feathers as a blue jay.
About how many feathers does the blue jay have?
Answers
Answer:
2,842 feathers
Step-by-step explanation:
An emperor penguin has 76,634 feathers. The penguin has about 27 times as many feathers as a blue jay. About how many feathers does the blue jay have?
76,634/27 = 2,842 feathers
The perimeter of a living room is 68 feet.if the length of the living room is 18 feet what is the width of the living room
Answers
the perimeter of a cuadrilateral is 2L+2W=68 where L is the length of the room
since L= 18 feet
we will have that
2(18)+2W=68
then
W=(68-36)/2= 32/2=16
Therefore, the width of the living room is 16 feet.the sales tax is 47 on the purchase of a dining room set for 940. find the sales tax rate.
Answers
The sales tax formula is used to determine how much businesses need to charge customers based on taxes in their area. State and local governments across the United States use a sales tax to pay for things like roads, healthcare and other government services. Sales tax applies to most consumer product purchases and exists in most states.
The sales tax formula is simply the sales tax percentage multiplied by the price of the item. It's important for businesses to know how to use the sales tax formula so that they can charge their customers the proper amount to cover the tax. For consumers, it's good to know how the sales tax formula works so that you can properly budget for your purchases
The sales tax formula is as shown below:
[tex]\text{sales tax=sales tax percentage }\times Price\text{ of the items}[/tex]Given that
Sales tax = 47
price of the dining room = 940
Sales tax rate= unknown
To find the sales tax rate, we would substitute into the formula above
[tex]\begin{gathered} 47=\text{sales tax rate }\times940 \\ \text{sales tax rate = }\frac{47}{940}\times100\text{ \%} \\ \text{sales tax rate = }\frac{1}{20}\times100=0.05\times100\text{ \%} \\ \text{sales tax rate= 5 \%} \end{gathered}[/tex]Hence, the sales tax rate is 5%
Shawna is making smoothies. The recipe calls for 2 parts yogurt to 3 parts
blueberries. Shawna wants to make 10 cups of smoothie mix. How many cups of
yogurt and blueberries does Shawna need?
Answers
Answer: 4 part yogurt 6 part blueberries
Step-by-step explanation: 2+3=5 5x2=10 3x2=6 2x2=4 6+4=10
The cost C (in dollars) of producing x units of a product is given by the following. C= 2.6. Square root of x + 600
Answers
The marginal cost in dollars of producing x units is given by the next equation:
[tex]C=2.6\sqrt[]{x}+600[/tex]a)
To find the marginal cost (in dollars per unit) when x= 9.
Then, we need to replace x=9 on the derivation of the cost equation:
So:
[tex]\frac{d}{dx}C=\frac{1.3}{\sqrt[]{x}}[/tex]Where:
[tex]\frac{d}{dx}2.6\sqrt[]{x}=2.6\frac{d}{dx}\sqrt[]{x}=2.6\frac{d}{dx}^{}x^{\frac{1}{2}}=2.6\cdot\frac{1}{2}x^{\frac{1}{2}-1}=1.3\cdot x^{-\frac{1}{2}}=\frac{1.3}{\sqrt[]{3}}[/tex]and, the derivate of a constant is equal to zero.
[tex]\frac{d}{dx}600=0[/tex]Replacing x= 9
[tex]\frac{d}{dx}C=\frac{1.3}{\sqrt[]{9}}[/tex]Hence, the marginal cost is equal to:
[tex]\frac{d}{dx}C=0.43[/tex]b) Now, when the production increases 9 to 10. It's the same as the cost of producing one more machine beyond 9.
Then, it would be x=10 on the cost equation:
[tex]C=2.6\sqrt[]{x}+600[/tex][tex]C=2.6\sqrt[]{10}+600[/tex][tex]C=608.22[/tex]and x= 9
[tex]C=2.6\sqrt[]{9}+600[/tex][tex]C=2.6(3)+600[/tex][tex]C=607.8[/tex]Then, we calculate C(10) - C(9) =
[tex]608.22-607.8[/tex][tex]=0.43[/tex]C)
Both results are equal.
Hence, the marginal cost when x=9 is equal to the additional cost when the production increases from 9 to 10.
what is 5/6 converted to decimal form
Answers
5/6 in decimal form is
[tex]\frac{5}{6}=0.8\bar{3}[/tex]A Geiger counter counts the number of alpha particles from radioactive material. Over a
long period of time, an average of 16 particles per minute occurs. Assume the arrival of
particles at the counter follows a Poisson distribution. Round your answers to four decimals.
a) Find the probability of exactly 21 particles arrive in a particular one minute period.
0.0426053 O
b) Find the probability of exactly one particle arrives in a particular one second period.
0.20424755689724
Answers
a) The probability of exactly 21 particles arrive in a particular one minute period is 0.0426
b) The probability of exactly one particle arrives in a particular one second period is 0.2042
What is probability?Probability is the ratio of the total number of conceivable outcomes to the number of outcomes in an exhaustive set of equally likely alternatives that result in a particular occurrence.
The probability is given by:
P(X=x) = (e⁻ⁿ nˣ)/x!
a) The probability of exactly 21 particles arrive in a particular one minute period is given by :
P(X=21) = (e⁻16 × 16²¹)/21!
P(X=21) = (2.176746853×10¹⁸)/51090942171709440000
P(X=21) = 0.0426
b) The probability of exactly one particle arrives in a particular one second period is :
1 min = 16 particles
1 sec = 16/60 particles
1 sec = 0.2667 particles
P(X=1) = (e⁻⁽¹⁶/⁶⁰⁾ × (16/60)¹)/1!
P(X=1) = 0.2042
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need help with this question please help
Answers
Let:
[tex]k\cdot RT=TU[/tex]Where:
k = Constant of proportionality
[tex]\begin{gathered} k\cdot4=6 \\ solve_{\text{ }}for_{\text{ }}k \\ k=\frac{6}{4} \\ k=\frac{3}{2} \end{gathered}[/tex]So:
[tex]\begin{gathered} k\cdot RS=UV \\ \frac{3}{2}(6)=UV \\ \frac{18}{2}=UV \\ UV=9 \end{gathered}[/tex]55mL of hardener to each container of resin. How much hardener should be added to 14 containers of resin?Write your answer in liters.
Answers
The amount of hardener added to 1 container of resin = 55mL.
The amount of hardener added to 14 containers is calculated as,
[tex]\begin{gathered} \text{Amount of hardener = 55 }\times\text{ 14 } \\ \text{Amount of hardener = 770 mL} \\ \text{Amount of hardener = 0.770 L} \end{gathered}[/tex]Thus 0.770 litres of hardener must be added to 14 containers of resin.
Find the equation of the line passing through the points (-11,-18) and (-22,-16). Write your answer in the
form y = mx + b.
Answer: y =
Write your answers as integers or as reduced fractions in the form A/B.
Answers
Answer:
y=−2/11x−20
Step-by-step explanation:
Determine whether the following relation is a function. Then state the domain and range of the relation or function.{(7,6), (4,-6), (0,-1), (3,3), (1,1)}Is this relation a function? Choose the correct answer below.A.Yes, because each first component corresponds to exactly one second component.B.No, because each first component corresponds to more than one second component.C.Yes, because each first component corresponds to more than one second component.D.No, because each first component corresponds to exactly one second component.
Answers
A function is a relation in which each possible input value leads to exactly one output value. We say “the output is a function of the input.”
The input values make up the domain, and the output values make up the range.
The relation is given to be:
[tex]\mleft\lbrace(7,6\mright),(4,-6),(0,-1),(3,3),(1,1)\}[/tex]To classify a function, get the input and output values. If each input value leads to only one output value, classify the relationship as a function. If any input value leads to two or more outputs, do not classify the relationship as a function.
The input values are: {7, 4, 0, 3, 1}
The output values are: {6, -6, -1, 3, 1}
Therefore, the relation is a function.
The correct option is OPTION A: Yes, because each first component corresponds to exactly one second component.
The domain is:
[tex]\mleft\{7,4,0,3,1\mright\}[/tex]The range is:
[tex]\mleft\{6,-6,-1,3,1\mright\}[/tex]Solve each system of equations please show your work! 3x+y-2z=22 x+5y+z=4 x=-3z
Answers
The solution of the system of equations are x = - 6 , y = 44 and z = 2
Given,
The system of equations;
3x + y - 2z = 22
x + 5y + z = 4
x = -3z
We have to solve the given equations;
Substitute x = -3z in both equations;
3x + y - 2z = 22
⇒ 3 × -3z + y - 2z = 22
⇒ - 9z + y - 2z = 22
⇒ - 11z + y = 22
And,
x + 5y + z = 4
⇒ - 3z + 5y + z = 4
⇒ 5y - 2z = 4
Solve the equations - 11z + y = 22 and 5y - 2z = 4
We get,
⇒ y = 44 and z = 2
So, x = - 3z
⇒ x = - 3 × 2
⇒ x = - 6
Thus, The solution of the system of equations are;
⇒ x = - 6 , y = 44 and z = 2
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Tristan tried his luck with the lottery. He can win $85 if he can correctly choose the 3 numbers drawn. If order matters and there are 13 numbers in the drawing, how many different ways could the winning numbers be drawn?
Answers
The ways to select the winning numbers is 286
How to determine the ways the winning numbers could be drawn?From the question, we have
Total numbers, n = 13Numbers to select, r = 3The winning numbers could be drawn is calculated using the following combination formula
Total = ⁿCᵣ
Where
n = 13 and r = 3
Substitute the known values in the above equation
Total = ¹³C₃
Apply the combination formula
ⁿCᵣ = n!/(n - r)!r!
So, we have
Total = 13!/10!3!
Evaluate
Total = 286
Hence, the number of ways is 286
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The pro shop at the Hidden Oaks Country Club ordered two brands of golf balls. Swinger balls cost$2.10 each and the Supra balls cost $1.00 each. The total cost of Swinger balls exceeded the total costof the Supra balls by $330.00. If an equal number of each brand was ordered, how many dozens ofeach brand were ordered?AnswerHow to enter your answer (opens in new window)KeypadKeyboard Shortcutdozen
Answers
The Solution:
Given that equal number of each brand of golf ball was ordered.
Let the number of each brand ordered be represented with n
Each swinger ball cost $2.10
So, the total cost of the swinger ball ordered is:
[tex]2.10n[/tex]Each Supra ball cost $1.00
So, the total cost of the supra ball ordered is:
[tex]\begin{gathered} 1.00\times n \\ \text{which becomes}\colon \\ n \end{gathered}[/tex]Given that the total cost of the Swinger balls exceeded the total cost of the Supra balls by $330.00. We have the linear equation below:
[tex]2.1n=n+330[/tex]We are required to find the number of dozens of each brand of golf balls that were ordered.
So, we shall solve for n and then divide the value by 12.
[tex]\begin{gathered} 2.1n=n+330 \\ \text{collecting the like terms, we get} \\ 2.1n-n=330 \\ 1.1n=330 \end{gathered}[/tex]Dividing both sides by 1.1, we get
[tex]\begin{gathered} \frac{1.1n}{1.1}=\frac{330}{1.1} \\ \\ n=300\text{ balls} \end{gathered}[/tex]Dividing 300 by 12 (since 1 dozen = 12 balls), we get
[tex]\frac{300}{12}=25\text{ dozens of each brand of golf balls were ordered.}[/tex]Therefore, the correct answer is 25 dozens.