C is a circle with centre the origin.
A tangent to C passes through the points (-20, 0) and (0, 10)
Work out an equation of C.
You must show all your working.
Answers
The equation of the circle C is x² + y² = 80 when it passes through the tangent points (-20, 0) and (0, 10).
Equation of circle:
Equation of the circle refer the position of a circle on a cartesian plane.
Given,
C is a circle with center the origin.
A tangent to C passes through the points (-20, 0) and (0, 10)
Here we need to find the equation of the circle C.
In order to find the equation of our line:
We have to identify the y-intercept which is (0,10).
Now, we need to find the gradient of our line which is:
Than can be calculated by,
=> (10 - 0) / (0- (-20))
=> 10/20 = 0.5
Now, the equation of our line is written as,
y = 0.5x + 10
But, here we know the center of our circle is the origin:
Which is in the following form:
=> x² + y² = r²
where r refers the radius
So, here we need the line to meet our circle at one point, we can substitute the equation for our straight line in for y.
Then the equation of the circle is written as,
=> x² + (0.5x + 10)² = r²
When we simplify it, then we get,
=> x² + (0.5x + 10)(0.5x + 10) = r²
Now, we can expand and simplify the brackets to leave us with:
=> 1.25x² + 10x + 100 = r²
=> 1.25x² + 10x + 100 - r² = 0
Here we know that this is a tangent and so it only meets the circle at one point.
So, this equation should only have one solution.
Then it can be written as,
=> b² - 4ac = 0
=> 10² - 4(1.25)(100 - r²) = 0
When we simplify this one then we get the value of
=> r² = 90
Therefore the equation of the circle is x^2 + y^2 = 80.
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Related Questions
Gordon is 5 years older than Krutika. David is 6 years younger than Krutika. If the total of their ages is 86, how old is the eldest of them?
Answers
Answer:
Step-by-step explanation:
Well let Krutica’s age = x
K = x, G = x+5, D = x-6
K+G+D = 86
x + x+5 + x-6 = 86
3x = 87
x = 29
Gordon is the eldest of the few:
29 or x+5=
34 years old
Answer: Gordon is 34 years old
21a5b³-14a³b² +63a²b
———————————-
7ab
Answers
Answer:
[tex]3a^4b^2-2a^2b+9a[/tex]
Step-by-step explanation:
Because multiplication is the inverse of division, we're able to split this fraction into three separate fractions, which will simplify nicely at the end:
[tex]\frac{21a^5b^3}{7ab}; \frac{-14a^3b^2}{7ab}; \frac{63a^2b}{7ab}[/tex]
We can start by simplifying each fraction since 21, -14, and 63 are all divisible by 7:
[tex]\frac{3a^5b^3}{ab}; \frac{-2a^3b^2}{ab}; \frac{63a^2b}{ab}[/tex]
Finally, when dividing quotients with exponents, we subtract the exponents and we put all of our answers back together since everything was multiplied by each other at the beginning of the problem:
[tex]3a^4b^2-2a^2b+9a[/tex]
Which of the following is equal to ?
Answers
The given expression, [tex][\frac{(x^{2} y^{3})^{-2} }{(x^{6} y^{3} z)^{2} } ]^{3}[/tex] is equal to [tex]\frac{(x^{2} y^{3})^{-6} }{(x^{6} y^{3} z)^{6} }[/tex].
What are Exponents?
The exponent of a number indicates how many times to multiply the number.For example, three to the power of two is represented as, [tex]3^{2}[/tex]. Here, 3 is the base number and 2 is the exponent. Also, [tex]3^{2} =3 \times 3[/tex].What are the Laws of Exponents?
There are major seven Laws of Exponents as follows:
Product Rule of Exponents: [tex]a^{m} \times a^{n} =a^{m+n}[/tex]Quotient Rule of Exponents: [tex]\frac{a^{m} }{a^{n} } =a^{m-n}[/tex]Power of a Power Rule: [tex](a^{m} )^{n} =a^{m \times n}[/tex]Product of a Power Rule: [tex]a^{m} \times b^{m} =(ab)^{m}[/tex]Power of a Quotient Rule: [tex]\frac{a^{m} }{b^{m} } =(\frac{a}{b})^{m}[/tex]Zero Power Rule: [tex]a^{0} =1[/tex]Negative Exponent Rule: [tex]a^{-m} =\frac{1}{a^{m}}[/tex]From the given question, we have the expression, [tex][\frac{(x^{2} y^{3})^{-2} }{(x^{6} y^{3} z)^{2} } ]^{3}[/tex] -----(1)
Here, we can use the power of a power rule to simplify the expression.
The power of a power rule is of the form, [tex](a^{m} )^{n} =a^{m \times n}[/tex]
Solving expression (1) using the power of a power rule, we get
[tex][\frac{(x^{2} y^{3})^{-2} }{(x^{6} y^{3} z)^{2} } ]^{3}=\frac{(x^{2} y^{3})^{-2 \times 3} }{(x^{6} y^{3} z)^{2 \times 3} }\\\implies [\frac{(x^{2} y^{3})^{-2} }{(x^{6} y^{3} z)^{2} } ]^{3}=\frac{(x^{2} y^{3})^{-6} }{(x^{6} y^{3} z)^{6} }[/tex]
Therefore, the given expression is equal to [tex]\frac{(x^{2} y^{3})^{-6} }{(x^{6} y^{3} z)^{6} }[/tex]
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a store put dress shirts on sale for 2 for $27.00. How much will
someone pay if they buy 5 shirts?
Answers
Answer:
$67.50
Step-by-step explanation:
27(2) = 54
27/2 = 13.5
54+13.5 = 67.5
Answer:
67.5
explanation:
27/2= 13.5
5*13.5= 67.5
A regular hexagon with a perimeter of 45 meters is dilated by a scale factor of 4/3 to create a new hexagon. What is the perimeter of the new hexagon?
Answers
The perimeter of the resulting hexagon is 60 meters.
What is the perimeter of the image of a hexagon as a result of a dilation?
In this problem we find the perimeter of a regular hexagon, which is dilated around its center. Regular hexagons are polygons with six sides of equal length. Dilations are rigid operations of the form:
P'(x, y) = O(x, y) + k · [P(x, y) - O(x, y)]
Where:
O(x, y) - Center of dilation.k - Scale factorP(x, y) - Original pointP'(x, y) - Resulting pointThe perimeter is the sum of the side lengths of the hexagon and by dilation we find the following relationship between original and resulting perimeters (p, p'):
p' = k · p
If we know that p = 45 and k = 4 / 3, then the resulting perimeter is:
p' = (4 / 3) · (45)
p' = 60
The resulting hexagon has a perimeter of 60 meters.
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Are the lines parallel? y=-3x+2 and 3x+y=6
A. Yes, they have the same slope and a different y intercept
B. No they are not parallell
Answers
Answer:
A. Yes, they have the same slope and a different y-intercept.
Step-by-step explanation:
Equation of line is given as y = mx + c, where m is the slope and c is the y-intercept.
Rearrange 3x + y = 6 to y = mx + c form.
y = -3x + 6
Parallel lines have the same slope.
Since slope of both lines is -3, the lines are parallel.
If f(x) = -8m + 3 and g(x) = -8m^2 + m - 5, what is (f times g)x?
A.) 64m^3 - 16m^2 + 43m - 15
B.) 64m^3 +43m - 15
C.) 64m^3 - 32m^2 + 43m - 15
D.) -16m^2 + 43m - 15
Answers
Answer: Choice C
[tex]64m^3-32m^2+43m-15[/tex]
==================================================
Explanation:
Multiply out the expressions using the distribution rule as shown below.
[tex](-8m+3)(-8m^2+m-5)\\\\n(-8m^2+m-5) \ \ \text{ ... see note1}\\\\-8m^2n+mn-5n\\\\-8m^2( n )+m( n )-5( n )\\\\-8m^2( -8m+3 )+m( -8m+3 )-5( -8m+3 ) \ \ \text{ ... see note2}\\\\( 64m^3-24m^2 )+( -8m^2+3m )+( 40m-15 )\\\\64m^3+(-24m^2-8m^2)+(3m+40m)-15 \\\\64m^3-32m^2+43m-15 \\\\[/tex]
I used the tool WolframAlpha to confirm the answer is correct. GeoGebra is another free tool that offers similar capabilities.
Footnotes:
Note1: I let n = -8m+3, and replaced -8m+3 with n. That way the distribution on the next step could happen.Note2: I plugged in n = -8m+3. After which distribution happens 3 more times.Which is the equation in slope-intercept form
for the graph of the line shown?
F. y=-3x - 2
G. y=-3x + 2
H. y = 3x - 2
I. y = 3x + 2
Please help me!
Answers
differentiative
2x-3
2x+1
Answers
The differential equation is 8x-4 when the equation is (2x-3)(2x+1).
Given that,
The equation is (2x-3)(2x+1)
We have to find the differentiative equation.
Calculus differentiates the process of locating a function's derivatives. A derivative is the pace at which one function changes in relation to another. Sir Isaac Newton set the foundations for the laws of differential calculus. Numerous scientific disciplines make use of the limits and derivatives ideas. The essential ideas of calculus are differentiation and integration.
Differentiation is the rate at which one quantity changes in relation to another.
Take the equation
(2x-3)(2x+1)
We can do in two ways
1st way
Multiply
4x²+2x-6x-3
Subtraction
4x²-4x-3
Differentiation
8x-4
The equation is 8x-4.
2nd way
We have a formula in differentiation that is
u.v=uv'+vu'
Take u as 2x-3 and v as 2x+1
(2x-3)(2)+(2x+1)(2)
4x-6+4x+2
8x-4
Therefore, The differential equation is 8x-4 when the equation is (2x-3)(2x+1).
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Given: <1 and <2 are complementary. <1 and <3 are complementary
Prove: 22 = 23
Complete the proof.
Answers
Answers:
1st box: [tex]\boldsymbol{90}[/tex]
2nd box: [tex]\boldsymbol{90}[/tex]
3rd box: [tex]\boldsymbol{\text{m}\angle 2 = \text{m}\angle 3}[/tex]
===============================================
Explanation:
Complementary angles form a corner. Note that "complementary" and "corner" both start with "C" to help remember the rule.
By "corner", I refer to a 90 degree angle.
Since [tex]\angle 1\text{ and } \angle 2[/tex] are complementary, we know that [tex]\text{m}\angle 1+\text{m}\angle 2 = 90[/tex] simply by the definition of what it means to be complementary.
---------------
Through similar logic, [tex]\angle 1\text{ and } \angle 3[/tex] are complementary which means [tex]\text{m}\angle 1+\text{m}\angle3 = 90[/tex]
We can then equate the left hand sides (LHS) of both equations since both LHS expressions equal 90. This is an example of the substitution property in action.
That's how we end up with [tex]\text{m}\angle 1+\text{m}\angle 2 = \text{m}\angle 1+\text{m}\angle 3[/tex]
After this point, subtract [tex]\text{m}\angle1[/tex] from both sides to cancel them out. We'll be left with [tex]\text{m}\angle 2 = \text{m}\angle 3[/tex], which concludes with [tex]\angle 2 \cong \angle 3[/tex] to show the two angles are congruent.
Body-mass index, or BMI, takes both weight and height into account when assessing whether an individual IS
underweight or overweight. BMI varies directly as one's weight, in pounds, and inversely as the square of one's height,
in inches. In adults, normal values for the BMI are between 20 and 25, inclusive. Values below 20 indicate that an
individual is
underweight and values above 25 indicate that an individual is overweight. A person who weighs 180
pounds and is 5 feet, or 60 inches, tall has a BMI of 35.15. Use the four-step procedure for solving variations to
determine what the BMI is, to the nearest tenth, for a 210-pound person who is 5 feet 9 inches tall. Is this person
underweight, normal, or overweight?
Answers
The person is overwieght as his BMI is 31.00 (approx)
What is Proportion?
In general, proportion refers to a part, share, or quantity assessed in comparison to a total. According to the notion of proportion, two ratios are in proportion when they are equivalent. It is a formula or statement that shows that two ratios or fractions are equivalent.
Solution:
Since, BMI is directly proportionate with weight and indirectly proportionate with square of height, we need to use a constant to make proportion to an equation.
According to the question,
BMI of a person who weighs 180 pounds and is 5 feet tall is 35.15.[tex]BMI = k*weight/height^{2}[/tex]
here, k is any constant value
35.15 = k*180/(60*60)
(35.15 * 3600)/180 = k
k = 703
Calculating BMI of a person weighing 210 pound and is 5 feet 9 inch tall.
First of all we need to convert 5 feet 9 inches into inches
5 feet = 60 inches
so, 5 feet 9 inch means 69 inches
BMI = 210*703/(69*69)
BMI = 31.00 (after rounding off)
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Luke claims two circles are always isometric because the shape never changes. Is he correct?
Group of answer choices
Yes - an isometry preserves shape.
Yes - all circles are similar.
No - the center may be located on a difference coordinate.
No - an isometry preserves both size and shape.
Answers
Luke asserts that since the shape is constant, two circles are always isometric. he is wrong. No, an isometry keeps the size and shape intact.
Given that,
Luke asserts that since the shape is constant, two circles are always isometric.
We have to say is he accurate.
The answer is
No, an isometry keeps the size and shape intact.
Because a shape-preserving transformation (movement) in the plane or in space is called an isometric transformation (or isometry). The isometric transformations include translation, rotation, and combinations thereof, such as the glide, which combines a translation with a reflection.
Therefore, Luke asserts that since the shape is constant, two circles are always isometric. he is wrong. No, an isometry keeps the size and shape intact.
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b) Simplify 18x³y³ + 2x³y
Answers
The GCF: 2x³y
answer: 2x³y2x³y(9y³+1)
please help with the geometry homework
Answers
The triangle RSY is congruent to triangle TSX which is triangle RSY ≅ triangle TSX as side SX ≅ side SY and side XR ≅ side YT.
First, let us understand the SAS congruency theorem:
This one denotes Side - Angle - Side and it means that if 2 sides and one angle of a triangle are equal to the corresponding 2 sides and one angle of another triangle, then they are both congruent.
We are given:
SR and ST are straight lines.
side SX ≅ side SY
side XR ≅ side YT
So,
side SR ≅ side ST
In triangle RSY and triangle TSX;
side SR ≅ side ST (above proved)
∠RSY ≅ ∠TSX (Common)
side SY ≅ side SX (Given)
So, triangle RSY ≅ triangle TSX by SAS congruency theorem.
Thus, the triangle RSY is congruent to triangle TSX which is triangle RSY ≅ triangle TSX as side SX ≅ side SY and side XR ≅ side YT.
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(Simplify your answer.)Find f(g(x)) and g(f(x)) and determine whether the pair of functions f and g are inverses of each other. x-8 f(x) = 4x + 8 and g(x) = 4
Answers
Answer:
x, x, yes==========================
Givenf(x) = 4x + 8,g(x) = (x - 8)/4.To find f(g(x)), g(f(x)),Determine if f and g are inverse functions.Solutionf(g(x)) = 4(x - 8)/4 + 8 = x - 8 + 8 = xg(f(x)) = (4x - 8 + 8)/4 = 4x/4 = xSince both composite functions result in same value of x, they are inverses of each other.19) For which value of s is the value of the expression s + 6 ÷s greatest?
A. 1
B. 2
C. 3
D. 4
Answers
The correct option A. 1, is the value of the variable 's' for which the solution of the expression is greatest .
Define the meaning of the greatest value?whenever one value exceeds another. We employ the "greater than" symbol. as in: 9 > 6.For the given question,
The expression is-
(s + 6) ÷ s
Use the options;
A : s = 1
= (s + 6) ÷ s
= (1 + 6) ÷ 1
= 7
B : s = 2
= (s + 6) ÷ s
= (2 + 6) ÷ 2
= 4
C : s = 3
= (s + 6) ÷ s
= (3 + 6) ÷ 3
= 3
D : s = 4
= (s + 6) ÷ 4
= (4 + 6) ÷ 4
= 2.5
Thus, the value of the variable 's' for which the solution of the expression is greatest is 1.
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Draw a square and it diagonals. Guess the measures of these angles.
Answers
All angles of a square are 90° regardless of the length of its sides. However, all sides must be the same length.
What is a square?A square is a term to refer to a geometric figure that is characterized by the following characteristics:
Has four sidesIt has four 90° anglesAll its sides have the same lengthOne key to identifying a square and the value of its angles is to add them together. The sum of the angles of any geometric figure must result in 360°.
90° + 90° + 90° + 90° = 360°Additionally, if we mark the diagonals of a square we must take into account that these angles will have a different value. To find this value we must divide the angle in half, because these diagonals divide the figure into two parts.
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I will mark brainliest!! help much needed >_<
Find the missing value so that the line passing through the 2 points has the given slope.
(-20, y) and (-11, 15); slope: 0
a. 15
b. -13
c. 11
Answers
The missing value of that makes the line that passes the given points have a slope of 0 is determined as: a. 15.
What is the Slope of a Line?The slope (m) is defined as the rise of a line over its run. The formula for finding the slope (m) of a line is given as:
Slope (m) = change in y / change in x = [tex]\frac{y_2 - y_1}{x_2 - x_1}[/tex].
Given the points (-20, y) and (-11, 15) has a slope (m) of 0. Therefore, let:
(-20, y) = [tex](x_1, y_1)[/tex]
(-11, 15) = [tex](x_2, y_2)[/tex]
m = 0
Substitute the values into the slope formula:
0 = (15 - y) / (-11 -(-20))
0 = (15 - y) / 9
Multiply both sides by 9
0 = 15 - y
0 + y = 15 - y + y
y = 15
The answer is: a. 15.
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Sarah mailed her computer to Store B for repairs. Her bill was:
Initial Repair Cost = $1,350
Sales Tax of 7 percent = $94.50
Total Cost of Computer Repair = $1,444.50
Shipping = 2 percent of Total Cost
What is the cost to ship this computer from Store B?
$
Answers
Answer: $28.89
Step-by-step explanation: 2% of 1,444.50 is $28.89
Isaac works at a deli. He made a 30-cup batch of potato salad, and now he is filling 2.5-cup
to-go containers to sell to customers.
Write an equation that shows how the number of cups of potato salad remaining,y, depends
on the number of to-go containers Isaac has filled, x.
y =
Help
Answers
The equation y = 30 - x shows the number of cups of potato salad remaining, and its solution is y = 27.5.
Isaac made a 30-cup batch of potato salad, and now he is filling 2.5-cup
to-go containers to sell to customers.
Let y represent the number of cups of potato salad remaining,
And x represents the number of to-go containers Isaac has filled,
As per the given situation, we can write the equation as:
y = 30 - x ....(i)
Substitute the value of x = 2.5 in the above equation,
y = 30 - 2.5
y = 27.5
Thus, the equation y = 30 - x shows the number of cups of potato salad remaining, and its solution is y = 27.5.
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A, B, C, D, E, F, G and H form a cuboid. AB = 6.7cm, BC = 2.3cm, and CG = 8.1cm. Find ED rounded to 1DP.
Answers
According to the Pythagoras theorem, the value of ED is 8.4 cm
Pythagoras theorem:
Pythagoras theorem defines that the right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Given,
A, B, C, D, E, F, G and H form a cuboid.
AB = 6.7cm, BC = 2.3cm, and CG = 8.1cm.
Now, we need to find the value of Ed and round off it into 1 decimal place.
Here we have the cuboid with the edges named as A, B, C, D, E, F, G and H.
Now, we have to use the value of BC and CG to find the value of hypotenuse of BG.
Because, BG and ED are opposite to each other.
Which means both have the same hypotenuse value.
So, according to the Pythagoras theorem,
=> BG² = BC² + CG²
=> BG² = (2.3)² + (8.1)²
=> BG² = 5.29 + 65.61
=> BG² = 70.9
So, the value of BG is 8.42.
Therefore, the value of ED is 8.4 When we round off this into one decimal place.
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can someone please help asap !!!
Answers
Answer:
a = -1
b = -9
Step-by-step explanation:
when x = -2 put -2 for x in the equation:
a(3- (-2)) = -2*(-2) + b and then solve
a(3+2) = 4+b
5a = 4+b
then look at the numbers in the box and try for which number this equation will work. when you put a = -1 and b = -9 you will see it is equal on both sides because each side is -5
Evaluate the expression −(4)power2− (−2)power4
Answers
Answer:
-32
Step-by-step explanation:
-(4)^2 - -2^4 = -16 - 16 = -32
Marie ran 842.4 meters in 12
minutes. How many
did she run?
A. 70.2
B. 70
C. 60.2
D. 72
meters/minute
Answers
Answer:
A. 70.2
Step-by-step explanation:
solve your equation step-by-step.
842.412=x1
Step 1: Cross-multiply.
842.412=x1
(842.4)*(1)=x*(12)
842.4=12x
Step 2: Flip the equation.
12x=842.4
Step 3: Divide both sides by 12.
12x12=842.412
x=70.2
Hope this helps! :D
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h - 2 > 10; h = 12
is that true?
Answers
Answer:
no, h=anything greater than 12
Step-by-step explanation
Answer:
No. It is false.
Step-by-step explanation:
h - 2 > 10; h = 12
Substitute 12 in place of h.
12 - 2 >? 10
Simplify.
10 >? 10
10 is NOT > 10
10 is not greater than 10. 10 is equal to 10. If the symbol had the "or equal to" underline under it, then it would be true. See image.
Arnulfo deposited $55 into a savings account. According to the rule of 72, what interest rate will cause his money to double in approximately 23 years?.
Answers
Interest rate will cause his money to double in approximately 23 years is 3.13%.
Given:
Arnulfo deposited $55 into a savings account and interest is in semi annually.
For 23 years the interest payments will be = 23*2
= 46
Interest rate = 2*72 / 46
= 144/46
= 72/23
= 3.13%
Therefore Interest rate will cause his money to double in approximately 23 years is 3.13%.
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HELP!!!!!!! LOTS OF POINTS
Answers
Answer:
1) 1 A. Yes B. Because the change in x and y is constant
2) 2 A. No B. Although the change in y is
constant, the change in x is not
3) 5. A. No B.Because the change in y is not constant
4) 4,5 A. yes B.Because the change in x and constant
Step-by-step explanation:
Someone please help me with this problem, thank uuu
Answers
Answer:
B
Step-by-step explanation:
Nathaniel invested $2,900 in an account paying an interest rate of 5.4% compounded continuously. Assuming no deposits or withdrawals are made, how much money, to the nearest cent, would be in the account after 11 years?
Answers
Answer:
$5252.53============================
GivenInvested amount P = $2900,Annual interest rate r = 5.4% = 0.054,Time t = 11 years,Compound number = continuous.Find the balance after 11 yearsUse equation for continuous compound:
[tex]P(t) = P_0e^{tr}[/tex],where P(t) - final amount, P₀ - initial amount, t - time, r - interest ratePlug in the values and calculate:
[tex]P(11) = 2900e^{11*0.054}=5252.53 \ rounded[/tex]Answer:
$5,252.53 (nearest cent)
Step-by-step explanation:
Continuous Compounding Formula
[tex]\large \text{$ \sf A=Pe^{rt} $}[/tex]
where:
A = Final amount.P = Principal amount.e = Euler's number (constant).r = Annual interest rate (in decimal form).t = Time (in years).Given values:
P = $2,900r = 5.4% = 0.054t = 11 yearsSubstitute the given values into the formula and solve for A:
[tex]\implies \sf A=2900 \cdot e^{(0.054 \cdot 11)}[/tex]
[tex]\implies \sf A=2900 \cdot e^{0.594}[/tex]
[tex]\implies \sf A=2900 \cdot 1.81121882[/tex]
[tex]\implies \sf A=5252.53457...[/tex]
Therefore, assuming no deposits or withdrawals are made, the amount of money in the account after 11 years would be $5,252.53 (nearest cent).
The ratio of green M & M's to yellow is 2:5
a. If there are only green and yellow M & M's in the bag, what is the smallest number of M & M's possible?
b. If there are 84 total green and yellow M & M's in the bag, how many are green?
c. If red M & M's were added to the bag in part b to get a total of 100, what is the ratio of green to yellow to red in simplest form? (enter ratio as green:yellow:red)
:
:
Answers
a. The smallest number of M & M's possible is 14.
b. The number of green is 24.
c. The ratio of green to yellow to red in 6: 15: 25.
How to calculate the value?a. When there are only green and yellow M & M's in the bag, the smallest number of M & M's possible will be:
= (2 × 2) + (2 × 5)
= 4 + 10
= 14
b. If there are 84 total green and yellow M & M's in the bag, the number of green will be:
= 2 / (2 + 5) × 84
= 2/7 × 84
= 24 green
c. If red M & M's were added to the bag in part b to get a total of 100, the ratio of green to yellow to red in simplest form will be:
= Green : Yello : Red
= 24 : 60 : 100
= 6 : 15 : 25
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