Final answer:
The exponential function that goes through the points (0,6) and (2,96) is y = 6 * 4^x, where 6 is the initial value (a) and 4 is the base growth rate (b).
Explanation:
The student is asking for an exponential function in the form y = ab^x that passes through the points (0,6) and (2,96).
To find this function, we must determine the values of 'a' and 'b'. Since the function passes through the point (0,6), if we substitute x = 0, we get y = a * b^0. Since any number raised to the power of 0 is 1, it simplifies to y = a. Therefore, a = 6.
Next, we substitute the second point (2,96) into the original equation with our known value for 'a': 96 = 6 * b^2. To solve for 'b', we divide both sides of the equation by 6, resulting in 16 = b^2. Finding the positive square root of 16 gives us b = 4.
Therefore, the exponential function is y = 6 * 4^x.
The exponential function that goes through the points [tex]\((0, 6)\)[/tex] and [tex]\((2, 96)\)[/tex] is [tex]y = 6 \times 4^x[/tex]
To find the exponential function in the form [tex]\( y = ab^x \)[/tex] that goes through the points [tex]\((0, 6)\)[/tex] and [tex]\((2, 96)\)[/tex], we need to determine the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex].
1. Use the point [tex]\((0, 6)\)[/tex]
When [tex]\( x = 0 \), \( y = 6 \).[/tex]
[tex]\[6 = ab^0\][/tex]
Since [tex]\( b^0 = 1 \)[/tex]
[tex]\[6 = a \times 1\][/tex]
Therefore, [tex]\( a = 6 \).[/tex]
2. Use the point [tex]\((2, 96)\)[/tex]
When [tex]\( x = 2 \), \( y = 96 \)[/tex]
[tex]\[96 = 6b^2\][/tex]
Solving for [tex]\( b^2 \)[/tex]
[tex]\[b^2 = \frac{96}{6} = 16\][/tex]
Taking the square root of both sides:
[tex]\[b = 4\][/tex]
Thus, the exponential function is:
[tex]\[y = 6 \times 4^x\][/tex]
To verify, we can check both points:
For [tex]\((0, 6)\)[/tex]
[tex]\[y = 6 \times 4^0 = 6 \times 1 = 6\][/tex]
For [tex]\((2, 96)\)[/tex]
[tex]\[y = 6 \times 4^2 = 6 \times 16 = 96\][/tex]
Triangle A B C has centroid G. Lines are drawn from each point through the centroid to the midpoint of the opposite side to form line segments A E, B F, and C D. The length of line segment A G is 2 x + 10, the length of line segment F G is 2 x minus 1, and the length of line segment G B is 3 x + 6.
G is the centroid of triangle ABC.
What is the length of AE?
units
Answer:
The length of AE is 39 units.
Step-by-step explanation:
Given:
G is the centroid of a triangle ABC.
Lines are drawn from each point through the centroid to the midpoint of the opposite side to form line segments AE, BF, and CD.
The length of line segment A G is 2 x + 10, the length of line segment F G is 2 x minus 1, and the length of line segment G B is 3 x + 6
To Find: Length of the AE.
The centroid of the triangle is always cut the line in the ratio of 2:1. For clarification, the image is provided.
Hence,
G divides the line BF into a 2:1 ratio.
Thus,
[tex]\begin{aligned}&2FG=BG\\&2(2x-1)=3x+6\\&4x-2=3x+6\\&x=8 \end{aligned}[/tex]
Also,
G divides the line AE into a 2:1 ratio.
Therefore,
[tex]AG=2GE\\GE=AG/2[/tex]
Line segment AE is the sum of AG and GE.
Thus,
[tex]AE=AG+GE\\AE=AG+AG/2\\AE=3AG/2\\AE=3(2x+10)/2\\AE=39[/tex]
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The length of AE is 9 units.
Given that G is the centroid, it divides each median into two segments with the ratio of 2:1, where the longer segment is toward the vertex and the shorter segment is toward the midpoint of the opposite side.
Let's denote the length of AG as a, the length of BG as b, and the length of FG as c. According to the given information:
1. AG = 2x + 10
2. FG = 2x - 1
3. BG = 3x + 6
We know that AG = 2GE and BG = 2GE. Thus:
[tex]\[ AG = 2GE \implies GE = \frac{AG}{2} = \frac{2x + 10}{2} = x + 5 \]\[ BG = 2GE \implies GE = \frac{BG}{2} = \frac{3x + 6}{2} = \frac{3}{2}x + 3 \][/tex]
Since GE is common, we can equate the expressions:
[tex]\[ x + 5 = \frac{3}{2}x + 3 \][/tex]
Solving for x:
[tex]\[ \frac{1}{2}x = 2 \][/tex]
x = 4
Now that we've found the value of x, we can substitute it back into the expressions for AG and FG to find their lengths:
AG = 2x + 10 = 2(4) + 10 = 18
FG = 2x - 1 = 2(4) - 1 = 7
Now, we know that AE = AG - GE:
AE = AG - GE = 18 - (x + 5) = 18 - (4 + 5) = 18 - 9 = 9
So, the length of AE is 9 units.
What is the length of the arc with a measure of 100 degrees in a circle of radius 15 inches
Answer:
30
Step-by-step explanation:
Find the area.
8 ft
7 ft
14 ft
Answer:
784
Step-by-step explanation:
I’m pretty sure I did this right I just multiplied them
What is the value of x ?
A : 15
B : 30
C: 45
D : 60
Answer:
A
Step-by-step explanation:
So that is a 90 degree angle and there is an angle of 30. 30-90 is 60. 60 divided by 4 is 15. So thus the answer is A,15.
Answer:
15
Step-by-step explanation:
90 is right angle
90 - 30 = 60
60 / 4 = 15 <---- theres the answer
hope i helped
What is the stretch or shrink of the graph g(x) = 2 |x + 3| − 5
It's a stretch by coefficient of two.
Hope this helps.
Philip ran 4 3/8 miles yesterday. Michael ran 1 5/8 miles yesterday.
How much farther did Philip run than Michael. fractions .with common denominators.
Philip ran 3 1/4 miles farther than Michael. We obtained this by subtracting the fraction and whole number of Michael's distance from Philip's and simplified the resultant fraction.
Explanation:To find out how much farther Philip ran than Michael, you need to subtract the distance run by Michael from Philip's distance. Since we are dealing with fractions with common denominators, this makes the subtraction process easier.
Philip's distance = 4 3/8 miles
Michael's distance = 1 5/8 miles
Subtract Michael's distance from Philip's distance: 4 3/8 - 1 5/8
Since the denominators are the same, you can simply subtract the numerators: 8 (the common denominator) stays the denominator, and subtract 3 from 5 which gives: 2/8
We subtract the whole numbers: 4 - 1 = 3. Therefore, the final difference is 3 2/8 miles. However, you can simplify 2/8 fraction to 1/4. So, the final and simplest difference is 3 1/4 miles.
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To find out how much farther Philip ran than Michael, we need to subtract the fractions representing their respective distances. After converting these to improper fractions, we subtract the numerators, giving us a result of 2 6/8 miles, which is the distance by which Philip ran farther than Michael.
Explanation:The problem involves subtracting fractions to find the difference between the distances that Philip and Michael ran. First, convert the mixed fractions into improper fractions. Philip ran 4 3/8 miles which equals 35/8 miles and Michael ran 1 5/8 miles which equals 13/8 miles. Subtracting fractions with like denominators is straightforward, so subtract the numerators: 35 - 13 equals 22. Therefore, Philip ran 22/8 or 2 6/8 miles more than Michael.
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whats 12-7-8-9+66=??????
Hey there!
12 - 7 - 8 - 9 + 66 = ?
12 - 7 = 5
5 - 7 - 8 - 9 + 66
5 - 8 = -3
-3 - 9 + 66
-3 - 9 = -12
-12 + 66 = 54
Answer: 54
Good luck on your assignment and enjoy your day!
~LoveYourselfFirst:)
Luis made a $150 bank deposit using $5 bills and $10 bills. He gave the teller a total of 24 bills. How many $10 bills were in the deposit?
Answer:
100 10's
Step-by-step explanation:
Answer: 6 10$ bills
Step-by-step explanation:
18 (5$) times 6 (10$) = $150
18 + 6 = 24 bills in total ^
Tony has $10 to buy apples and bananas for a fruit salad. Apples cost $2 per pound and bananas cost $1 per pound. Write and graph an inequality to describe the situation. Then give two possible combinations of pounds of apples and bananas that Tony can buy. HELP
The equation of inequality would be = 2 x + y ≤ 10 for the process of purchase.
Explanation:
Total amount of money tony have , T = $10
Thus the cost of apple, (We can represent as Ca) = $2
Also the cost of banana,(We can represent as Cb) = $1
Let x be the number of apples bought
Let y be the number of bananas bought
Total expenditure to buy fruits, E = $2 x + $1 y
So,The equation of inequality would be = 2 x + y ≤ 10 for the process of purchase.
A sequence is defined by the recursive function f(n + 1) =1/3 f(n). If f(3) = 9 , what is f(1) ?
1
3
27
81
Solution:
Given that,
A sequence is defined by the recursive function:
[tex]f(n+1) = \frac{1}{3} \times f(n)[/tex] ------ eqn 1
Given that,
[tex]f(3) = 9[/tex]
Substitute n = 2 in eqn 1
[tex]f(2 + 1) = \frac{1}{3} \times f(2)\\\\f(3) = \frac{1}{3} \times f(2)\\\\Substitute\ f(3) = 9\\\\9 = \frac{1}{3} \times f(2)\\\\f(2) = 27[/tex]
Substitute n = 1 in eqn 1
[tex]f(1 + 1) = \frac{1}{3} \times f(1)\\\\f(2) = \frac{1}{3} \times f(1)\\\\Substitute\ f(2) = 27\\\\27 = \frac{1}{3} \times f(1)\\\\f(1) = 27 \times 3\\\\f(1) = 81[/tex]
Thus f(1) is 81
Answer:
D 81
Step-by-step explanation:
The length of a rectangle is 4 more than 3 times the width. If the perimeter of the rectangle is 18.4 cm, what is the area of the rectangle?
Area of the rectangle is 10.27 cm²
Step-by-step explanation:
Step 1: Let the width of the rectangle be x, then length = 4 + 3x. Find the dimensions if perimeter = 18.4 cmPerimeter = 2(length + width)
⇒ 18.4 = 2 (x + 4 + 3x)
⇒ 18.4 = 2(4x + 4)
⇒ 18.4 = 8x + 8
⇒ 8x = 10.4
⇒ x = 10.4/8 = 1.3
∴ 4 + 3x = 4 + 3×1.3 = 7.9
Step 2: Find area of the rectangle.Length = 7.9 cm and width = 1.3 cm
⇒ Area = length × width
= 7.9 × 1.3 = 10.27 cm²
what is 2b = 20, what does the b equal
Answer:
b = 10
Step-by-step explanation:
Divide 20 by 2 to get b by itself:
20/2 = b
10 = b
MARK AS BRAINLIEST!
Answer:
B=10
Step-by-step explanation:
2(b)=20
2(10)=20
Or
2b÷2=20÷2
B=10
What must be the length of ZY in order for ZY to be tangent
to circle X at point Y?
14 units
15 units
16 units
17 units
Answer:
15 units
Step-by-step explanation:
The picture of the question in the attached figure
we know that
Triangle XYZ is a right triangle, because segment ZY is tangent to circle X at point Y
so
Applying the Pythagorean Theorem
[tex]XZ^2=XY^2+ZY^2[/tex]
we have
[tex]XZ=8+9=17\ units\\XY=8\ units[/tex]
substitute the given values
[tex]17^2=8^2+ZY^2[/tex]
[tex]ZY^2=289-64\\ZY^2=225\\ZY=15\ units[/tex]
Answer:
15
Step-by-step explanation:
Gradpoint-What is the value of y?
25
35
50
100
Answer:
35
Step-by-step explanation:
60 x 70 ÷ 50 yyyyyygsgsgsvsvshsh
If i have 6 bottles of water and each weigh 3/5 pounds how much do i have
You have 3 and 3/5 pounds of water.
To find this, simply multiply 3/5 by 6/1 (multiplying straight across). 6 times 3 is 18 and 5 times 1 is 5. Thus meaning you have 18/5 pounds, which simplified is 3 and 3/5 pounds.
I hope this helps!
Answer:
if you multiply 3/5 ×6 that would be 3.6 or in mix number that would be 3⅗
solve for k
10/3= k/9
Answer:
k = 30
Step-by-step explanation:
[tex]\frac{10}{3}=\frac{k}{9}\\\\\frac{k}{9}=\frac{10}{3}\\\\k=\frac{10}{3}*9\\\\k=10*3=30[/tex]
Answer:
k = 30
Step-by-step explanation:
[tex]\frac{10}{3}[/tex] = [tex]\frac{k}{9}[/tex]
We can easily solve this by cross multiplying. Multiply 10 by 9, and multiply the variable (k) by 3.
10 x 9 = 90
3 x k = 3k
These two (90 and 3k) are equal to one another and thus can both be set up into an equation.
3k = 90
Solve for k.
3k/3 = 90/3
k = 30
What is the quotient in simplest form? 8/3 divided by 2/3
Answer:
[tex] \frac{8}{3} \div \frac{2}{3} \\ \frac{8}{3} \times \frac{3}{2} = \frac{4}{1} \\ = 4[/tex]
Answer is 4
Answer: 4
Step-by-step explanation:
8/3 ÷ 2/3 = 4
In Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction.
The first step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 2/3 is 3/2.
8/3 × 3/2
Next, multiply the two numerators. Then, multiply the two denominators. In this step, we first cancel by a common factor of 3, which then gives 8/2, then cancel by another common factor of 2, which gives 4.
Thus 8/3 x 3/2 = 4
I hope this helps.
What is the change that occurs to the parent function f(x) = x^2 given the function f(x) = (x + 1)^2 - 5.
Solve proportion S/84=3/21
Answer:
S=21
Step-by-step explanation:
First cross multiply, next divide. I hope this helped!
Answer: S = 12
Step-by-step explanation:
S/84 =3/21
Cross multiply to find the value of S
21 x S= 84 x 3
21S= 252
S= 252/21
S= 12
One yard crew works four times faster than the other yard crew. If they work together they can finish a job in 60 minutes. How fast each yard crew works?
Time taken by both crew is 48 min and 12 min , when they work together !
Step-by-step explanation:
Here we have , One yard crew works four times faster than the other yard crew. If they work together they can finish a job in 60 minutes. We need to find How fast each yard crew works . Let's find out:
One yard crew works four times faster than the other yard crew
Let speed of one crew be x , So speed of other crew is four times i.e.
⇒ [tex]4x[/tex]
If they work together they can finish a job in 60 minutes.
According to above statement , both can finish work in 60 min i.e.
⇒ [tex]4x+x=60[/tex]
⇒ [tex]5x=60[/tex]
⇒ [tex]x=12[/tex]
So , time taken by other crew is 4x = 4(12) = 48 min . Therefore , Time taken by both crew is 48 min and 12 min , when they work together !
The second yard crew works for 48 minutes.
The first yard crew works for 12 minutes which is four times faster that the second crew.
Step-by-step explanation:
Let us assume,
The second yard crew completes the work in 'x' minutes.
It is given that,
The first yard crew works four times faster than the other yard crew. Therefore, first yard crew will complete the work in 'x/4' minutes.If they work together they can finish a job in 60 minutes.
⇒ x + x/4 = 60 minutes.
⇒ 5x / 4 = 60
⇒ x = 240/5
⇒ x = 48 minutes.
Therefore, the second yard crew works for 48 minutes.
To find the first yard crew :
We already know that, they are 4 times faster that the second yard crew.
Substitute x = 48 in x/4,
⇒ 48 ÷ by 4
⇒ 12 minutes.
Therefore, the first yard crew works for 12 minutes which is four times faster that the second crew.
What is the measure of a straight angle?
A less than 90°
Bgreater than 90°
C 90°
D 180°
Answer:
D 180°
Step-by-step explanation:
The measure of a straight angle is 180°.
The probability of NOT drawing a red marble out of a bag containing 3 blue, 2 green , 8 yellow , 4 red , and 3 orange marbles.
Answer:
[tex] \frac{4}{5} [/tex]
Step-by-step explanation:
Total number of marbles
= 3+2+8+4+3
= 20 marbles
P(not red)
=
[tex] \frac{3 + 2 + 8 + 3}{20} \\ = \frac{16}{20} \\ = \frac{4}{5} [/tex]
or P(not red)
=
[tex]1 - \frac{4}{20} \\ = 1 - \frac{1}{5} \\ = \frac{4}{5} [/tex]
A movie theater sells small and large boxes of popcorn.
• A small box of popcorn costs $1.50.
• A large box of popcorn costs $4.00.
• A total of 25 boxes of popcorn were sold totaling $75.00.
How many large boxes of popcorn were sold?
A 6
B. 10
C.
12
D.
15
There 15 large boxes of popcorn were sold. The option D is correct.
We need to determine the number of large boxes of popcorn sold. Let's denote the number of small boxes as s and the number of large boxes as l. We are given the following information:
The cost of a small box of popcorn is $1.50.
The cost of a large box of popcorn is $4.00.
A total of 25 boxes of popcorn were sold.
The total revenue from popcorn sales was $75.00.
We can set up two equations based on this information:
s + l = 25 (the total number of boxes sold)
1.50s + 4.00l = 75.00 (total revenue from sales)
Let's solve for l using these equations. Multiplying the first equation by 1.50 gives us:
1.50s + 1.50l = 37.50
Subtracting this from the second equation:
(1.50s + 4.00l) - (1.50s + 1.50l) = 75.00 - 37.50
2.50l = 37.50
l = 37.50 / 2.50
l = 15
The sum of three consecutive even numbers is 522.find the number
Answer: 172, 174 and 176
Step-by-step explanation:
Since they are even numbers,
Let x be the number, x + 2 be the second number and x +4 the third number
can think of this as x+(x+2)+(x+4)=522
so you get 3x=522-6
3x= 516
so x=172
If x is 172, second number is 174 and third number is 176
2x+15=145
Solve for x
Answer:
Step-by-step explanation:2x=145-15
2x=130
X=130/2
X=65
A 12inch diameter pizza has 22 calories per square inch. About how many calories are in the entire pizza? Use 3.14 for pi. Select from the drop-down menu to correctly complete the statement
Step-by-step explanation:
Diameter of pizza = 12 inch
Therefore, radius r = 6 inch
[tex]Area \: of \: entire \: pizza \\ = \pi {r}^{2} \\ = 3.14 \times {6}^{2} \\ = 3.14 \times 36 \\ = 113.04 \: {inch}^{2} \\ \\ amount \: of \: calories \: in \: entire \: pizza \\ = area \: of \: pizza \\ \times calories \: per \: square \: inch \\ = 113.04 \times 22 \\ = 2,486.88 \\ [/tex]
I need help!!! please help me out as quick as possible
Answer:
D) 262,144
Step-by-step explanation:
The given sequence is given explicitly as:
[tex]A_n = {4}^{n - 1} [/tex]
We want to find the tenth term of this sequence.
We substitute n=10 to get:
[tex]A_ {10}= {4}^{10 - 1} [/tex]
Simplify the exponent
[tex]A_ {10}= {4}^{9} [/tex]
Simplify the power
[tex]A_ {10}= 262144[/tex]
Therefore the 10th term is 262,144
Which statement is correct?
A kilometer is 100 times the length of a meter.
A kilometer is 1,000 times the length of a meter.
A centimeter is 100 times the length of a meter.
A meter is 10 times the length of a centimeter.
Answer:
A centimeter is 100 times the length of a meter.
A kilometer is 1,000 times the length of a meter.
The statement that asserts a true claim would be:
B). A kilometer is [tex]1,000[/tex] times the length of a meter.
Metric System
The metric system is a measurement system that employs the meter, liter, and gram as base units of length (distance), capacity (volume), and weight (mass).Kilometers are [tex]1,000[/tex] meters. In the metric system, the meter is the basic unit for measuring length or distance.[tex]1[/tex] kilometer [tex]=1000[/tex] metersSo, a kilometer is [tex]\boldsymbol{1,000}[/tex] times the length of a meter.Find out more information about "mathematical units" here:
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solve for x. x + 1/2 = 3/4
X=_______________________
Answer:
[tex]x=\frac{1}{4}[/tex]
Step-by-step explanation:
In order to get the answer to this question you will have to subtract 1/2 on both sides:
[tex]x+\frac{1}{2} =\frac{3}{4}[/tex]
Subtract 1/2 on both sides:
[tex]-\frac{1}{2} -\frac{1}{2}[/tex]
Subtract 1/2 from 3/4:
[tex]\frac{3}{4} -\frac{1}{2}[/tex]
Find the common denominator:
[tex]CD=4[/tex]
[tex]2\times2=4[/tex]
[tex]1\times2=2[/tex]
[tex]=\frac{2}{4}[/tex]
Do the same with the second fraction:
[tex]4\times1=4[/tex]
[tex]3\times1=3[/tex]
[tex]=\frac{3}{4}[/tex]
Subtract the numerators:
[tex]3-2=1[/tex]
Keep the denominator the same:
[tex]=\frac{1}{4}[/tex]
[tex]x=\frac{1}{4}[/tex]
Hope this helps.
Angles 1 and 3 are directly across from each other.
They are called vertical angles. They share only a
vertex. They always measure the same. Therefore,
if angle 1 measures 27 degrees, angle 3 will
measure
degrees.
The measure of angle 3 is 27 degrees.
Step-by-step explanation:
The angles 1 and 3 are vertical angles and share same vertex.Therefore, the measure of the angle 1 and angle 3 are equal.Given,
angle 1 = 27°
Since angle 3 is same as angle 1, the measure of angle 3 is also 27 degrees
angle 3 = 27°.