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1. Vanessa invested $2500 into an account that will increase in value by 3.5% each year. Write an exponential function to model this situation, then find when the account will have $5000?
2. The average price of a movie ticket in 1990 was $4.22. Since then, the price has increased by approximately 3.1% each year. Write an exponential function to model this situation, then find how many years until tickets cost $9.33.
The exponential function that model this situation is [tex]A(t) = 2500(1 + 0.035)^t.[/tex]
The account will have $5000 in 20 years.
What is the exponential function for Vanessa's investment growth?Let A be the amount in the account after t years.
Then, we can model this situation with the function A(t) = 2500(1 + 0.035)^t with the use of compound intererst formula which is [tex]P = A*(1+r)^t[/tex]
To find when the account will have $5000, we can set A(t) = 5000 and solve:
5000 = 2500(1 + 0.035)^t
2 = (1.035)^t
Taking the natural logarithm:
ln(2) = t ln(1.035)
t = ln(2)/ln(1.035)
t = 20.148791684
t = 20 years.
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Answer:
1) 21 years
2) 26 years
Step-by-step explanation:
Question 1To model the account balance of Vanessa's account at t years, we can use an exponential function in the form:
[tex]\large\boxed{A(t) = A_0(1 + r)^t}[/tex]
where:
A(t) is the value of the investment after t years.A₀ is the initial amount of the investment.r is the annual interest rate (as a decimal).t is the time elapsed (in years).Given Vanessa invested $2500 into the account and it will increase in value by 3.5% each year:
A₀ = $2500r = 3.5% = 0.035Substitute these values into the formula to create an equation for A in terms of t:
[tex]A(t) = 2500(1 + 0.035)^t[/tex]
[tex]A(t) = 2500(1.035)^t[/tex]
To find when the account balance will be $5000, set A(t) equal to $5000 and solve for t:
[tex]A(t)=5000[/tex]
[tex]2500(1.035)^t=5000[/tex]
[tex](1.035)^t=\dfrac{5000}{2500}[/tex]
[tex](1.035)^t=2[/tex]
[tex]\ln (1.035)^t=\ln 2[/tex]
[tex]t \ln 1.035=\ln 2[/tex]
[tex]t=\dfrac{\ln 2}{ \ln 1.035}[/tex]
[tex]t=20.1487916...[/tex]
[tex]t=20.15\; \sf years\;(2\;d.p.)[/tex]
Therefore, it will take approximately 20.15 years for Vanessa's account to reach a value of $5000.
Since the interest rate is an annual rate of 3.5%, it means that the interest is applied once per year, at the end of the year. Therefore, we need to round up the number of years to the next whole number.
So Vanessa's account will have $5,000 after 21 years.
Note: After 20 years, the account balance will be $4,974.47. After 21 years, the account balance will be $5,148.58.
[tex]\hrulefill[/tex]
Question 2To model the increase in movie ticket prices over time, we can use an exponential function in the form:
[tex]\large\boxed{P(t) = P_0(1 + r)^t}[/tex]
where:
P(t) is the price of the ticket (in dollars) after t years.P₀ is the initial price of the ticket (in dollars).r is the annual growth rate (as a decimal).t is the time elapsed (in years).Given the initial price of the ticket was $4.22 and the price has increased by 3.1% each year:
P₀ = $4.22r = 3.1% = 0.031Substitute these values into the formula to create an equation for P in terms of t:
[tex]P(t) = 4.22(1 + 0.031)^t[/tex]
[tex]P(t) = 4.22(1.031)^t[/tex]
To find how many years until tickets cost $9.33, we can set P(t) equal to $9.33 and solve for t:
[tex]P(t)=9.33[/tex]
[tex]4.22(1.031)^t=9.33[/tex]
[tex](1.031)^t=\dfrac{9.33}{4.22}[/tex]
[tex]\ln (1.031)^t=\ln \left(\dfrac{9.33}{4.22}\right)[/tex]
[tex]t \ln (1.031)=\ln \left(\dfrac{9.33}{4.22}\right)[/tex]
[tex]t =\dfrac{\ln \left(\dfrac{9.33}{4.22}\right)}{\ln (1.031)}[/tex]
[tex]t=25.9882262...[/tex]
Therefore, it will take approximately 26 years for movie ticket prices to reach $9.33, assuming the annual growth rate remains constant at 3.1%.
A top travels 8 centimeters each time it is spun. if it is spun 7 times what distance does it travel?
If a top travels 8 centimeters each time it is spun and it is spun 7 times, the total distance it travels is 56 centimeters.
How the total distance is determined:The total distance is determined by multiplication of the distance traveled per spin and the number of spins.
Multiplication involves the multiplicand, the multiplier, and the product.
The traveling distance per spun = 8 centimeters
The number of spinning of the top = 7 times
The total distance = 56 centimeters (8 x 7)
Thus, using multiplication, the total distance the top travels after the 7th spin is 56 centimeters.
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The number 1.3 is both a(n) __________ and a(n) __________ number.
The number 1.3 is both a rational and an irrational number.
What is the number 1.3?The number 1.3 is a rational number because it can be expressed as the quotient of two integers, namely 13/10.
The number 1.3 an irrational number because it cannot be expressed as the ratio of two integers, without repeating or terminating decimals, and its decimal representation goes on forever without repeating.
So we can conclude that the number 1.3 is both rational and irrational number.
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determine what type of model bets fits the given situation: A $500 raise in salary each year
The type of model that best fits the situation of a $500 raise in salary each year is a linear model.
In a linear model, the dependent variable changes a constant amount for constant increments of the independent variable.
In the given case, the dependent variable is the salary and the independent variable is the year.
You may build a table to show that for increments of 1 year the increments of the salary is $500:
Year Salary Change in year Change in salary
2010 A - -
2011 A + 500 2011 - 2010 = 1 A + 500 - 500 = 500
2012 A + 1,000 2012 - 2011 = 1 A + 1,000 - (A + 500) = 500
So, you can see that every year the salary increases the same amount ($500).
In general, a linear model is represented by the general equation y = mx + b, where x is the change of y per unit change of x, and b is the initial value (y-intercept).
In this case, m = $500 and b is the starting salary: y = 500x + b.
Teena uses 1/4 cup of oil for a cake. How many cakes can she make if she has 6 cups of oil?
Answer:
24 cakes.
Step-by-step explanation:
6 cups of oil divided by 1/4 cup oil per cake = 24 cakes
6/(1/4) = 24
or 6/(0.25) = 24
She can make 24 cakes with 6 cups of oil.
Out of 1000 students who appeared in an examination,60% passed the examination.60% of the failing students failed in mathematics and 50% of the failing students failed in English.If the students failed in English and Mathematics only, find the number of students who failed in both subjects.
The value of number of students who failed in both mathematics and English is 40.
Since, Given that;
60% of the 1000 students passed the examination,
Hence, we can calculate the number of students who passed the exam as follows:
60/100 x 1000 = 600
So, 600 students passed the examination.
Now, let's find the number of students who failed the examination.
Since 60% of the students passed, the remaining 40% must have failed. Therefore, the number of students who failed the examination is:
40/100 x 1000 = 400
Of the 400 failing students, we know that 60% failed in mathematics.
So, the number of students who failed in mathematics is:
60/100 x 400 = 240
Similarly, we know that 50% of the failing students failed in English.
So, the number of students who failed in English is:
50/100 x 400 = 200
Now, we need to find the number of students who failed in both subjects.
We can use the formula:
Total = A + B - Both
Where A is the number of students who failed in mathematics, B is the number of students who failed in English, and Both is the number of students who failed in both subjects.
Substituting the values we have, we get:
400 = 240 + 200 - Both
Solving for Both, we get:
Both = 240 + 200 - 400
Both = 40
Therefore, the number of students who failed in both mathematics and English is 40.
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Need help asap, please and thank you
If the population in the year 2007 is 111.3 million, then the population in the year 2044 will be 148.37 million.
In order to find the population in the year 2044, we use the population growth formula; which is : P = P₀ × (1 + r)ⁿ;
where P = future population, P₀ = initial population, r = annual growth rate, and n = number of years;
Substituting the values,
We get;
⇒ P = (111.3 million) × (1 + 0.0078)²⁰⁴⁴⁻²⁰⁰⁷;
Simplifying this expression,
We get;
⇒ P = (111.3 million) × (1.0078)³⁷;
⇒ P ≈ 148.37 million;
Therefore, the population in the year 2044 is estimated to be approximately 148.37 million.
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7. Given right triangle ABC below, determine sin(A).
The value of Sin A is 5/13.
Option A is the correct answer.
We have,
Sin A = Perpendicular / Hypotenuse
Sin A = BC / AB
And,
BC = 5
AB = 13
Substituting.
Sin A = 5/13
Thus,
The value of Sin A is 5/13.
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Instructions: Find the missing probability.
P(B)=1/2P(A|B)=11/25P(AandB)=
Please help. Is the answer even there?
The critical values t₀ for a two-sample t-test is ± 2.0.6
To find the critical values t₀ for a two-sample t-test to test the claim that the population means are equal (i.e., µ₁ = µ₂), we need to use the following formula:
t₀ = ± t_(α/2, df)
where t_(α/2, df) is the critical t-value with α/2 area in the right tail and df degrees of freedom.
The degrees of freedom are calculated as:
df = (s₁²/n₁ + s₂²/n₂)² / [(s₁²/n₁)²/(n₁-1) + (s₂²/n₂)²/(n₂-1)]
n₁ = 14, n₂ = 12, X₁ = 6,X₂ = 7, s₁ = 2.5 and s₂ = 2.8
α = 0.05 (two-tailed)
First, we need to calculate the degrees of freedom:
df = (s₁²/n₁ + s₂²/n₂)² / [(s₁²/n₁)²/(n₁-1) + (s₂²/n₂)²/(n₂-1)]
= (2.5²/14 + 2.8²/12)² / [(2.5²/14)²/13 + (2.8²/12)²/11]
= 24.27
Since this is a two-tailed test with α = 0.05, we need to find the t-value with an area of 0.025 in each tail and df = 24.27.
From a t-distribution table, we find:
t_(0.025, 24.27) = 2.0639 (rounded to four decimal places)
Finally, we can calculate the critical values t₀:
t₀ = ± t_(α/2, df) = ± 2.0639
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