Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Fred started studying how the number of branches on his tree grows over time.
The relationship between the elapsed time
t, in years, since Fred started studying the tree, and the number of its branches,
B_("year ")(t), is modeled by the following function:

B_("year ")(t)=20*(2.5)^(t)
Complete the following sentence about the monthly rate of change in the number of branches.
Round your answer to two decimal places.
Every month, the number of branches increases by a factor of

Fred started studying how the number of branches on his tree grows over time. $\newline$ The relationship between the elapsed time $t$ , in years, since Fred started studying the tree, and the number of its branches, $B_{\text {year }}(t)$ , is modeled by the following function: $\newline$ $B_{\text {year }}(t)=20 \cdot(2.5)^{t}$ $\newline$ Complete the following sentence about the monthly rate of change in the number of branches. $\newline$ Round your answer to two decimal places. $\newline$ Every month, the number of branches increases by a factor of

View step-by-step help

View step-by-step help

Compound interest: word problems

Full solution

Q. Fred started studying how the number of branches on his tree grows over time.

\newline

The relationship between the elapsed time

t

, in years, since Fred started studying the tree, and the number of its branches,

B_{\text {year }}(t)

, is modeled by the following function:

\newline

B_{\text {year }}(t)=20 \cdot(2.5)^{t}

\newline

Complete the following sentence about the monthly rate of change in the number of branches.

\newline

Round your answer to two decimal places.

\newline

Every month, the number of branches increases by a factor of

Understand function representation: Understand the given function and what it represents. $\newline$ The function $B_{\text{year}}(t)=20\times(2.5)^{t}$ represents the number of branches on the tree after $t$ years. To find the monthly rate of change, we need to express $t$ in months instead of years.
Convert time to months: Convert the time from years to months. $\newline$ Since there are $12$ months in a year, we need to find the equivalent monthly growth factor. Let's denote $m$ as the number of months. Then, $t$ (in years) is equal to $\frac{m}{12}$ (since $m$ months is $\frac{m}{12}$ years).
Rewrite function in terms of months: Rewrite the function in terms of months. $\newline$ The new function in terms of $m$ (months) will be $B_{\text{month}}(m)=20\cdot(2.5)^{\frac{m}{12}}$ .
Calculate monthly growth factor: Calculate the monthly growth factor. $\newline$ We need to find the value of $(2.5)^{\frac{1}{12}}$ to determine by what factor the number of branches increases each month.
Evaluate growth factor: Evaluate $(2.5)^{1/12}$ using a calculator. $\newline$ $(2.5)^{1/12} \approx 1.0912$ (rounded to four decimal places for intermediate calculation).
Round growth factor: Round the monthly growth factor to two decimal places as instructed. $\newline$ The monthly growth factor rounded to two decimal places is approximately $1.09$ .

More problems from Compound interest: word problems

Question

Use the following function rule to find

f(2)

\newline

f(x) = 11 \cdot (9)^x + 8

\newline

f(2) =

Posted 2 months ago

Question

Raymond works for a pharmaceutical company that is testing the effectiveness of a new medication. He gave one of the study participants

400

milligrams of the medication, and he knows the amount remaining in the participant's body will decrease by a factor of

1/3

\newline

Write an exponential equation in the form

y = a(b)^x

that can model the amount of medication,

y

, remaining in the participant's body after

x

\newline

Use whole numbers, decimals, or simplified fractions for the values of

a

b

\newline

y =

\newline

Posted 2 months ago

Question

100

bacteria doubles in size every

60

hours. What will the population be

180

hours from now?

\newline

Posted 2 months ago

Question

Charlotte opened a savings account and deposited

\$800.00

as principal. The account earns

8\%

interest, compounded annually. What is the balance after

10

\newline

Use the formula

A = P(1 + \frac{r}{n})^{nt}

A

is the balance (final amount),

P

is the principal (starting amount),

r

is the interest rate expressed as a decimal,

n

is the number of times per year that the interest is compounded, and

t

is the time in years.

\newline

Round your answer to the nearest cent.

\newline

\$

\newline

Posted 2 months ago

Question

Bob rented a car for a flat fee of `$150` and an additional `$10` per day. Write a linear function that represents the total cost,

f(x)

x

Posted 3 months ago

Question

Sally purchased a laptop for `$800` and has to pay a service fee of `$20` for each day until it is delivered. Write a linear function to represent the total cost,

h(x)

x

Posted 3 months ago

Question

\$1,000

in a savings account with a daily interest rate of

0.1\%

. Write an exponential function to represent the total amount,

s(x)

, in his account after

x

Posted 3 months ago

Question

A=2450+740 s

\newline

The total amount,

A

, in thousands of dollars, of scholarship money given out by a college throughout its history at a time of

s

semesters after the summer of the year

2000

is given by the equation. What was the total amount of scholarship money, in thousands of dollars, given out by the college up to the summer of the year

2000

Posted 1 month ago

Question

T=235+980 d

\newline

T

, in newton-meters, on a diving board with a

100

kilogram weight placed

d

meters from the diving board's fulcrum is given by the equation. How much torque in newton-meters is on the diving board when the weight is placed at the fulcrum?

Posted 2 months ago

Question

R=42-0.7 t

\newline

Quinn returned home one summer's day to find it extremely hot. He turned the air conditioner on, and the room's temperature began decreasing at a constant rate. The given equation shown gives the room's temperature,

R

, in degrees Celsius,

t

minutes after Quinn turned on the air conditioner. What was the room's temperature, in degrees Celsius, when Quinn returned home?

Posted 1 month ago