The total number of gallons of water in a tank at time t is modeled by the expression A(t)=30+8t-2/3(t+1)^3/2

Answer:

193 gallons

Step-by-step explanation:

Given

[tex]A(t) = 30+8t-\frac{2}{3}(t+1)^{\frac{3}{2}}[/tex]

Required

Determine the maximum amount of water the tank can hold --- Missing from the question

Start by differentiating A w.r.t t

[tex]A'(t) = 0 + 8 + \frac{d}{dt}[-\frac{2}{3}(t+1)^{\frac{3}{2}}][/tex]

Solving: [tex]\frac{d}{dt}[-\frac{2}{3}(t+1)^{\frac{3}{2}}[/tex]

[tex]\frac{d}{dt}[-\frac{2}{3}(t+1)^{\frac{3}{2}} = -\frac{2}{3}\frac{d}{dt}[(t+1)^{\frac{3}{2}}[/tex]

Apply power rule:

[tex]\frac{d}{dt}[-\frac{2}{3}(t+1)^{\frac{3}{2}} = -\frac{2}{3}[\frac{3}{2}(t + 1)^{\frac{3}{2}-1} * \frac{d}{dt}[t+1][/tex]

[tex]\frac{d}{dt}[-\frac{2}{3}(t+1)^{\frac{3}{2}} = -(t + 1)^{\frac{3}{2}-1} * \frac{d}{dt}[t+1][/tex]

[tex]\frac{d}{dt}[-\frac{2}{3}(t+1)^{\frac{3}{2}} = -(t + 1)^{\frac{3}{2}-1} * [1+0][/tex]

[tex]\frac{d}{dt}[-\frac{2}{3}(t+1)^{\frac{3}{2}} = -(t + 1)^{\frac{3}{2}-1} * [1][/tex]

[tex]\frac{d}{dt}[-\frac{2}{3}(t+1)^{\frac{3}{2}} = -(t + 1)^{\frac{3}{2}-1}[/tex]

[tex]\frac{d}{dt}[-\frac{2}{3}(t+1)^{\frac{3}{2}} = -(t + 1)^{\frac{1}{2}}[/tex]

So:

[tex]A'(t) = 0 + 8 + \frac{d}{dt}[-\frac{2}{3}(t+1)^{\frac{3}{2}}][/tex]

A'(t) = 0 + 8 + \frac{d}{dt}[-\frac{2}{3}(t+1)^{\frac{3}{2}}]

[tex]A'(t) = 0 +8 -(t + 1)^{\frac{1}{2}}[/tex]

[tex]A'(t) = 8 -(t + 1)^{\frac{1}{2}}[/tex]

Equate to 0 to solve for t

[tex]A'(t) = 0[/tex]

[tex]8 -(t + 1)^{\frac{1}{2}} = 0[/tex]

Collect Like Term

[tex]-(t + 1)^{\frac{1}{2}} = -8[/tex]

Square both sides

[tex](-(t + 1)^{\frac{1}{2}})^2 = (-8)^2[/tex]

[tex]t +1 = 64[/tex]

Make t the subject:

[tex]t = 64 - 1[/tex]

[tex]t = 63[/tex]

So, the tank is at maximum when t = 63.

Substitute 63 for t in: [tex]A(t) = 30+8t-\frac{2}{3}(t+1)^{\frac{3}{2}}[/tex]

[tex]A(63) = 30+8*63-\frac{2}{3}(63+1)^{\frac{3}{2}}[/tex]

[tex]A(63) = 30+8*63-\frac{2}{3}(64)^{\frac{3}{2}}[/tex]

[tex]A(63) = 30+8*63-\frac{2}{3}*512}[/tex]

[tex]A(63) = 30+8*63-\frac{2*512}{3}}[/tex]

[tex]A(63) = 30+8*63-\frac{1024}{3}}[/tex]

[tex]A(63) = 30+8*63-341.33[/tex]

[tex]A(63) = 192.67[/tex]

Approximate:

[tex]A(63) = 193[/tex]

(SAT PREP) Find the value of x.

(I have several, please help!!)

Find all solutions to (x+5)(2x-3). Explain or show your reasoning.

Select a number shown by the model . Check all that apply .

Round to the tens place to estimate the answer.

172 x 18 =

what is 3 1/5 minus 2 1/4 equal

4b5+4b3+16b2

Please factor this

-20x(-5)-50 evaluate ​

In order to get to work, Lindsay takes access roads that require a toll for drivers to use. She pays $4 each day to use the access roads. She also pays $5 each day to park her car at the company lot. Assuming the toll payments are used to plot the x-coordinate and the parking lot payments are used to plot the y-coordinate, which ordered pair represents how much money Lindsay spends during a 5-day work week?

A.
(
4
,
5
)
B.
(
12
,
15
)
C.
(
15
,
20
)
D.
(
20
,
25
)

Kwame fills a bird feeder with seeds. As
birds visit the feeder, the height of the
seeds changes from 13.6 cm to 11.1 cm
over a period of 6 1/4 hours. What is the
average change in the height of the seeds
each hour?

A piece of cake has 347 calories in it.How many calories are there in 8 piece of cake.​

I would love it if you could help with this! Thank you so much! Have a good day!

a quotient that is between 8÷0.001 and 8÷110.

!!!I’ll mark brainlest and it’s worth 30 points, it’s for a math test and i’m really confused.!!!

If angle 1 = 55°, what is angle 5?
A. 55°
B. 40°
C. 35°
D. 25°

thanks ;)

Sally has four continents 2 dime 6 nickels and 10 pennies can you please help me​

Could someone help me with this? It would be greatly appreciated! (Photo Attached)

Zoe buys a new car that is valued at $24,000. The car will depreciate at
a rate of 1% per month. What is the value of the car after 6 months?

PLEASE HELP ME GIVE ME THE ANSWER AND TELL MME STEP BY STEP HOW U GOT IT ILL MARK U BRAINLIEST

Write in Expanded Notation 45,231

TIME REMAINING
59:50
The table shows the mean number of basketball goals made by four random samples of players from the school team
during this year's season.
Sample #
Sample Mean
Number of Goals
7
1
4
2
3
5
4
8
Is a valid prediction for the mean of the population possible using these samples?
O No, there are not enough samples.
Yes, the sample means are all less than 10.
Yes, the variation of the sample means is small.
O No, the variation of the sample means is too great.
Mark this and retum
Save and Exit
Next
Submit

PLZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ HELP I'LL GIVE BRAINLEST

A circle has a diameter of 28 mm. What is its circumference?

Use 3.14 for 1, and do not round your answer. Be sure to include the correct unit in your answer.

Can 12m , 16m ,20m firm a right angle

Solve 2(x + 1) = 2x + 5. (1 point)

What is the volume of the sphere with the radius of 9 and round to the hundredths​

Which is the approximate measure of this angle?

If f(x) = 3x - 2 then what is f(10)/​

Shawna bought 2 notebooks that cost $2.19 each. She paid for them with a
$5 bill. Which number sentence can be used to find how much money
Shawna received in change?
A. (2 x 5.00) + 2.19 =
B.
5.00 + (2 x 2.19) =
C.
(2 X 5.00)
2.19
OD 5.00
(2 x 2.19)

Last month Albert earned half of what Betty earned, while Betty earned $5000 more than what Calvin earned. If Donna earned twice what Calvin earned and Albert and Betty together earned $900 more than what Calvin and Donna earned together, how much did each person earn?

Solve for x.
71°
X°
42°

Find the value of x
PIC IS BELOW