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3 years ago

What is the area of the largest rectangle with lower base on the x-axis and upper vertices on the curve y = 27 - x2?

Mathematics

2 answers:

Likurg_2 [28]3 years ago

6 0

Answer:

108 units squared

Step-by-step explanation:

Length is 2x , width is y

area = 2x * y

sub for y

area = 2x(27 - x^2)

area = 54x - 2x^3

1st derivative: 54 - 6x^2

54 - 6x^2 = 0

6(9 - x^2) = 0

6(3 - x)(3 + x) = 0

so, x = 3

solve for y: 27 - 3^2 = 18

area = (3*2)* 18 = 108 units squared

statuscvo [17]3 years ago

5 0

The answer is 54 square units.
let the vertex in quadrant I be (x,y)
then the vertex in quadratnt II is (-x,y) 
base of the rectangle = 2x 
height of the rectangle = y 
Area = xy 
= x(27 - x²) 
= -x³ + 27x 
d(area)/dx = 3x² - 27 = 0 for a maximum of area 
3x² = 3 x 3² = 27 
x² = 9 
x = ±3 
y = 27-9 = 18 
So, the largest area = 3 x 18 = 54 square units

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Which of the following is a true statement?

Nikolay [14]

Answer:

The last choice: 68/5 - 22/5 = 9 1/5

Step-by-step explanation:

Solve each problem:

9 3/7 = 10 3/7

The fractions are the same so look at the whole numbers.

Does 9 equal 10? No, it doesn't so this is a false statement.

332/4 = 1/83

Simplify 332/4:

332/4 = 83/1

83 does not equal 1/83 so this is a false statement.

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Convert the improper fraction into a mixed number:

7 2/5 = 5 2/5

These numbers do not equal each other so this is a false.

68/5 - 22/5 = 9 1/5

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3 years ago

Toby buys a new MP3 player for a price of $45.50. What is the total amount his credit card is

Fittoniya [83]

Answer:

$48.69

Step-by-step explanation:

45.50 + (45.50 * 0.07) = 48.69

hope this helps!

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3 years ago

Find the common difference for the sequence below. 1/4, 5/16, 3/8,..

vredina [299]

Answer:

the common difference of the sequence is $(\frac{1}{16})$

Step-by-step explanation:

A sequence of numbers is said to be in arithmetic progression of each term of the sequence has same common difference.

Now, here first term a1 = 1/4

Second term a2 = 5/16

Third term a3 = 3/8

Now, Common difference (d) = a2 - a1 = a3 - a2

Now, a2 - a1 = $\frac{5}{16} - \frac{1}{4} = \frac{5 - 4}{16} = \frac{1}{16}$

Now, a3 - a2 = $\frac{3}{8} - \frac{5}{16} = \frac{6 - 5}{16} = \frac{1}{16}$

Hence, the common difference of the sequence is $(\frac{1}{16})$

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3 years ago

Which situation would be represented by the integer -8? A hole 8 feet deep. An 8-yard pass. You earn $8. A temperature of 8°F.

frozen [14]

A hole 8 feet deep because it’s down making it negative

5 0

2 years ago

Read 2 more answers

A line tangent to the curve f(x)=1/(2^2x) at the point (a, f(a)) has a slope of -1. What is the x-intercept of this tangent?

kirza4 [7]

Answer:

x-intercept = 0.956

Step-by-step explanation:

You have the function f(x) given by:

$f(x)=\frac{1}{2^{2x}}$ (1)

Furthermore you have that at the point (a,f(a)) the tangent line to that point has a slope of -1.

You first derivative the function f(x):

$\frac{df}{dx}=\frac{d}{dx}[\frac{1}{2^{2x}}]$ (2)

To solve this derivative you use the following derivative formula:

$\frac{d}{dx}b^u=b^ulnb\frac{du}{dx}$

For the derivative in (2) you have that b=2 and u=2x. You use the last expression in (2) and you obtain:

$\frac{d}{dx}[2^{-2x}]=2^{-2x}(ln2)(-2)$

You equal the last result to the value of the slope of the tangent line, because the derivative of a function is also its slope.

$-2(ln2)2^{-2x}=-1$

Next, from the last equation you can calculate the value of "a", by doing x=a. Furhtermore, by applying properties of logarithms you obtain:

$-2(ln2)2^{-2a}=-1 \\\\2^{2a}=2(ln2)=1.386\\\\log_22^{2a}=log_2(1.386)\\\\2a=\frac{log(1.386)}{log(2)}\\\\a=0.235$

With this value you calculate f(a):

$f(a)=\frac{1}{2^{2(0.235)}}=0.721$

Next, you use the general equation of line:

$y-y_o=m(x-x_o)$

for xo = a = 0.235 and yo = f(a) = 0.721:

$y-0.721=(-1)(x-0.235)\\\\y=-x+0.956$

The last is the equation of the tangent line at the point (a,f(a)).

Finally, to find the x-intercept you equal the function y to zero and calculate x:

$0=-x+0.956\\\\x=0.956$

hence, the x-intercept of the tangent line is 0.956

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3 years ago