MATHEMATICS HIGH SCHOOL

The hypotenuse AB of a right triangle ABC is 5 ft, and one leg, AC, is decreasing at the rate of 2 ft/sec. The rate, in square feet per second, at which the area is changing when AC = 3 is?

Answers

Answer 1

Answer:

Answer: $-(7)/(4) \quad \text{ft}^(2)/\sec$

Step-by-step explanation:

Since ABC is a right triangle, at any moment it holds that

$5^2=(AC)^2+(BC)^2$

Moreover, the area A of the triangle is given by

$A= (1)/(2)(AC)(BC)$

and we know that the rate of change of the length (AC) is

constant decreasing 2, which may be written using the Leibniz

notation as

$(d(AC))/(dt)=-2$ .

Using the chain rule and the product rule for derivation, the two

first equations tell us

that

$0 = 2 (d(AC))/(dt)(AC) + 2 (d(BC))/(dt)(BC)$

and

$(dA)/(dt) = (1)/(2) \left( (d(AC))/(dt) \cdot (BC) + (d(BC))/(dt) \cdot (AC)\right)$

Moreover, using the first of the last two equations we get

$(AC)(d(AC))/(dt) = -(BC)(d(BC))/(dt) \Rightarrow\n\n\n(AC)(-2) = -(BC) (d(BC))/(dt) \quad \Rightarrow \quad (d(BC))/(dt)=2 ((AC))/((BC))$

Now, when (AC)=3, we have that

$25=(AC)^2 + (BC)^2 \quad \Rightarrow 25 = 9 + (BC)^2\n \n\Rightarrow \quad 16=(BC)^2 \quad \Rightarrow (BC)=4$

and

$(d(BC))/(dt) = 2 ((AC))/((BC))=2 (3)/(4)=(3)/(2)$ .

Hence, at this moment the rate of change of the area of the triangle is

$(dA)/(dt) = (1)/(2) \left( (d(AC))/(dt) \cdot (BC) + (d(BC))/(dt) \cdot(AC) \right)=(1)/(2)\left( -2 \cdot 4 + (3)/(2)\cdot 3\right ) = -(7)/(4)$

Answer 2

Answer: okay
we have the rate of change of AC = d(AC)/dt = -2
the rate of change od BC = d(BC)/dt
area = (1/2) *AC) (BC)
taking differential on both sides we ge
d(A)/dt = 1/2){ (BC) d(AC)/dt + (AC) d(BC)/dt)}....(1)
again
when AC= 3
applying pythagorous thm
we get
(5)^2 =(3)^2 +(BC)^2
hence we get BC = 4
now we need to find d(BC)/dt
we have
(5)^2 = (AC)^2 +(BC)^2
taking differenial
0=2(AC) d(AC/dt) +2BC d(BC)/dt
that is
d(BC)/dt = -(3) *(-2)/4 ..(at AC =3)
hence
d(BC)/dt = 3/2
substituting these values in equation (1)
d(A)/dt = (1/2) {4 * -2 + 3 *3/2}

which gives
d(A)/dt = -7/4

The rate, in square feet per second, at which the area is changing when AC = 3 is -7/4 ft/sec.

I hope my answer has come to your help. Thank you for posting your question here in Brainly.

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1.) Solve 10(3x – 5) = 40*

Answers

Answer:

$x = 3$

Step-by-step explanation:

$10(3x - 5) = 40 \n first \: you \: need \: to \: distribute \n \n 30x - 50 = 40 \n add \: 50 \: from \: both \: sides \n \n 30x = 90 \n divide \: both \: sides \: by \: 30 \n \n x = 3$

Any help is appreciated I will mark you as Brinley

Answers

Answer:

1,3

Step-by-step explanation:

What is f(x) = 7x2 + 42x written in vertex form?

Answers

Answer:

$\boxed{\boxed{y = 7(x+3)^2-63}}$

Step-by-step explanation:

The general vertex form of parabola is,

$y=a(x-h)+k$

where,

(h, k) is the vertex of the parabola.

The given function is,

$\Rightarrow f(x) = 7x^2 + 42x$

$\Rightarrow y = 7x^2 + 42x$

$\Rightarrow y = 7(x^2 + 6x)$

$\Rightarrow y = 7(x^2 + 2\cdot x\cdot 3)$

$\Rightarrow y = 7(x^2 + 2\cdot x\cdot 3+3^3-3^2)$

$\Rightarrow y = 7(x^2 + 2\cdot x\cdot 3+3^3)-7\cdot 3^2$

$\Rightarrow y = 7(x+3)^2-7\cdot 9$

$\Rightarrow y = 7(x+3)^2-63$

This is vertex form the given function.

Hello,

y=7x²+42x
==>y=7(x²+2*3*x+9)-63
==>y=7(x+3)²-63

Picture is included please help

Answers

Answer:

B cannot be factored into a perfect square

Determine whether the sequence converges or diverges. if it converges give the limit. 48, 12, 3, 3/4, ...

Answers

The sequence converges.
The limit of the sequence is 0.

-------------------------------------------------------------

This is a geometric sequence with the first term a = 48 and r = 1/4 = 0.25 as the common ratio. Since |r| < 1, this means that the terms are steadily getting smaller and smaller. The terms will approach 0 but not actually get there.

Note: if you added up all the terms of this infinite sequence, then the sum would be S = a/(1-r) = 48/(1-0.25) = 64

Select the coordinates of two points on the line y = 8. (2 points) (8, 8) and (8, 0) (8, −8) and (8, 8) (0, 8) and (8, 0) (0, 8) and (8, 8)

Answers

(8, 8) and (0, 8). If the equation is y = any number, the second coordinate will always be that number. :)

I took the test and got that one wrong, the true answer is 0,8 and 8,8. It was just backwards. Your answer was just needing to be flipped, first last and last first. to get 0,8 and 8,8.

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