The altitude of a triangle is increasing at a rate of

2
centimeters/minute while the area of the triangle is increasing at a rate of
1.5
square centimeters/minute. At what rate is the base of the triangle changing when the altitude is
11.5
centimeters and the area is
91
square centimeters?

im a little confused on the formula so the answer is always wrong

Area of triangle is (1/2) base * height

given: dh/dt = 2 , dA/dt = 1.5,
find: db/dt when A = 91 and h = 11.5 = 23/2
(skipping the units, since that will all work out)

at A = 91 and h = 11.5
91 = (1/2)(b)(11.5)
b = 364/23 = appr 15.826

A = (1/2)bh
dA/dt = (1/2)b dh/dt + (1/2)h db/dt
1.5 = (1/2)(364/23)(2) + (1/2)(23/2) db/dt
3/2 = 364/23 + 23/4 db/dt
23/4 db/dt = -659/46
db/dt = - 1318/526 = appr - 2.49 cm/sec

the base is decreasing at appr 2.5 cm/s

dA = 1.5 cm² / min

dh = 2 cm / min

h = 11.5 cm

A = 91 cm²

Area of a triangle:

A = b ∙ h / 2

91 = b ∙ 11.5 / 2

Multiply both sides by 2

182 = 11.5 b

b = 182 / 11.5

Differentiate equation for area of a triangle to find rate of change of the area of a triangle ( dA ):

A = ( 1 / 2) b ∙ h

dA / dt = ( 1 / 2 ) ( b ∙ dh / dt + h ∙ db / dt )

The altitude of the triangle is increasing at a rate of 2 cm / min while the area of a triangle is increasing at a rate of 1.5 cm² / min means:

dh / dt = 2

When the altitude is 11.5 cm and the area is 88 cm², the base is b = 182 / 11.5

This gives:

1.5 = ( 1 / 2 ) [ ( 182 / 11.5 ) ∙ 2 + 11.5 ∙ db / dt )

1.5 = ( 1 / 2 ) ( 364 / 11.5 + 11.5 db / dt )

Multiply both sides by 2

3 = 364 / 11.5 + 11.5 db / dt

Subtract 364 / 11.5 to both sides

3 - 364 / 11.5 = 11.5 db / dt

34.5 / 11.5 - 364 / 11.5 = 11.5 db / dt

Multiply both sides by 11.5

34.5 - 364 = 132.25 db / dt

- 329.5 = 132.25 db / dt

- 329.5 / 132.25 = db / dt

db / dt = - 329.5 / 132.25 = - 32950 / 13225 = - 25 ∙ 1318 / 25 ∙ 529 = - 1318 / 529 cm / min

The length of the base of the triangle is decreasing at the rate 2.4915 cm / min ≈ 2.5 cm / min

To solve this problem, we can use the formula for the area of a triangle:

Area = (1/2) * base * altitude

First, differentiate both sides of the equation with respect to time (t) using the chain rule:

d(Area)/dt = (1/2) * (d(base)/dt * altitude + base * d(altitude)/dt)

The problem provides:

- d(altitude)/dt = 2 cm/min (rate at which the altitude is increasing)
- d(Area)/dt = 1.5 cm²/min (rate at which the area is increasing)
- altitude = 11.5 cm
- Area = 91 cm²

Let's plug in the given values into the equation:

1.5 = (1/2) * (d(base)/dt * 11.5 + base * 2)

Simplify the equation:

3 = d(base)/dt * 11.5 + 2base

Now we have an equation with two variables, d(base)/dt (rate of change of the base) and base. We need to find the value of d(base)/dt when base = ?.

To proceed further, we need more information about the triangle, specifically the value of the base.

To solve this problem, we need to use the formula for the area of a triangle, which is given by:

Area = (1/2) * base * altitude

We are given that the altitude is increasing at a rate of 2 centimeters/minute and the area is increasing at a rate of 1.5 square centimeters/minute.

Let's set up the known rates:

d(altitude)/dt = 2 cm/min
d(area)/dt = 1.5 cm^2/min

We are asked to find the rate at which the base of the triangle is changing, which is d(base)/dt. To find this, we can use the chain rule of differentiation.

First, differentiate the area equation with respect to time (t):

d(area)/dt = (1/2) * d(base)/dt * altitude + (1/2) * base * d(altitude)/dt

Now, we can substitute the given values:

1.5 = (1/2) * d(base)/dt * 11.5 + (1/2) * base * 2

Simplify the equation:

1.5 = 5.75 * d(base)/dt + base

Rearrange the equation to solve for d(base)/dt:

5.75 * d(base)/dt = 1.5 - base

Now, substitute the given area value to find base:

91 = (1/2) * base * 11.5

base = 182/11.5 = 15.8261

Now, substitute the base value back into the equation:

5.75 * d(base)/dt = 1.5 - 15.8261

Simplify:

d(base)/dt = (1.5 - 15.8261) / 5.75

Calculate:

d(base)/dt ≈ -2.82 cm/min

Therefore, the rate at which the base of the triangle is changing when the altitude is 11.5 cm and the area is 91 cm^2 is approximately -2.82 cm/min.