Ask question

Ask question

All categories

2 years ago

6

(12 points) A 13 foot long ladder leans against a wall and the foot of the ladder is sliding away from the wall at a constant ra

te of 3 feet/sec. Meanwhile, a firefighter is climbing up the ladder at a constant rate of 2 feet/sec. At the instant the firefighter has climbed 4 feet of the ladder, the foot of the ladder is 5 feet from the wall. Let h be the vertical height of the firefighter above the ground. Find dh/dt at that instant.

1 answer:

Viefleur [7K]2 years ago

3 0

Answer:

19/13 ft/s ≈ 1.4615 ft/s

Step-by-step explanation:

It may be convenient to choose the time reference as the instant where we want to compute dh/dt. That is, we want to find h'(0).

If w represents the height of the ladder on the wall, we can write two relations: one for w, and one for h, the height of the firefighter.

w² + x² = 13² . . . . . . where x=5+3t is the distance of the foot of the ladder from the wall

h = w·(4 +2t)/13 . . . . where 4+2t is the length along the ladder that the firefighter has climbed

__

Differentiating with respect to time, we get ...

2ww' +2xx' = 0

w' = (-x/w)x' = -3x/w . . . . . solving for w'

and ...

h' = w'(4 +2t)/13 +w(2/13) . . . . . . . using the product rule

h' = (-3x/w)(4 +2t)/13 + w(2/13) . . . . filling in w'

__

At t=0, x=5, w=12. and we have ...

h' = (-3·5/12)(4/13) +(12·2)/13 = 19/13 ≈ 1.4615

dh/dt = 19/13 ft/s ≈ 1.4615 ft/s at the given instant

_____

The attachment shows a solution using a graphing calculator. It gives the same result.

You might be interested in

What is the piecewise-defined function that expresses the cost of the order, C(x), in terms of the number of photos ordered, x?

Murljashka [212]

Answer:

See below

Step-by-step explanation:

The values in the table change by $3.80 from one line to the next. Since each change from line to line changes in number of photos by 20, the average cost per photo is $3.80/20 = $0.19. There is apparently a $5.80 -3.80 = $2.00 shipping charge for numbers of photos in the range shown in the table.

Thus, the piecewise function has 3 pieces:

for x < 100: $2.00 + 0.19x

for x < 200: $4.50 + 0.17x

for x ≥ 200: $0.15x

This could be written as ...

$C(x)=\left\{\begin{array}{rcl}\$2.00+0.19x&\text{for}&0\le x$

8 0

2 years ago

What is the volume rounded to the nearest tenth of the largest sphere that can fit inside of the box ?

RSB [31]

Answer:

36 cubic units.........................

3 0

2 years ago

Which statements are true about ΔJKL and ΔLMN? Select all that apply.

svp [43]

Answer:

A. ΔJKL and ΔLMN have only one pair of angles that are congruent

C. ΔJKL has angles that measure $50^o,60^o,70^o$

Step-by-step explanation:

step 1

Find the value of x

we know that

$m\angle KLJ=m\angle MLN$ ----> by vertical angles

substitute the given values

$(6x-2)^o=(5x+10)^o$

solve for x

$6x-5x=10+2\\x=12$

step 2

Find the measure of the interior angles of triangle JKL

we have

$m\angle KLJ=(6(12)-2)=70^o$

$m\angle KJL=(5(12))=60^o$

Remember that the sum of the interior angles in any triangle must be equal to 180 degrees

so

$m\angle JKL=180^o-130^o=50^o$

therefore

Triangle JKL has angles that measure $50^o,60^o,70^o$

step 3

Find the measure of the interior angles of triangle LMN

we have

$m\angle MLN=(5(12)+10)=70^o$

$m\angle LMN=(6(12)-7)=65^o$

Remember that the sum of the interior angles in any triangle must be equal to 180 degrees

so

$m\angle LNM=180^o-135^o=45^o$

Triangle LMN has angles that measure $45^o,65^o,70^o$

we know that

If two triangles are similar, then the ratio of its corresponding sides is proportional and its corresponding angles are congruent

The corresponding angles of triangle JKL and LMN are not congruent

therefore

The triangles are not similar

<u><em>Verify each statement</em></u>

Part a) ΔJKL and ΔLMN have only one pair of angles that are congruent

The statement is true (see the explanation)

Part b) ΔJKL and ΔLMN have two pairs of angles that are congruent

The statement is false

Because have only one pair of angles that are congruent

Part c) ΔJKL has angles that measure $50^o,60^o,70^o$

The statement is true (see the explanation)

Part d) ΔLMN has angles that measure $50^o,60^o,70^o$

The statement is false

Because, Triangle LMN has angles that measure $45^o,65^o,70^o$

Part e) ΔJKL and ΔLMN are similar

The statement is false

Because, the corresponding angles are not congruent

6 0

3 years ago

From a point 100 m from a building the angles of elevation of the top and bottom of a flagpole atop a building are 54.5 degrees

AnnZ [28]

Answer:60ft

Step-by-step explanation:The height of the flagpole is approximately

60

feet.

Explanation:

Always try to draw a diagram.

enter image source here

We know that there is a right angle between the ground and the building. Therefore, we can use the 3 basic trig ratios instead of the sine or cosine law to solve this problem.

Since the angle in the corner of the larger right triangle measures

42

˚

, the top angle in this triangle measures

180

˚

−

90

˚

−

42

˚

=

48

˚

.

By basic trig ratios, we can find the height of the building with the flag pole on top, call it

H

.

tan

42

˚

1

=

H

500

H

=

500

tan

42

˚

I would keep it in exact form until the last step.

We now devise an expression for the height of the building (without the flag pole). Call it

a

tan

38

˚

1

=

a

500

a

=

500

tan

38

˚

We can now state that

h

=

H

−

a

h

=

500

tan

42

˚

−

500

tan

38

˚

h

≈

59.559

≈

60

feet

Hopefully this helps!

Answer link

EET-AP

Apr 10, 2017

The flagpole is

60

f

t

in height to the nearest foot.

Explanation:

1) The flagpole is on top of a building.

2)Angles of elevation both measured from point

500

f

t

from building

3) Angle of elevation to the top of building is

38

d

e

g

4) Angle of elevation to the top of flagpole is

42

d

e

g

The information above will provide us with two right angle triangles, one smaller one inside a larger one.

Both will have a base of

500

f

t

.

The smaller triangle will have a base angle

β

of

38

deg opposite the

90

deg, and the larger triangle will have a base angle

β

of

42

deg.

From this information we can find the heights of the building and the building + pole using the definition of the tangent of the two base angles

β

:

tan

(

β

)

=

o

p

p

a

d

j

where the

o

p

p

is the height and the

a

d

j

is the

500

f

t

o

p

p

(

b

u

i

l

d

)

=

tan

(

38

)

⋅

(

500

f

t

)

=

390.6

f

t

=

height of building

o

p

p

(

f

l

a

g

)

=

tan

(

42

)

⋅

(

500

f

t

)

=

450.2

f

t

=

height of building + pole

Then to the nearest foot the height of the flagpole is:

450.2

f

t

−

390.6

f

t

=

60

f

t

4 0

3 years ago

Find the product of 21 and (6.3x10^-4)

katen-ka-za [31]

The answer would end up being 0.01323

5 0

2 years ago