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larisa [96]
3 years ago
6

A 10 foot ladder is leaning against a wall. Call x the distance from the top of the ladder to the ground, and call y the distanc

e from the wall to the foot of the ladder. At the instant that the foot of the ladder is 6 feet away from the wall, the foot of the ladder is moving away from the wall at a rate of 4 ft/sec. At what rate is the top of the ladder falling down the wall at this time (in feet/sec)
Mathematics
1 answer:
liq [111]3 years ago
5 0

-5.3\:\text{ft/s}

Step-by-step explanation:

We start by applying the Pythagorean theorem to the ladder, with its length L as the hypotenuse:

L^2 = 100\:\text{ft}^2 = x^2 + y^2

where x is the vertical distance from the top of the ladder to the ground and y is the horizontal distance from the bottom of the ladder to the wall. Taking the derivative of the above expression with respect to time, we get

0 = 2x\dfrac{dx}{dt} + 2y\dfrac{dy}{dt}

Solving for dx/dt, we get

\dfrac{dx}{dt} = -\left(\dfrac{y}{x}\right)\dfrac{dy}{dt} = -\left(\dfrac{\sqrt{L^2 - x^2}}{x}\right)\dfrac{dy}{dt}

We know that

\dfrac{dy}{dt} = 4\:\text{ft/s}

when x = 6 ft. So the rate at which the top of the ladder is going down is

\dfrac{dx}{dt} = -\left(\dfrac{\sqrt{100\:\text{ft}^2 - (6\:\text{ft})^2}}{6\:\text{ft}}\right)(4\:\text{ft/s})

\:\:\:\:\:\:\:= -5.3\:\text{ft/s}

The negative sign means that the distance x is decreasing as y is increasing.

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what is the value of the reciprocal of $\frac{-2}{5}$? express your answer as a decimal to the nearest tenth.
olga2289 [7]

The reciprocal of -2/5, expressed as a decimal to the nearest tenth is: -2.5.

<h3>What is the Reciprocal of a Number?</h3>

The reciprocal of a number can be described as the inverse of that number. For example, if a is given as a real number, therefore, the reciprocal of a would be 1/a.

Another example is a/b. The reciprocal of a/b is b/a.

Thus, the reciprocal of -2/5 is -5/2.

Expressing this as a decimal, we would have:

-5/2 = 2.5 (nearest tenth).

Therefore, the reciprocal of -2/5, expressed as a decimal to the nearest tenth is: -2.5.

Learn more about the reciprocal of a number on:

brainly.com/question/20896748

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Ryan spent 1/3 of his monthly salary for rent and 1/7 of his monthly salary for his utility bill. If $759 was left, what was his
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Step-by-step explanation:

<h3><u>G</u><u>i</u><u>v</u><u>e</u><u>n</u><u> </u><u>Inform</u><u>ation</u><u> </u><u>:</u></h3>

  • Ryan spent 1/3 of his monthly salary for rent and 1/7 of his monthly salary for his utility bill.
  • Remaining money = $759

<h3><u>T</u><u>o</u><u> </u><u>calculate</u><u> </u><u>:</u></h3>

  • His monthly salary.

<h3><u>C</u><u>a</u><u>l</u><u>c</u><u>u</u><u>l</u><u>a</u><u>t</u><u>i</u><u>o</u><u>n</u><u> </u><u>:</u></h3>

Let us assume his monthly salary as x. According to the question,

➝ Money spent on rent + Money spent for utility bill + Remaining money = His salary

\longrightarrow\sf {\dfrac{1}{3}x + \dfrac{1}{7}x + 759 = x} \\

\longrightarrow\sf {\dfrac{7x + 3x + 15939}{21}= x} \\

\longrightarrow\sf {\dfrac{10x+ 15939}{21}= x} \\

\longrightarrow\sf {10x+ 15939= 21x} \\

\longrightarrow\sf {15939= 21x - 10x} \\

\longrightarrow\sf {15939= 11x} \\

\longrightarrow\sf {\cancel{\dfrac{15939}{11}}= x} \\

\longrightarrow\underline{\boxed{\bf {1449= x}}} \\

<u>Therefore,</u><u> </u><u>his</u><u> </u><u>monthly</u><u> </u><u>income</u><u> </u><u>is</u><u> </u><u>$</u><u>1</u><u>4</u><u>4</u><u>9</u><u>.</u>

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