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marysya [2.9K]
3 years ago
13

A canister is dropped from a helicopter 500m above the ground. Its parachute does not open, but the canister has been designed t

o withstand an impact velocity of 100 m/s. Will it burst?
Mathematics
1 answer:
solmaris [256]3 years ago
5 0

Answer:

The impact speed 98.995 m/s is less than 100 m/s and the canister will not burst.

Step-by-step explanation:

A function<em> F</em> is called an antiderivative of <em>f</em> on an interval <em>I</em> if F'(x) = f(x) for all x in <em>I.</em>

Recall that if the object has position function s=f(t), then the velocity function is v(t)=s'(t). This means that the position function is an antiderivative of the velocity function. Likewise, the acceleration function is a(t)=v'(t), so the velocity function is an antiderivative of the acceleration.

An object near the surface of the earth is subject to a gravitational force that produces a downward acceleration denoted by g. For motion close to the ground we may assume that g is constant, its value being about 9.8 \:{\frac{m}{s^2}}.

We know that the acceleration due to gravity is given by

a(t)=-9.8

and the antiderivative is velocity

v(t)=\int a(t)\,dt\\v(t)=\int -9.8\,dt\\v(t)=-9.8t +C

We know that the canister was dropped, so the initial velocity at t = 0 is zero, this fact let us know the value of C.

v(0)=9.8(0)+C\\C=0

The antiderivative of velocity is the position

s(t)=\int v(t) \, dt\\s(t)=\int -9.8t \, dt\\s(t)=-4.9t^2+C

To find the value of the constant C, we know that the height was 500 m at t  = 0, this means s(0)=500

500=-4.9(0)^2+C\\C=500

s(t)=-4.9t^2+500

Using the fact that at the time of impact the height s(t) is zero we can compute the total time of the fall:

s(t)=-4.9t^2+500=0\\\\-4.9t^2=-500\\\\t^2=\frac{5000}{49}\\\\\mathrm{For\:}x^2=f\left(a\right)\mathrm{\:the\:solutions\:are\:}x=\sqrt{f\left(a\right)},\:\:-\sqrt{f\left(a\right)}\\\\t=\sqrt{\frac{5000}{49}},\:t=-\sqrt{\frac{5000}{49}}

A negative time does not make sense, so we only take as a possible solution

t=\sqrt{\frac{5000}{49}}=\frac{50\sqrt{2}}{7}\approx 10.102

Now the final velocity is

v(\frac{50\sqrt{2}}{7})=-9.8(\frac{50\sqrt{2}}{7})\approx -98.995

The impact speed 98.995 m/s is less than 100 m/s and the canister will not burst.

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marin [14]

Answer:

y=-3x+5

Explanation:

Given a line L such that:

• L has y-intercept (0,5); and

,

• L is perpendicular to the line with equation y=(1/3)x+1.

We want to find the equation of the line in the slope-intercept form.

The slope-intercept form of the equation of a straight line is given as:

\begin{equation} y=mx+b\text{ where }\begin{cases}m={slope} \\ b={y-intercept}\end{cases} \end{equation}

Comparing the given line with the form above:

y=\frac{1}{3}x+1\implies Slope,m=\frac{1}{3}

Next, we find the slope of the perpendicular line L.

• Two lines are perpendicular if the product of their slopes is -1.

Let the slope of L = m1.

Since L and y=(1/3)x+1 are perpendicular, therefore:

\begin{gathered} m_1\times\frac{1}{3}=-1 \\ \implies Slope\text{ of line L}=-3 \end{gathered}

The y-intercept of L is at (0,5), therefore:

y-intercept,b=5

Substitute the slope, m=-3, and y-intercept, b=5 into the slope-intercept form.

\begin{gathered} y=mx+b \\ y=-3x+5 \end{gathered}

The equation of line L is:

y=-3x+5

7 0
1 year ago
I WILL MARKED BRAINLIEST IF YOU COULD ANSWER THIS!
tester [92]

Answer:

s = -2400t + 17400

Step-by-step explanation:

Let's say t is the x value on a coordinate plane, and s is the y. Then, we have the points (0, 17400) and (6, 3000). The slope of these is 14400/-6 or -2400.

Now we just have the equation y = -2400x + b, and from the point (0, 17400) we can find that b is 17400. So, we have y = -2400x + 17400. Convert these back into t and s and you get your answer, s = -2400t + 17400.

6 0
3 years ago
List all possible rational roots. Then use synthetic division to confirm which rational roots exist:
Kisachek [45]

Answer:

\boxed{(1) \, x = \, \pm \dfrac{1}{2}, \pm 1, \pm2, \pm \dfrac{5}{2}, \pm 5, \pm 10; (2) \, x = -2}

Step-by-step explanation:

2x³+ 6x² - x - 10 = 0

(1) Possible roots

The Rational Roots Theorem states that, if a polynomial has any rational roots, they will have the form p/q, where p is a factor of the constant term  and q is a factor of the leading coefficient.

\text{Possible rational root} = \dfrac{ p }{ q } = \dfrac{\text{factor of constant term}}{\text{factor of leading coefficient}}

In your function, the constant term is -10 and the leading coefficient is 2, so

\text{Possible root} = \dfrac{\text{factor of 10}}{\text{factor of 2}}

Factors of 10 = ±1, ±2, ±5, ±10

Factors of 2 = ±1, ±2

\text{Possible roots are } \large \boxed{\mathbf{x = \pm \dfrac{1}{2}, \pm 1, \pm2, \pm \dfrac{5}{2}, \pm 5, \pm 10}}

(2) Synthetic division

Rather than work through all 12 possibilities, I will do one that works.

\begin{array}{r|rrrr}-2 & 2 & 6 & -1 & -10\\& & -4& -4 & 10\\& 2 & 2& -5 & 0\\\end{array}

So, x = -2 is a root, and the quotient is 2x² + 2x - 5.

(3) Check for other rational roots

2x² + 2x - 5 = 0

D = b² - 4ac =2²- 4(2)(-5) = 4 + 40 = 44

√44 = 2√11, which is irrational.

Since irrational roots come in pairs, the cubic equation has two real, irrational roots and one rational root at x = -2.

6 0
3 years ago
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1 year ago
Which of the following are incorrect expressions for slope?
vfiekz [6]

Answer:

Option B and C are correct.

\frac{x_2-x_1}{y_2-y_1}

\frac{run}{rise} are the expression incorrect for slope

Step-by-step explanation:

Slope is defined as the change in the dependent variable  relative to the change in the dependent variable

or the ratio of the horizontal changes to vertical changes between any two points on the graph of the line.

The vertical changes between any two points is rise

The horizontal changes between any two points is run.

Formula for slope is given by:

For any two points (x_1, y_1) and (x_2, y_2)

then slope is:

\text{Slope} =\frac{rise}{run}= \frac{y_2-y_1}{x_2-x_1}

or we can write this as:

Δy = y_2-y_1

Δx = x_2-x_1

⇒\text{Slope} = \frac{\triangle y}{\triangle x}

Therefore, the expression which are incorrect for slope  are;

\frac{x_2-x_1}{y_2-y_1}

\frac{run}{rise}

8 0
3 years ago
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