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worty [1.4K]
3 years ago
8

two brothers marco and fernando are training for a marathon marco runs at a rate of 6miles per hor. Fernando funs at at a rate o

f 7.2 miles per hour in the same direction.how much time will they be 0.3 miles apart
Mathematics
1 answer:
satela [25.4K]3 years ago
6 0
We know that
speed=distance/time
solve for time
time=distance/speed

in this problem
<span>Marco runs at a rate of 6 miles per hour. 
</span><span>Fernando funs at a rate of 7.2 miles per hour
Difference=7.2-6=1.2 miles/hour
so
speed=1.2 miles/hour
distance=0.3 miles
time=?
</span>time=distance/speed-----> 0.3/1.2-----> 0.25 hour-----> 0.25*60=15 minutes
<span>
the answer is
0.25 hour (15 minutes)

Alternative Method
Let 
x---------> Fernando's distance when Marco is 0.3 miles apart

</span>Fernando funs at a rate of 7.2 miles per hour
<span>for distance =x
time=x/7.2------> equation 1

</span>Marco runs at a rate of 6 miles per hour. 
for distance=x-0.30
time=(x-0.30)/6------> equation 2

equate equation 1 and equation 2
 7.2*(x-0.3)=6x-----> 7.2x-2.16=6x
7.2x-6x=2.16------> x=2.16/1.2-------> x=1.8 miles

time=x/7.2-----1.8/7.2=0.25 hour
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Area of the bounded curves y=x^2, y=√(7+x)
N76 [4]

Answer:

\displaystyle \int\limits^{1.718}_{-1.529} {\sqrt{7 + x} - x^2} \, dx = 5.74773

General Formulas and Concepts:

<u>Calculus</u>

Differentiation

  • Derivatives
  • Derivative Notation

Derivative Property [Addition/Subtraction]:                                                         \displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)]  

Basic Power Rule:

  1. f(x) = cxⁿ
  2. f’(x) = c·nxⁿ⁻¹

Integration

  • Integrals

Integration Rule [Reverse Power Rule]:                                                               \displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C

Integration Rule [Fundamental Theorem of Calculus 1]:                                     \displaystyle \int\limits^b_a {f(x)} \, dx = F(b) - F(a)

Integration Property [Addition/Subtraction]:                                                       \displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx

U-Substitution

Area of a Region Formula:                                                                                     \displaystyle A = \int\limits^b_a {[f(x) - g(x)]} \, dx

Step-by-step explanation:

<u>Step 1: Define</u>

\displaystyle \left \{ {{y = x^2} \atop {y = \sqrt{7 + x}}} \right.

<u>Step 2: Identify</u>

<em>Graph the systems of equations - see attachment.</em>

Top Function:  \displaystyle y = \sqrt{7 + x}

Bottom Function:  \displaystyle y = x^2

Bounds of Integration: [-1.529, 1.718]

<u>Step 3: Integrate Pt. 1</u>

  1. Substitute in variables [Area of a Region Formula]:                                   \displaystyle \int\limits^{1.718}_{-1.529} {\sqrt{7 + x} - x^2} \, dx
  2. [Integral] Rewrite [Integration Property - Addition/Subtraction]:               \displaystyle \int\limits^{1.718}_{-1.529} {\sqrt{7 + x} - x^2} \, dx= \int\limits^{1.718}_{-1.529} {\sqrt{7 + x}} \, dx - \int\limits^{1.718}_{-1.529} {x^2} \, dx
  3. [Right Integral] Integration Rule [Reverse Power Rule]:                             \displaystyle \int\limits^{1.718}_{-1.529} {\sqrt{7 + x} - x^2} \, dx= \int\limits^{1.718}_{-1.529} {\sqrt{7 + x}} \, dx - \frac{x^3}{3} \bigg| \limits^{1.718}_{-1.529}
  4. Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]:           \displaystyle \int\limits^{1.718}_{-1.529} {\sqrt{7 + x} - x^2} \, dx= \int\limits^{1.718}_{-1.529} {\sqrt{7 + x}} \, dx - 2.88176

<u>Step 4: Integrate Pt. 2</u>

<em>Identify variables for u-substitution.</em>

  1. Set <em>u</em>:                                                                                                             \displaystyle u = 7 + x
  2. [<em>u</em>] Basic Power Rule [Derivative Rule - Addition/Subtraction]:                 \displaystyle du = dx
  3. [Limits] Switch:                                                                                               \displaystyle \left \{ {{x = 1.718 ,\ u = 7 + 1.718 = 8.718} \atop {x = -1.529 ,\ u = 7 - 1.529 = 5.471}} \right.

<u>Step 5: Integrate Pt. 3</u>

  1. [Integral] U-Substitution:                                                                               \displaystyle \int\limits^{1.718}_{-1.529} {\sqrt{7 + x} - x^2} \, dx= \int\limits^{8.718}_{5.471} {\sqrt{u}} \, du - 2.88176
  2. [Integral] Integration Rule [Reverse Power Rule]:                                       \displaystyle \int\limits^{1.718}_{-1.529} {\sqrt{7 + x} - x^2} \, dx = \frac{2x^\Big{\frac{3}{2}}}{3} \bigg| \limits^{8.718}_{5.471} - 2.88176
  3. Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]:           \displaystyle \int\limits^{1.718}_{-1.529} {\sqrt{7 + x} - x^2} \, dx = 8.62949 - 2.88176
  4. Simplify:                                                                                                         \displaystyle \int\limits^{1.718}_{-1.529} {\sqrt{7 + x} - x^2} \, dx = 5.74773

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Integration

5 0
2 years ago
Select all that apply. Solve for x , 0 &lt; x &lt; 360;
Vesna [10]

Answer:

All options are correct

Step-by-step explanation:

1. Note that

\cos 2x=2\cos ^2x-1.

2. Substitute previous expression into the equation:

2\cos ^2x-1+\cos^2x=1,\\ \\3\cos^2x=2,\\ \\\cos^2x=\dfrac{2}{3},\\ \\\cos x=\pm\sqrt{\dfrac{2}{3}}.

3. If 0° < x < 360°, then

x=\arccos \left(\sqrt{\dfrac{2}{3}}\right)\approx 35^{\circ};;

x=180^{\circ}-35^{\circ}=145^{\circ};

x=180^{\circ}+35^{\circ}=215^{\circ};

x=360^{\circ}-35^{\circ}=325^{\circ}.

All options are true.

Or you can solve this equation graphically. Plot the graph of the function y=\cos2x+cos^2x that represents the left side of the equation and the graph of the function y=1 that represents the right side of the equation. their common points are solutions.

8 0
3 years ago
If you know how to do this please help me and if you don't PLEASE DONT SUBMIT AN ANSWER please let others to answer it int the c
masya89 [10]
16/34 is the answer : CAH is adjacent/ hypotenuse
6 0
3 years ago
PLEASE HURRY <br> What is the measure of M?<br> 52<br> Р<br> 78°<br> M
vodomira [7]

Answer:

50 degree

Step-by-step explanation:

the angle sum property of the equilateral triangle is =180 degree

since, 52 degree +78 degree +x =180 degree

130 degree + x =180 degree

x= (180-130) degree

<h3> x= 50 degree</h3>

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4 0
3 years ago
Plz help.......................
kondaur [170]
The asnwer is 3u-4

just divide like terms
4 0
2 years ago
Read 2 more answers
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