Wize High School Algebra I Textbook (Common Core) > Radical Equations and Functions

Applications of Radical Functions

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Example: Applications of Radical Functions
A rectangular field has a perimeter of 24 meters with a length of 4 meters and a width of x+2\sqrt{x+2}x+2​ meters. Find the area of the rectangle. 

The perimeter of a rectangle is:
P=2w+2lP=2w+2lP=2w+2l
where w = width and l = length ad l,w≥0l,w\geq0l,w≥0. Thus, 
P=2w+2l24=2(x+2)+2(4)16=2x+28=x+282=x+264−2=x∴  x=62\begin{array}{rcl}
P&=&2w+2l\\\\
24&=&2(\sqrt{x+2})+2(4)\\\\
16&=&2\sqrt{x+2}\\\\
8&=&\sqrt{x+2}\\\\
8^2&=&x+2\\\\
64-2&=&x\\\\
\therefore~~x&=&62
\end{array}P241688264−2∴  x​=======​2w+2l2(x+2​)+2(4)2x+2​x+2​x+2x62​

Verify:
24=2(62+2)+2(4)16=262+28=62+28=8✓\begin{array}{rclc}
24&=&2(\sqrt{62+2})+2(4)\\\\
16&=&2\sqrt{62+2}\\\\
8&=&\sqrt{62+2}\\\\
8&=&8&\checkmark
\end{array}241688​====​2(62+2​)+2(4)262+2​62+2​8​✓​

So, the width is x+2=62+2=8\sqrt{x+2}=\sqrt{62+2}=8x+2​=62+2​=8. 

Therefore, the area of the rectangular field is:
A=8⋅4=32m2A=8\cdot4=32m^2A=8⋅4=32m2

Practice: Applications of Radical Functions
Solve for VVV 
r=3V34πr=\sqrt{\frac{3V^3}{4\pi}}r=4π3V3​​

A)4πr233\displaystyle\sqrt[3]{\frac{4\pi{}r^2}{3}}334πr2​​ 

B) 4πr23\displaystyle\sqrt{\frac{4\pi{}r^2}{3}}34πr2​​ 

 C) 12πr233\displaystyle\sqrt[3]{12\pi{}r^23}312πr23​

D) 34πr23\displaystyle\sqrt[3]{\frac{3}{4\pi{}r^2}}34πr23​​

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Practice: Applications of Radical Functions
A garden is in the shape of an isosceles right triangle. The hypotenuse is 10ft. long. The cost of fencing is $1.25/ft. How much will it cost to fence in the garden?

Enter your answer in dollars rounded to 2 decimal places, do not include units.

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Extra Practice

Applications of Radical Functions

Practice: Applications of Radical Functions
A 3-foot ladder is leaning against a wall creating a 45∘45^{\circ}45∘ with the ground. How far away is the foot of the ladder from the foot of the wall?

Applications of Radical Functions

Practice: Applications of Radical Functions
Sally leaves her house to go for a jog. She leaves point A and runs 8km to point B. She takes a 2-minute water break and then continues along to point C. She takes another 2-minute water break at point C and then runs home to point A. 
If the distance from BC=AC\text{BC}=\text{AC}BC=AC and she runs at an average of 11km/hr11\text{km/hr}11km/hr, how long did Sally go for a run?

Applications of Radical Functions

Practice: Applications of Radical Functions
A small sandpit is in the shape of an isosceles right triangle. The hypotenuse is 25 inches. long. The cost of the materials for the sandpit is $2.46/in. How much will it cost to make that sandpit?

Applications of Radical Functions

Practice: Applications of Radical Functions
Solve for AAA 
B=35a7−623+4B=^3\sqrt{\dfrac{5a^7-6}{23}}+4B=3235a7−6​​+4

Applications of Radical Functions

Practice: Applications of Radical Functions
Solve for y.y.y. 
x=25−25y236πx=\sqrt{\dfrac{25-25y^2}{36\pi}}x=36π25−25y2​​

Applications of Radical Functions

Practice: Applications of Radical Functions
Solve for bbb.
a=4−10b36πa=\sqrt{\frac{4-10b^3}{6\pi}}a=6π4−10b3​​