Applied Use of Radical Equations (2024)

Applied Use of Radical Equations (1)

Applied Use of Radical Equations
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Applied Use of Radical Equations (2)

Applied Use of Radical Equations (3)

Directions: Solve the following problems dealing with radical equations. Show your work algebraically. Remember that you can use your graphing calculator to check your answers.

1.

A cube measures 1½ feet on each side.
A (straight) spider web connects the bottom corner of the cube to the opposite top corner, as shown. How far will a spider have to climb along this web?

Solution

Applied Use of Radical Equations (4)

2.

The speed traveled by a tidal wave can be modeled by the equation Applied Use of Radical Equations (5) where S is the speed in kilometers per hour, and d is the average depth of the water in kilometers.
a) Solve the equation for d.
b) A tidal wave is traveling at 115 kilometers per hour. What is the average depth of the water, to the nearest thousandths of a kilometer.

Applied Use of Radical Equations (6)

Solution


3.

The lateral surface area of a right circular cone, s, is represented by the equation Applied Use of Radical Equations (7), where r is the radius of the circular base and h is the height of the cone. If the lateral surface area of a large cone-shaped funnel is 364.25 square centimeters and its radius is 5.75 centimeters, find its height, to the nearest hundredth of a centimeter.

Applied Use of Radical Equations (8)

Solution

4.

A rectangle has a perimeter of 24 inches with a length of 4 inches and a width of Applied Use of Radical Equations (9) inches. Find x.

Solution


Applied Use of Radical Equations (10)

5.

A wrecking ball, when suspended from a crane, models the movement of a pendulum. The relationship between R (the rate of speed of the ball), m (the mass of the ball), L (the length of the chain), and F, (the force) is represented by of the Applied Use of Radical Equations (11). Determine F when L = 15, m = 60 and
R
= 0.7. (Express answer to the nearest tenth.)
Applied Use of Radical Equations (12)

Solution


6.

The escape velocity from a planet is modeled by Applied Use of Radical Equations (13), where Vesc is the escape velocity, G is the gravitational constant, M is the mass of the planet, and R is the radius of the planet.

Solve the equation for the radius of the planet.

Applied Use of Radical Equations (14)
Photo courtesy of NASA.

Solution


7.

In the Earth's atmosphere, the chief factor affecting the speed of sound is the temperature. The speed of sound (v), in knots, can be calculated using the formula:
Applied Use of Radical Equations (15)
where T is the temperature in Kelvin (K).
To convert between temperature Fahrenheit (Tf ) and Kelvin (TK ):
Applied Use of Radical Equations (16)
To convert between knots, kts, and miles per hour, mph:
Applied Use of Radical Equations (17)

Calculate the speed of sound, in miles per hour, when the temperature is 68º Fahrenheit.
(Express answer to nearest tenth.)

Solution


Applied Use of Radical Equations (18)

8.

A real estate agent is determining the inflation rate of homes in her area.
Inflation rates of homes can be modeled by: Applied Use of Radical Equations (19)
where i is the annual inflation rate (decimal),
w1 is the home's worth before inflation,
w2 is the home's worth after inflation,
n is the number of years from w1 to w2 .

If a home sells for $80,000 in 2010 and $120,000 in 2015, what was the annual rate of inflation (to nearest tenth of a percent)?

Applied Use of Radical Equations (20)

Solution

9.

The formula for determining if a sailboat qualifies to compete in the America's Cup race is:
Applied Use of Radical Equations (21)

where L is the length of the boat in meters,
S is the area of the sails in square meters, and
DSP is the volume of the water displaced by the boat in cubic meters.

A boat has a length of 22 meters and a sail area of 310 square meters. What is the minimum allowable value for the water displacement to the nearest tenth of a cubic meter?

Applied Use of Radical Equations (22)

Solution

10.

The horizon (skyline) is an apparent line that separates the earth from the sky. The distance to the horizon from an observer close to the Earth's surface (ignoring atmospheric refraction) can be approximated by:
Applied Use of Radical Equations (23)
where d is the distance in kilometers and h is the height above ground level in meters.
Express answers to nearest tenth.

Applied Use of Radical Equations (24)

Solution

a) What is the distance to the horizon at an average eye-level height of 5 ft. 7 in. (1.7 meters)? Express answer in kilometers and in miles.

b) What is the distance to the horizon from the top of Mount Everest, 8,848 meters in altitude? Express answer in kilometers and in miles.

Applied Use of Radical Equations (25)

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Applied Use of Radical Equations (2024)

FAQs

How can radical equations be used in real life? ›

Radicals are also used to calculate the velocity of objects moving in circular paths, which has practical applications in roller coaster and satellite design. Engineers use radicals to calculate the stress and tension on bridges, ensuring they can withstand daily traffic and environmental conditions.

What is a practical application of radical equations? ›

Any expression that contains the square root of a number is a radical expression. Both have real world applications in fields like architecture, carpentry and masonry. Radical expressions are utilized in financial industries to calculate formulas for depreciation, home inflation and interest.

What is a real life example of a radical function? ›

Real World Applications of Radical Functions

The financial industry uses rational functions to find interest, depreciation and other kinds of regular calculations. For example, to calculate the inflation of a home that increases in value the financier would use radical functions to find the value of the home in time.

Where are radical expressions used? ›

In a scientific context, radical expressions and equations are used in a range of applications, including physics, chemistry, and biology. For example, the formula for calculating the energy of a particle in quantum mechanics involves the use of radical expressions and equations.

What is an example of a radical in everyday life? ›

European paper sizes are a good example of real world usage of a radical. The ratio of the length of the longer side of A4 paper to the shorter side is a good approximation of √2 .

What are some real life occupations that use radical functions? ›

Radical expressions and functions are used in real life scenarios such as engineering, architecture, and healthcare, where they are used to calculate stress and strain on materials, structural loads and forces, and measure radioactive decay or decrease of drug concentration.

What is the importance of radical equation? ›

Radicals in mathematics are important. By using radicals as inverse operations to exponents, you can solve almost any exponential equation. Radicals such as the square root have been used for thousands of years.

How is radical used in math? ›

Radical - The √ symbol that is used to denote square root or nth roots.

What is the objective of solving radical equations? ›

Objective. Students will be able to solve radical equations (square root) by isolating the radical and then squaring both sides of the equation. They will also learn to identify extraneous solutions and understand how and why they arise.

How do biologists use radicals? ›

Radicals are actually found throughout biology. They play a central role in cofactor biosynthesis used to expand the repertoire of enzyme catalyzed reactions6 and are essential for many metabolic transformations in primary and secondary metabolism and in cell signaling.

What are the real life applications of rational exponents? ›

Where are rational exponents and radicals used in the real world? Rational exponents and radicals show up in a lot of places! For example, they're used in physics to calculate things like electrical resistance and wave frequencies. They're also used in finance to calculate compound interest.

What is the real world application of square root function? ›

Square roots are applied in my disciplines such as engineering, finance, probability, architecture, and physics. The rate of return on an asset in a period of 2 units ( 2 years, 2 months, etc.) is found by using square roots. V2 denoted the money value after 2 years.

How can simplifying radicals be used in real life? ›

Simplifying radical expressions helps in real-world problem-solving by making complex calculations more manageable and understandable. Whether dealing with rational exponents or radical functions, simplification streamlines equations, making it easier to identify patterns, analyze data, and find practical solutions.

How are rational functions used in the real-world? ›

Rational functions can be used in a variety of ways to model real-world situations. For example, we can use a rational function to model the speed of a car that's braking, or the amount of a drug in a person's system over time.

How do you use radical expressions in operations? ›

Use the rule x√a⋅x√b=x√ab a x ⋅ b x = a b x to multiply the radicands. Look for perfect squares in the radicand, and rewrite the radicand as the product of two factors. Identify perfect squares. Rewrite as the product of two radicals.

Where are rational equations used in real life? ›

Work problems

Rational equations can be used to solve a variety of problems that involve rates, times and work. Using rational expressions and equations can help you answer questions about how to combine workers or machines to complete a job on schedule.

Where do you use equations in real life? ›

However, whether you realize it or not, mathematical equations are part of just about every area of your life, from architecture and cooking to tomorrow's weather forecast. It can even be found in online security and evaluating the lives at risk in the event of a natural disaster.

How are square root functions used in real life? ›

Square roots are applied in my disciplines such as engineering, finance, probability, architecture, and physics. The rate of return on an asset in a period of 2 units ( 2 years, 2 months, etc.) is found by using square roots. V2 denoted the money value after 2 years.

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