Name _______________________________________
Date ___________________________
COURSE: MSC V
MODULE 1: Essentials of Algebra
UNIT 5: Solving Literal Equations
As you work through the tutorial, complete the following questions.
1. The Coneyville water tank is built as a section of a cone known as a(n)
______________________________________________________________ .
2. In the formula for the volume of the water tank, what do each of these
variables represent?
a. h 5 __________________________________________________________
b. r 5 __________________________________________________________
c. R 5 __________________________________________________________
d. v 5 __________________________________________________________
3. The __________ of a circle is the length of any line drawn from
the center of a circle to any point on the __________ of the circle.
4. What is the relationship between the radius and the diameter of a circle?
________________________________________________________________
5. For the water tank that Jesús is rebuilding, the radius of the
Key Words:
Frustum
Cone
Volume
Radius
Circumference
Diameter
Like terms
Learning
Objectives:
• Identifying the
variables in the
formula for the
volume of the
frustum of a cone
• Recognizing the
radius and
diameter of a
circle
• Using substitution
to express one
radius in terms of
the other
• Simplifying
algebraic
expressions by
multiplying and
combining like
terms
© RIVERDEEP, Inc.
_____________ base is twice the radius of the ______________ base.
6. Literal equations can be simplified by using __________________ to
express one ________________ in terms of another and by multiplying
and combining ___________________ terms.
destination
MATH
41
Name _______________________________________
Date ___________________________
COURSE: MSC V
MODULE 1: Essentials of Algebra
UNIT 5: Solving Literal Equations
1. The equation d 5 rt is used to find the distance traveled at a known rate
of speed for a certain amount of time.
a. In the formula for distance, list the variables and tell what they represent.
____________________ ____________________ ____________________
b. Express the rate in terms of time and distance. In other words, rewrite
the formula with r as the subject. __________
2. The area of a rectangle is equal to the length of the rectangle multiplied
by its width. Use variables to write a literal equation for the area of a
rectangle. __________
3. The diameter of a circle is 30 cm. What is the radius? __________
4. The diameter of a small circle is equal to the radius of a larger circle. The
diameter of the small circle is 5 cm. What is the diameter of the larger
circle? __________
5. What mathematical operation is implied in the expression πr ?
________________________________________________________________
© RIVERDEEP, Inc.
6. Dijit knows that the formula for the volume of a frustum is
v 5 1}3} πh (r2 1 rR 1 R2). Help Dijit rewrite the equation for a frustum that
has a height of 12 and a top radius of 4. Simplify your answer.
________________________________________________________________
destination
MATH
42
Name _______________________________________
Date ___________________________
COURSE: MSC V
MODULE 1: Essentials of Algebra
UNIT 5: Solving Literal Equations
As you work through the tutorial, complete the following questions.
1. In the literal equation v 5
1
}}
3
2
πh(7r ), what two values does Jesús
know? __________________________________________________________
2. Which value can he find by sending someone to the site? ______________
3. Which variable in the equation above is not known?
4. What must be done to the equation v 5
1
}}
3
________________
πh(7r 2) to remove the
1
}}
3
Key Words:
Pi (π)
Volume
Isolate
Inverse operation
Learning
Objectives:
• Using the
properties of
equality to rewrite
a formula for a
particular variable
from the right side, while keeping the formula balanced? ______________
5. What must be done to the equation 3v 5 πh(7r 2) to remove π from the
right side, while keeping the formula balanced? ______________________
6. Dividing by 7r 2 is the same as multiplying by __________ .
© RIVERDEEP, Inc.
7. Which equation represents the height of the tank Jesús is working on?
2
a. h 5
π7r
}}
3v
b. h 5
3v
}}
π7r 2
c. h 5
πv
}}
3(7r 2)
d. h 5
3(7r 2)
}}
πv
8. To isolate a particular variable in a literal equation, use _____________
operations so that the particular variable is the only term on one
side of the ______________ .
destination
MATH
43
Name _______________________________________
Date ___________________________
COURSE: MSC V
MODULE 1: Essentials of Algebra
UNIT 5: Solving Literal Equations
1. The perimeter of a rectangle is found using the formula p 5 2(l 1 w).
a. List and identify the variables in the formula.
_________________ _________________ _________________
b. Express the length of a rectangle in terms of the perimeter and the
width. ________________________________________________________
c. Express the width of a rectangle in terms of the perimeter and the
length. ________________________________________________________
2. The formula for the circumference of a circle is C 5 πd.
a. Express the diameter of a circle in terms of the circumference.
__________
b. Express the radius of a circle in terms of the circumference. __________
2
3. The formula for the area of a circle is A 5 πr . Express the radius of a
circle in terms of the area. __________
4. Last weekend, members of the Coneyville Running Club participated in a
5-kilometer race. The fastest runner in the club finished the race in 17.2
minutes. The slowest runner in the club finished the race in 35.6 minutes.
Use the formula for distance, d 5 rt.
© RIVERDEEP, Inc.
a. Find the rate of speed of the fastest runner in the club. __________
b. Find the rate of speed of the slowest runner in the club. __________
c. How long would it take the fastest runner to run 7 kilometers?
______
d. How long would it take the slowest runner to run 2 kilometers?
______
e. How far could the fastest runner run in 12 minutes? __________
f. How far could the slowest runner run in 45 minutes? __________
destination
MATH
44
Name _______________________________________
Date ___________________________
COURSE: MSC V
MODULE 1: Essentials of Algebra
UNIT 5: Solving Literal Equations
As you work through the tutorial, complete the following questions.
1. Dijit now has an equation to solve for the height of the frustum, h 5
Key Words:
3v
}}2.
π 7r
What is the value of each variable on the right side of the equation?
a. v 5 __________________________________________________________
b. r 5 __________________________________________________________
c. π 5 __________________________________________________________
2. Rewrite the literal equation, substituting the known values.
________________________________________________________________
3. What is the height of the Coneyville water tank? __________
4. How do Dijit and Jesús verify that the height measurement is correct?
Substitute
Simplify
Improper fraction
Common factor
Learning
Objectives:
• Substituting
values in a literal
equation to solve
for a particular
variable
• Applying the
order of
operations to
simplify
expressions
• Checking a
solution in the
original formula
________________________________________________________________
________________________________________________________________
5. a. To solve a literal equation for a specific variable, substitute the known
values for the other __________________________________________ .
© RIVERDEEP, Inc.
b. Use the _______________ of _______________ to solve for the subject
of the equation.
6. To check the solution, ________________ the values into the original literal
equation and see if both sides of the equation ______________________ .
destination
MATH
45
Name _______________________________________
Date ___________________________
COURSE: MSC V
MODULE 1: Essentials of Algebra
UNIT 5: Solving Literal Equations
}}. The
1. Density is equal to an object’s mass divided by its volume, or d 5 m
v
recycling center receives a container of crushed aluminum cans with a
mass of 15 kg and a volume of 5,550 cm3. Find the density of the
3
aluminum cans in g/cm . __________
2. a. Rewrite the formula for density to solve for m. __________
b. Find the mass of an object with a density of 19.3 g/cm3 and a
3
volume of 115 cm . __________
3. A park in Coneyville has a large circular jogging trail around a field. The
diameter of the trail is 120 m.
a. What is the radius of the trail? __________
b. The formula for the circumference of a circle is C 5 πd. Substitute the
known values to find the circumference of the jogging trail. Use 3.14
for π. __________
2
c. The formula for the area of a circle is A 5 πr . Substitute the known
values to find the area of the field surrounded by the trail. __________
4. The volume of a cone is given by v 5
1
}}
3
2
πr h.
a. Rearrange this expression to solve for height. _________________
© RIVERDEEP, Inc.
b. Calculate the height of an ice-cream cone that holds 98 cm3 of ice
cream and whose base has a radius of 2.5 cm. Use 3.14 for π.
Round your answer to the nearest whole cm. _________________
destination
MATH
46
Name _______________________________________
Date ___________________________
COURSE: MSC V
MODULE 1: Essentials of Algebra
UNIT 5: Solving Literal Equations
Identifying the Variables in a Given Formula
1. Rockridge has a water tank similar to the one Jesús rebuilt for Coneyville.
The tank is also a frustum, but the bottom radius is 8 times the top radius.
The formula for the volume of a frustum is v 5 1}3} πh (r2 1 rR 1 R2).
a. How can you express the bottom radius in terms of the top radius if
R 5 the bottom radius and r 5 the top radius? __________
b. Simplify the formula for the volume of the frustum by substituting the
value for R you found in part a, then multiply and combine like terms.
__________
c. Express the diameter of the bottom circle of the frustum in terms of the
diameter of the top circle of the frustum, if D 5 the bottom diameter and
d 5 the top diameter. __________
© RIVERDEEP, Inc.
Rewriting a Formula in Terms of a Different Variable
2. The track at Coneyville needs a new
surface. The track is in the shape
of an oval. The straight portions are
rectangles, each 100 m long and
8 m wide. The curved portions are
two halves of an annulus. The area
of a rectangle is given by A 5 lw.
The area of an annulus is given by
A 5 π(R2 2 r 2), where r is the radius
of the smaller circle and R is the
radius of the larger circle.
100 m
32 m
a. If r 5 5}4}R, what is an expression
for the area of the curved portion
of the track in terms of R and π? __________
destination
MATH
47
Name _______________________________________
b. If π 5
22
}}
7
Date ___________________________
and R 5 40 m, calculate the area of the curved portion of
the track. ______________________________________________________
______________________________________________________________
c. What is the total area of both straight portions of the track? __________
d. How many square meters of new surface will be needed for the track?
__________
Substituting Values & Solving an Equation
3. Garson’s farm has ordered a new grain tower to hold the feed for the
horses. This tower will be a right circular cylinder. The lateral surface
area of a cylinder is given by L 5 2πrh, where L is the lateral surface
area, r is the radius, and h is the height.
a. Rewrite the formula L 5 2πrh to solve for r, the radius. __________
b. How many meters long is the radius of the grain tower if the height is
9.75 m and the lateral surface area is 600 m2? Use π 5 3.14 and
round your answer to the nearest
1
}}
100
of a meter. ____________________
Putting It All Together
4. The volume, v, of a cylinder is equal to the area, A, of the circular base
times the height, h. The area of a circle is πr 2. These formulas can be
expressed as literal equations: v 5 Ah, A 5 πr 2.
© RIVERDEEP, Inc.
a. Use substitution to combine these two formulas into a single expression
to find the volume of a cylinder. __________
b. Rewrite this expression to solve for the height. __________
c. Find the height of a cylinder that has a radius of 4 m and a volume of
3
500 m . Round your answer to the nearest
1
}}
10
of a meter. __________
destination
MATH
48
Name _______________________________________
Date ___________________________
COURSE: MSC V
MODULE 1: Essentials of Algebra
UNIT 5: Solving Literal Equations
1. The circumference of a circle is C 5 2πr. Rewrite this literal equation to
solve for r. __________
2. The perimeter of a square of side length s is found using the formula
p 5 4s.
a. Write a formula for the side length, s, of a square in terms of its
perimeter, p. __________________________________________________
b. Find the side length of a square with perimeter of 36 cm. ____________
3. The formula for simple interest is I 5 prt, where I is the interest paid, p is
the original principal, and t is the time. Rearrange this literal expression
so that the rate, r, is the only variable on the right side of the equation.
__________
4. The area of a triangle is 2}1} the base times the height, or A 5 2}1} bh. If you
know the area of a triangle and the measure of the height, what
operations must you apply to this literal equation to find the
length of the base of the triangle? __________________________________
__________________________________________________________________
5. The formula for the volume of a spherical ball is v 5
4
}}
3
πr3.
a. Calculate the volume if r3 5 8 cubic inches. Use 3.14 for π, and
© RIVERDEEP, Inc.
round your answer to the nearest
1
}} .
100
__________
b. Isolate r3 on the left side of the formula. __________
destination
MATH
49
Name _______________________________________
Date ___________________________
h
l
w
6. The volume of a right rectangular prism is given by the formula
V 5 l 3 w 3 h.
a. Find a formula for the height of a right rectangular prism in terms of its
volume, length, and width. ______________________________________
b. If a right rectangular prism has a length of 10 cm, a width of 5 cm,
3
and a volume of 1,000 cm , what is its height? ____________________
7. The pyramid of Quetzalcoatl in Cholula, Mexico, was the largest building
constructed in pre-Columbian Mexico. The volume of the structure is
estimated to be about 3 million cubic meters. The base of the pyramid
is a square with 350 m on each side.
a. Use the formula for the volume of a pyramid, V 5 3}1} Bh, where B is the
area of the base, to write a formula for height in terms of volume and
area of the base. ______________________________________________
b. Use the formula from part a to calculate the height of this enormous
© RIVERDEEP, Inc.
structure. ______________________________________________________
______________________________________________________________
destination
MATH
50