Math 7
Mathematics homework help
Lesson 3.7
Introduction
Course Objectives
This lesson will address the following course outcomes:
· 12. Simplify algebraic expressions by using the distributive property, combining like terms, and factoring out a greatest common factor.
· 13. Evaluate formulas with multiple variables in a variety of contexts, including science, statistics, geometry, and financial math. Solve simple formulas for a specified variable.
Specific Objectives
· Formulas are relations between variables
· Solving all equations follows the basic rules of undoing and keeping the equation balanced.
Students will be able to
· Evaluate the dependent variable given values for the independent variables.
· Solve for a variable in a linear equation in terms of another variable.
· Factor out a greatest common factor.
In this lesson we will be exploring formulas, which are general relationships between two or more variables. We have already explored several formulas in this course, and will revisit those as well as explore some new formulas.
One thing new we will do in this section is solve a formula for a variable. The process is similar to solving an equation to get a numerical answer, but instead of a number, we get a differently written formula. To get a feel for how to do this, watch these videos:
· Solving Formulas – Part 1 [+]
· Solving Formulas – Part 2 [+]
BAC Revisited
Problem Situation 1: The Blood Alcohol Content Formula
Recall that Widmark’s formula for BAC is B=−0.015t+2.84NWgB=-0.015t+2.84NWg , where
B = percentage of BAC
N = number of standard drinks
W = weight in pounds
g = gender constant (0.68 for men, 0.55 for women)
t = number of hours since first drink
For an average male who weighs 190 lbs, the formula can be simplified to
B=−0.015t+0.022NB=-0.015t+0.022N
Forensic scientists often use this equation at the time of an accident to determine how many drinks someone had. In these cases, time (t) and BAC (B) are known from the police report. The crime lab uses this equation to estimate the number of drinks (N).
#1 Points possible: 5. Total attempts: 5
For a 190 pound man, find the number of drinks if the BAC is 0.09 and the time is 2 hours. Give your answer to 1 decimal place.
drinks
The crime lab needs to use the formula frequently to find the number of drinks N for various 190 pound males who have different values of BAC and hours since drinking. For example, the next case may involve BAC of 0.11 and time of 3 hours. They could each time go through the sequence of steps as in question 1. However, it would be more efficient to solve the formula for the variable N. That means: keeping the relationship among the variables the same and the equation balanced, rewrite the formula so that the variable N is alone on one side, while the other side of the equation has variables B and t, and numbers and operations.
#2 Points possible: 5. Total attempts: 5
Solve the formula B=−0.015t+0.022NB=-0.015t+0.022N for NN .
NN =
#3 Points possible: 5. Total attempts: 5
Use the formula for N that you just found to find the value of N when B = 0.09 and t = 2 to verify that the value you get for N is the same as the value found in the first question. Give your answer to 1 decimal place.
drinks
#4 Points possible: 5. Total attempts: 5
Use the formula for N that you found in question 2 to find the number of drinks when the 190 pound man has a BAC of 0.11 and the time was 3 hours. Give your answer to 1 decimal place.
drinks
Z-Score
Problem Situation 2: Statistics z-score Formula
A formula widely used in statistics is z=x−μσz=x-μσ , where
z = z-score
x is a specific data value
μ = mean value (μ is pronounced “mu”. It is Greek lower case mu.)
σ = population standard deviation (σ is pronounced “sigma”. It is Greek lower case sigma.)
It is not necessary that you understand what these terms mean. Rather we will use the formula both to evaluate values and to solve for values.
#5 Points possible: 5. Total attempts: 5
Find the value for z if x = 2.3, μ = 1.9, σ = 0.2
z =
#6 Points possible: 5. Total attempts: 5
Solve for the value of x if z = 1.5, μ = 1.9, σ = 0.2
x =
#7 Points possible: 5. Total attempts: 5
Use the z-score formula to solve for the variable x in terms z, μ, and σ.
xx =
To enter μ type mu, and to enter σ type sigma.
Factoring
Factoring
Recall from simplifying expressions the distribution property:
a(b+c)=ab+aca(b+c)=ab+ac
Sometimes when solving equations or formulas it can be necessary to use this property in reverse. When we do this, we call it factoring the greatest common factor (GCF):
ab+ac=a(b+c)ab+ac=a(b+c)
Examples:
x+0.15x=(1+0.15)x=1.15xx+0.15x=(1+0.15)x=1.15x
5x+tx=(5+t)x5x+tx=(5+t)x
4x+32=4(x+8)4x+32=4(x+8)
For some more examples, watch this video:
· Factor GCF [+]
This can be used for solving formulas.
Example: Solve qt+pt=cqt+pt=c for t.
| (q+p)t=c(q+p)t=c | By factoring out the GCF | |
| t=cq+pt=cq+p | Dividing both sides by the quantity (q+p)(q+p) |
#8 Points possible: 16. Total attempts: 5
Rewrite each of the following expressions by factoring out the Greatest Common Factor
a. 3x+273x+27 =
b. 12x−812x-8 =
c. x2−8xx2-8x =
d. 4×2+4x4x2+4x =
Simple Interest
Problem Situation 3: Simple Interest Formula
The formula A=P+PrtA=P+Prt allows us to find the amount A in a bank account that has an initial investment P at an annual interest rate of r percent (in decimal form) for t years. When simple interest is used, the interest earned is not paid into the account until the end of the time period.
#9 Points possible: 5. Total attempts: 5
Find the value for A if P = 2000, r = 0.01, and t = 5
A =
#10 Points possible: 5. Total attempts: 5
Find t if A = 2200, P = 2000, and r = 0.01
t =
#11 Points possible: 5. Total attempts: 5
Solve for t in terms of A, P, and r.
t =
#12 Points possible: 5. Total attempts: 5
Solve for P in terms of A, r, and t. Suggestion: first factor the right side of the original formula.
P =
HW 3.7
#1 Points possible: 5. Total attempts: 5
The formula for finding the area of a rectangle is A=LWA=LW .
If the units for length are in feet and the units for width are in feet, what are the units for A?
Solve this formula for L. L =
If the length of a rectangle is 13 feet and the width is 6 inches, what is the area of the rectangle in units of square feet? ft2
#2 Points possible: 5. Total attempts: 5
The formula for finding the perimeter of a rectangle is P=2L+2WP=2L+2W .
If the units for length are in feet and the units for width are in feet, what are the units for P?
Solve this formula for L. L =
#3 Points possible: 5. Total attempts: 5
The formula for finding the distance traveled, based on speed and time, is D=RTD=RT , where
D is distance R is rate (speed) T is time
Units must be consistent. If the unit for D is miles and the unit for T is minutes, what must the units for R be?
Solve this formula for R. R =
If a bicyclist rides for 170 minutes at an average speed of 10 miles per hour, how far was the ride, to 1 decimal place? miles.
At what speed must a bicyclist ride to cover 60 miles in 5 hours, to 1 decimal place? miles/hour.
#4 Points possible: 5. Total attempts: 5
The formula for determining the flow rate of a stream is Q=AVQ=AV , where
Q is the flow rate A is the cross sectional area of the stream v is the velocity of the water
Units must be consistent. If the units for V are meters per second and the units for A are square meters, what must the units for Q be?
Solve this formula for V. V =
A stream has a cross sectional area of 8 square meters. The velocity of the water past this point is 2.6 meters per second What is the flow rate, to 1 decimal place? m^3/s.
What is the speed of the water if the flow rate is 11 m^3/s and the cross sectional area is 5 square feet, to 1 decimal place? meters/second.
#5 Points possible: 5. Total attempts: 5
The formula for determining simple interest is i=prti=prt , where
i is the interest p is the principal r is the interest rate as a decimal. For example, if the interest rate is 3%, then r = 0.03. t is the time
Units must be consistent. If the unit for p is dollars, the unit for i is dollars and the unit for t is years, what must the units for r be?
Solve this formula for r. r =
10000 dollars are invested at a simple interest rate of 2 percent for 3 years. How much interest is earned, to the nearest cent? dollars.
What is the interest rate, as a percent, that is necessary to earn 475 dollars in simple interest if 10000 dollars are invested for 1 years, to 1 decimal place? %/year.
#6 Points possible: 5. Total attempts: 5
Acceleration is the rate at which velocity changes. The formula is a=v−v0t−t0a=v-v0t-t0, where
a is acceleration v0 is initial velocity (speed), v is final velocity t0 is initial time, t is final time
Units must be consistent.If the units for v and v0 are feet per second and the units for t and t0 are seconds, what must the units for a be?
Solve this formula for v.
· v=a−(t−t0)+v0v=a-(t-t0)+v0
· v=a(t−t0)v0v=a(t-t0)v0
· v=at−t0v0v=at-t0v0
· v=a(t−t0)+v0v=a(t-t0)+v0
A car was stopped at a red light. When the light turns green, the car reaches a speed of 55 miles per hour in 4 seconds. What is the acceleration, to 1 decimal place? feet/s2.
What is the final speed of a car that accelerates at 70 ft/s2 for 2 seconds if it’s initial speed was 12 miles per hour, to 1 decimal place? miles/hour.
#7 Points possible: 5. Total attempts: 5
Solve for the specified variable in each equation
a. A group of French students plan to visit the United States for two weeks. They are trying to pack appropriate clothing, but are not familiar with Fahrenheit. One student remembers this formula: F=95C+32F=95C+32 where F is the temperature in Fahrenheit and C is the temperature in Celsius. Solve the equation for C. C =
b. The simple interest formula is A = P + Prt. In the formula
· A = the full amount paid for the loan
· P = the principle or the amount borrowed
· r = the interest rate as a decimal
· t = time in years
A car dealership wants to use the formula to find the rate needed for certain values of the other variables. Solve the formula for r. r =
#8 Points possible: 5. Total attempts: 5
Rewrite each of the following expressions by factoring out the Greatest Common Factor
a. 6x+486x+48 =
b. 15x−1015x-10 =
c. x2−2xx2-2x =
d. 5×2+5x5x2+5x =
#9 Points possible: 5. Total attempts: 5
Use Greatest Common Factoring to solve the simple interest formula, A = P + Prt, for P.
P =
#10 Points possible: 5. Total attempts: 5
Sometimes linear equations are written in forms other than y=mx+by=mx+b . One common form looks like Ax+By=CAx+By=C . When you see an equation like this, you can find the slope and y-intercept by solving the equation for y, rewriting it into slope-intercept form.
Rewrite 15x+3y=3015x+3y=30 into slope-intercept form by solving for y.
y=y=
From that we can see that:
m =
b =
#11 Points possible: 5. Total attempts: 5
Sketch a graph of −6x−4y=−8-6x-4y=-8
Clear All Draw: