Task 2:
Use the table to complete each part.
a. Copy the table. Extend the table by writing expressions for y when x = 5, 6, and 7.
b. Simplify each expression for y in the table from part (a). What pattern do you notice in the simplified expressions?
c. Write an equation that relates x and y. Us your equation to find the value of y when x = 15.
The Table:
x | y
1 | 1
2 | 1 + 2
3 | 1 + 2 +4
4 | 1 + 2 + 4 + 8
plz help!
let's get the actual answers for the y values
x | y
1 | 1 = 1
2 | 1 + 2 = 3
3 | 1 + 2 +4 = 7
4 | 1 + 2 + 4 + 8 = 15
5 | 1+2+4+8+16 = 32
6 | 1+2+4+8+16+32 = 63
Did you notice that all the sums are one less than a power of 2
e.g for x = 3, the sum = 7 = 2^3 - 1
for x = 6 , the sum = 63 = 2^6 - 1
so when x = 15, the sum should be 2^15 - 1 = 32767
Okay thanks can you help with this one?
Task 3:
Solve. Show all your work and explain your steps.
You are buying gifts for 10 people. You decide to buy each person either a CD or a DVD. A CD costs $12 and a DVD cost $20.
a. Let c = the number of CDs you decide to buy. What is an expression in terms of c for the number of DVDs you buy?
b. What is an expression in terms of c for the cost of the CDs? For the cost of the DVDs?
c. Write and simplify an expression in terms of c for the total cost of all the gifts you buy.
Let c = the number of CDs you decide to buy
Then the number of DVDs is 10 - c
cost of CD = 12c
cost of DVD = 20(10-c)
total cost = 12c + 20(10-c)
= 12c + 200 - 20c
= 200 - 8c
thank you ssssoooooo much !
The property used is Distributive correct? Would we say it has commutative property of addition as well or just Distributive?
thanks
a. To extend the table by writing expressions for y when x = 5, 6, and 7, we need to find the pattern in the table and continue it.
Based on the given table, we can see that each value of y seems to be the sum of powers of 2: 1, 2, 4, 8, ...
So, we can write the expressions for y when x = 5, 6, and 7 as follows:
When x = 5:
y = 1 + 2 + 4 + 8 + 16 = 31
When x = 6:
y = 1 + 2 + 4 + 8 + 16 + 32 = 63
When x = 7:
y = 1 + 2 + 4 + 8 + 16 + 32 + 64 = 127
b. To simplify the expressions for y in the extended table, we can use the formula for the sum of a geometric series:
Sum = a(1 - r^n) / (1 - r),
where 'a' is the first term, 'r' is the common ratio, and 'n' is the number of terms.
In this case, the first term 'a' is 1, the common ratio 'r' is 2, and the number of terms 'n' is equal to the value of x.
Let's simplify each expression using the formula:
When x = 5:
y = 1(1 - 2^5) / (1 - 2) = 1(1 - 32) / (1 - 2) = 1(-31) / (-1) = 31
When x = 6:
y = 1(1 - 2^6) / (1 - 2) = 1(1 - 64) / (1 - 2) = 1(-63) / (-1) = 63
When x = 7:
y = 1(1 - 2^7) / (1 - 2) = 1(1 - 128) / (1 - 2) = 1(-127) / (-1) = 127
By simplifying, we can see that the expressions for y match the values we found in part (a). Each expression gives us the sum of the powers of 2 up to the given value of x.
c. To write an equation that relates x and y, we can see that the pattern in the simplified expressions is:
y = 2^x - 1
Hence, the equation is y = 2^x - 1.
To find the value of y when x = 15, we can substitute x = 15 into the equation:
y = 2^15 - 1 = 32768 - 1 = 32767
Therefore, when x = 15, y = 32767.