Answer:
since lisa earns half of what mary does divide 62 into 2 which is 31
and she has 8 students.
she is paid twice as much in students, so lisa would earn more due to getting paid more with having more students, as she gets paid twice the amount from students she has, and if she had 20 she would get paid 40 but if you add 2x of the mass it will be 80, and mary will have 40.
if its incorrect i apologize.
Let g(x) = x^2 and h(x) = mx+b
The piecewise function f(x) is basically a combination of g(x) and h(x) depending on what x is.
If x is equal to 3 or smaller, then f(x) = g(x) = x^2
If x is larger than 3, then f(x) = h(x) = mx+b
Plug in x = 3 into g(x)
g(x) = x^2
g(3) = 3^2
g(3) = 9
And do the same for h(x)
h(x) = mx+b
h(3) = m*3+b
h(3) = 3m+b
In order for f(x) to be differentiable at x = 3, the two functions g(x) and h(x) must meet up at this x value. The function f(x) must be continuous at x = 3. In other words, g(x) = h(x) must be true when x = 3
So we can equate the two function outputs
g(3) = h(3)
9 = 3m+b
Solve for b to get
b = 9-3m
---------------------------------------------------------------
Now differentiate each function g(x) and h(x) with respect to x. Plug in x = 3 after you differentiated
g(x) = x^2
g ' (x) = 2x
g ' (3) = 2*3
g ' (3) = 6
h(x) = mx+b
h ' (x) = m
h ' (3) = m
If f(x) is differentiable at x = 3, then f ' (x) must be continuous at x = 3
This means,
g ' (x) = h ' (x)
g ' (3) = h ' (3)
6 = m
m = 6
Now use this value of m to find b
b = 9 - 3m
b = 9 - 3*6
b = 9 - 18
b = -9
---------------------------------------------------------------
In summary, we found that
m = 6
b = -9
which are the values needed to make f(x) differentiable
Answer:
11 feet (Option C)
Step-by-step explanation:
Let the longer side be l and the shorter side be b.
We know that,
→ Perimeter of rectangle = 2 ( longer side + shorter side )
Here,
- Perimeter of rectangle is 32 feet.
→ 32 = 2 (l + b)
→ 32 = 2l + 2(5)
→ 32 = 2l + 10
→ 32 - 10 = 2l
→ 22 = 2l
→ = l
→ 11 = l
→<u> 11 feet = longer side</u>
<u>Length</u><u> </u><u>of</u><u> </u><u>the</u><u> </u><u>longer</u><u> </u><u>side</u><u> </u><u>is</u><u> </u><u>1</u><u>1</u><u> </u><u>feet</u><u>.</u>
Ryan was 8 when his parents invested $4000 in a certificate of deposit that pays 6%. If Ryan leaves the account alone until the investment doubles, how old will he be? (Assume that the interest is not compounded.)
<span>This one has a twist to it in that it tells you that the interest is not compounded. </span>
<span>They need to earn $8,000 - $4,000 = $4,000 in interest. </span>
<span>They earn $4,000 X 6% = $240 in interest per year. </span>
<span>$4,000 / $240 = 16.67 </span>
<span>16.67 + 8 = 24.67 </span>
<span>Ryan will be 24 and 8 months old when the investment doubles. </span>
<span>2.)Benjamin has $6000 invested in two accounts. One earns 8% interest per year, and the other pays 7.5% interest per year. If his total interest for the year is $472.50, how much is invested at 8%? </span>
<span>X = the amount invested at 8% </span>
<span>($6,000 - X) = the amount invested at 7.5% </span>
<span>So: </span>
<span>.08X + .075(6,000 - X) = $472.50 </span>
<span>Now solve: </span>
<span>.08X + 450 -.075X = $472.50, this reduces to </span>
<span>.005X = $22.50, which finally reduces to </span>
<span>X = $4,500 </span>
<span>So $6,000 - X = $1,500 </span>
<span>So the answer is: $4,500 is invested at 8% </span>
<span>Test the result </span>
<span>$4,500 X 8% = $360.00 </span>
<span>$1,500 X 7.5% = $112.50 </span>
<span>$360.00 + $112.50 = $472.50, and you've proved your answer</span>