Incline mats, or triangle mats, are offered with different levels of incline to help gymnasts learn basic moves. As the name may suggest, two sides of the mat are right triangles. If the height of the mat is 14 inches shorter than the length of the mat and the hypotenuse is 4 inches longer than the length of the mat, what is the length of the mat?

The length of the mat is 30 inches.

Step-by-step explanation:

Let H be the hypotenuse of the incline mat, L its length and h its height.

It is given that H = L + 4 and h = L - 14. Now, using Pythagoras' theorem for right-angled triangles, H² = L² + h²

(L + 4)² = L² + (L - 14)²

Expanding, we have

L² + 8L + 16 = L² + L² -28L + 196

collecting all the terms to the right side, we have

L² + L² -28L + 196 - L² - 8L - 16 = 0

L² - 36L + 180 = 0

Using the quadratic formula, L =

L

We use the larger answer L = 30 inches since L = 6 would give h = L -14 = 6 - 14 = -8 .

So, the length of the mat is 30 inches.

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step-by-step explanation:

if gloria has "carry over" minutes, (that is, any minutes paid for but not used in a month can be applied to her payment for minutes in the next month,) her total usage for the three months is: total minutes = 320 + 243 + 489 = 1052 minutes number of 200 minute blocks = 1052/200 = 5 r52 (that is 5 with a remainder of 52) rounding up the number of 200 minute blocks gives her usage for the three months as 6 blocks. the cost for six blocks is 6 x $25 = $150 she also has 200 - 52 or 148 carry over minutes if gloria pays each month and unused minutes are lost, applying the rounding up procedure to each month gives: first month usage of 320 minutes rounds up to 2 blocks of 200 minutes for a payment of 2 x $25 = $50 second month usage of 243 minutes rounds up to 2 blocks of 200 minutes for a payment of 2 x $25 = $50 third month usage of 489 minutes rounds up to 3 blocks of 200 minutes for a payment of 3 x $25 = $75 total payments for the three months are $50 + $50 + $75 = $175 get the plan with carry over minutes!